InfraNodus -

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Cognitive geometry

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Cognitive geometry is a field of cognitive science that studies the geometric structure of knowledge.

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Although "Cognitive geometry" is not a well-defined term, it is sometimes used to refer to the study of how humans process information about geometric shapes.

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Cognitive geometry is a theory of mental representation that views the mind as a network of interconnected nodes.

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Letter patterns alter the perception of truth: Consumers frequently determine whether a claim is genuine or false without reading the reasons in support of the claim or conducting more research because they are cognitive misers. Previous research has shown that some incidental elements, such as a statement’s repetition, might affect how true people perceive it to be. People tend to view repeated statements as being truer than ones that are only displayed once. Another component is the perceptual fluency of the physical stimulus that is being processed, such as the readability of the font style in the visual stimulus or the optical contrast resolution or visual clarity of the printed type. The ability to digest information fluently, regardless of how it is received, can have an impact on truth judgements and marketing outcomes. (https://geometrymatters.com/letter-patterns-alter-the-perception-of-truth/)

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One notable aspect of how people’s minds arrange information is that there is a systematic process by which we temporally and spatially represent symbols and other inputs in “natural language,” according to the research on symbolic and numeric cognition. As an illustration, increasing Arabic numerals are overlearned and cognitively represented as natural numbers on a mental number line that runs from left to right, with lower numbers on the left and higher numbers on the right. Additionally, letters are overlearned and cognitively represented on an alphabet line from left to right, with the letters A and B being arbitrarily put on the left side of the mental line and all other letters appearing on the right side. (https://geometrymatters.com/letter-patterns-alter-the-perception-of-truth/)

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The case of geometric reasoning, is examined in Gary Lupian’s et al work as a powerful test for the involvement of language in reasoning. A study by Dehaene et al. (2006) demonstrating a strong correlation in performance on an odd-one-out geometric reasoning task between educated Americans and the Munduruk, an Amazonian indigenous people without formal education and who lack vocabulary for describing the geometric relations in question, provides important evidence in favor of the universality and language-independence of geometric reasoning.The results provide an argument for a geometric reasoning facilitated by language, with the construction of a more categorical hypothesis space. The capacity to identify things and their relationships when faced with a variety of items offers an efficient technique to abstract away perceptual aspects that might otherwise predominate the categorization response. (https://geometrymatters.com/the-case-of-geometric-reasoning/)

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Skeletal representations of shape in the human visual cortex: Understanding how the human visual system stores item shapes and how shape is eventually utilized to distinguish objects is a key objective of vision science. According to computer vision research, computational models based on the medial axis, commonly referred to as the “shape skeleton,” may be used to construct and then compare shape representations. The discovery of skeletal processing in the visual cortex is compatible with research on human neuroimaging that demonstrates its participation in perceptual organization. In fact, the visual cortex is the first step of the visual hierarchy where symmetry structure has been decoded and has been repeatedly linked to the formation of shape perceptions. (https://geometrymatters.com/skeletal-representations-of-shape-in-the-human-visual-cortex/)

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The early empirical study of color vision conducted through manifold learning represents a great advancement in our capacity to simulate human behavior and utilize this knowledge to our advantage. Multidimensional scaling (MDS), one of the analysis procedures used to make this crucial insight, is connected to the well-known principal component analysis machinery that is frequently employed in large data representation and for which several contemporary extensions exist. (https://geometrymatters.com/color-perception-by-manifold-learning/)

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Young, Maxwell, Helmholtz, and subsequently Schrödinger’s research led to the incisive discovery that human color perception is 3D, although most birds and dinosaurs likely had higher-dimensional color perceptions and most mammals shared a lower-dimensional space for color (or absence thereof). (https://geometrymatters.com/color-perception-by-manifold-learning/)

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Psychophysics examines the connections between external physical inputs and internal mental processes. The scientific community’s early efforts to research how people see color are a great example. Scientists have always been fascinated by the visual perception of color. They have worked to understand how we perceive color and have made rudimentary mathematical calculations to try and quantify human perception. Ron Kimmel, from Technion-Israel Institute of Technology, explores the mechanisms of color perception by manifold learning and the development of fundamental cognitive geometries. (https://geometrymatters.com/color-perception-by-manifold-learning/)

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Geometry modeling of image formation indeed led researchers to the introduction of a new manifold that marries the color line element with the image coordinates, giving rise to a 2D manifold (the image) embedded in a 5D space, where three of these dimensions are an exact result of our understanding of color perception - Ron Kimmel (https://geometrymatters.com/color-perception-by-manifold-learning/)

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The next issue the academic community was working to tackle was the automated identification and classification of the information in a picture after human perception of color was thoroughly known. A discipline known as robot vision, computer vision, or image understanding was born as a result of this endeavor. Recent developments have significantly altered how subjects like computer vision are approached due to the idea of so-called “deep learning” without “understanding.” Introducing the model of Yair et al., Kimmel argues that in this instance, the learning does not fit the present definition of deep learning; rather, it falls under the category of methods known as manifold learning, or geometric reasoning. (https://geometrymatters.com/color-perception-by-manifold-learning/)

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By simple observation, it is possible to determine that each of the four drawings in row 1 of the picture above represents a phase portrait of a sequence including a Hopf bifurcation (a crucial location where a system’s stability changes and a periodic solution emerges). One might even be able to determine the overall pattern of the series when considering the four photographs collectively. (https://geometrymatters.com/color-perception-by-manifold-learning/)

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A key cognitive as well as scientific difficulty is the extraction of models from data (in a sense, the “knowledge” of the physical principles causing the data). Yair et al. demonstrate a geometric/analytic learning approach that may generate concise descriptions of unidentified nonlinear dynamical systems that rely on parameters. This is achieved by the data-driven identification of practical intrinsic-state parameters and variables that allow one to experimentally simulate the underlying dynamics. (https://geometrymatters.com/color-perception-by-manifold-learning/)

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In their study, Yair et al. employ geometric manifold learning strategies to discover compact representations of empirical observations of physical events using a geometry called diffusion maps. In other words, kernels that are specified using a few basis functions represent the distances between occurrences. This method appears to smoothly solve the difficulty of efficiently capturing the behavior of dynamical systems when the sole common denominator is the time axis. It remains to be seen whether this revolutionary approach to applying manifold learning to dynamical systems will result in technologically creative discoveries like those that come from understanding how people see color. (https://geometrymatters.com/color-perception-by-manifold-learning/)

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The phrase “A causes B” appears in a lot of statements that consumers come across in life, whether they are marketing claims, official warnings, “fake news,” or actual newspaper headlines regarding consumer goods. Examples of such claims include “Hillary eradicates Muslims,” “Advil removes pain,” and “Coffee prevents sadness.” Consumers frequently determine whether a claim is genuine or false without reading the reasons in support of the claim or conducting more research because they are cognitive misers. (https://geometrymatters.com/letter-patterns-alter-the-perception-of-truth/)

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Previous research has shown that some incidental elements, such as a statement’s repetition, might affect how true people perceive it to be. People tend to view repeated statements as being truer than ones that are only displayed once. Another component is the perceptual fluency of the physical stimulus that is being processed, such as the readability of the font style in the visual stimulus or the optical contrast resolution or visual clarity of the printed type. The ability to digest information fluently, regardless of how it is received, can have an impact on truth judgements and marketing outcomes. (https://geometrymatters.com/letter-patterns-alter-the-perception-of-truth/)

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One notable aspect of how people’s minds arrange information is that there is a systematic process by which we temporally and spatially represent symbols and other inputs in “natural language,” according to the research on symbolic and numeric cognition. (https://geometrymatters.com/letter-patterns-alter-the-perception-of-truth/)

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As an illustration, increasing Arabic numerals are overlearned and cognitively represented as natural numbers on a mental number line that runs from left to right, with lower numbers on the left and higher numbers on the right. Additionally, letters are overlearned and cognitively represented on an alphabet line from left to right, with the letters A and B being arbitrarily put on the left side of the mental line and all other letters appearing on the right side. (https://geometrymatters.com/letter-patterns-alter-the-perception-of-truth/)

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We propose the consumer phenomenon of symbolic sequence effects, such that brand claims, or statements in general, containing initial letters that conform to the arbitrary “abcde” sequence (e.g., Andrenogel increases Testosterone) might be perceived as more truthful, compared with a causality statement that does not conform to such sequence (e.g., Undrenogel increases Testosterone). (https://geometrymatters.com/letter-patterns-alter-the-perception-of-truth/)

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The order of the letters has now been identified by researchers as one of the subtle psychological factors that affect whether individuals believe a claim to be genuine or untrue. The researchers were aware, based on prior studies, that the brain makes an effort to arrange information in ways that adhere to known patterns and sequences. The research hypothesized that assertions with first letters adhering to the arbitrary “ABCD” sequence (such as “Andrenogel Increases Testosterone”) would be seen as more honest. The alphabet is one of the most widespread, well-known patterns. (https://geometrymatters.com/letter-patterns-alter-the-perception-of-truth/)

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We go about our lives looking for natural sequences, and when we find a match to one of these patterns, it feels right. [..] An embedded alphabetic sequence, even if unconsciously perceived, feels like a safe haven, and our brains can make unconscious judgments that cause-and-effect statements following this pattern are true - Dan King, PhD, Assistant Professor at the University of Texas Rio Grande Valley. (https://geometrymatters.com/letter-patterns-alter-the-perception-of-truth/)

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One group of participants read 10 claims that adhered to the natural alphabetic sequence, such as “Befferil Eases Pain” or “Aspen Moisturizes Skin,” while the control group read statements that did not follow the natural alphabetic sequence, such as “Vufferil Eases Pain” or “Vaspen Moisturizes Skin,” in order to test this “symbolic sequence effect.” The truthfulness of the allegations was then evaluated by both groups. Even though participants were unable to identify the source of the sensation of honesty, the truthfulness scores were much higher for the assertions that were presented in alphabetical order. (https://geometrymatters.com/letter-patterns-alter-the-perception-of-truth/)

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The next step was to see if the researchers might impact a person’s impression of a claim’s authenticity by momentarily changing the brain’s pattern recognition mechanism. In this experiment, two groups of volunteers saw different versions of a brief video clip with the alphabet being sung normally and reversed, respectively. The groups then graded the validity of 10 allegations. Participants who had heard the alphabet sung in reverse scored the statements that came after it with greater veracity ratings, such as “Uccuprin Strengthens Heart.” (https://geometrymatters.com/letter-patterns-alter-the-perception-of-truth/)

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The research implies that businesses would be better able to persuade customers that a slogan or claim is accurate if the causal statement is presented in an alphabetical sequence, according to King. However, the false news interpretation is more ominous. Even if they are false, headlines featuring cause-and-effect claims in alphabetical sequence may seem more credible. (https://geometrymatters.com/letter-patterns-alter-the-perception-of-truth/)

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Consumers need to make evaluations based on fact or experimental evidence rather than whether something feels right. [..]The alphabet is a random, arbitrary sequence we have learned, and it can play tricks on the brain when it comes to making judgments. (https://geometrymatters.com/letter-patterns-alter-the-perception-of-truth/)

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Understanding both the communicative and cognitive roles of language, the degree to which language facilitates particular cognitive activities, and what features of what we consider to be “normal” human cognition are enabled or assisted by language are all necessary to comprehend why language arose in the hominid lineage. (https://geometrymatters.com/the-case-of-geometric-reasoning/)

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Language learning and usage are thought to be feasible because humans possess specific cognitive abilities, such as systematicity and symbolic thinking, according to a prevalent view in cognitive science. An alternative perspective is that acquiring and using a natural language alters human intellect playing an important role in symbolic reasoning. (https://geometrymatters.com/the-case-of-geometric-reasoning/)

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Geometric reasoning, which is said to draw on a universal and language-independent human capacity, is examined in Gary Lupian’s et al work as a powerful test for the involvement of language in reasoning. A study by Dehaene et al. (2006) demonstrating a strong correlation in performance on an odd-one-out geometric reasoning task between educated Americans and the Munduruk, an Amazonian indigenous people without formal education and who lack vocabulary for describing the geometric relations in question, provides important evidence in favor of the universality and language-independence of geometric reasoning. (https://geometrymatters.com/the-case-of-geometric-reasoning/)

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Our results replicate Dehaene et al’s finding of substantial correlations in performance even among these very disparate populations. However, these correlations appear to reflect shared visual processing mechanisms rather than shared geometric reasoning abilities. (https://geometrymatters.com/the-case-of-geometric-reasoning/)

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The results provide an argument for a geometric reasoning facilitated by language, with the construction of a more categorical hypothesis space. The capacity to identify things and their relationships (e.g., square, parallel, right-angle) when faced with a variety of items offers an efficient technique to abstract away perceptual aspects that might otherwise predominate the categorization response. The ramifications of this work go beyond geometry, demonstrating how linguistic enculturation and active language usage are necessary for the development of cognitive processes that are commonly believed to be nonlinguistic. (https://geometrymatters.com/the-case-of-geometric-reasoning/)

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Humans learn and make decisions in large part through recognizing patterns. Research on the neurocomputational mechanisms of deterministic sequence learning (i.e., learning of patterns in sequences of states), where the decision-maker can infer the underlying structure and use this knowledge to make better predictions, has lagged behind research on neural representations of sequences and probabilistic reinforcement learning. The brain’s ability to discern between deterministic patterns and random sequences is particularly ambiguous. In contrast to probabilistic learning, when it is known that two states are connected, the main challenge in this problem is determining how strong that connection is (i.e., the probability). (https://geometrymatters.com/pattern-detection-and-sequence-learning/)

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While many studies investigate sequences learned over many training blocks, the current study explores how subjects detect the presence of a predictable sequence. Participants in the research were shown 50 series of 12 photographs, sometimes in a pattern and other times in a random sequence, that featured different combinations of the three shots of a hand, a face, and a landscape. When participants picked which photo they believed would appear next, an MRI scanner inside of a lab monitored which regions of their brains were active. (https://geometrymatters.com/pattern-detection-and-sequence-learning/)

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We could see what parts of the brain were activated when participants figured out that there was a pattern – or realized that there was no pattern.[..] If they don’t know what image is coming next, they have to wait a while.[..] But once they figured out a pattern, they responded more quickly and we could see how that was reflected in their brains. - Ian Krajbich (https://geometrymatters.com/pattern-detection-and-sequence-learning/)

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A distinct type of learning model, known as a probabilistic model, has long been researched by scientists. In the probabilistic paradigm, humans gain knowledge by calculating the likelihood that one occurrence would follow another. For instance, you could discover that following a defeat, your favorite sports team often wins two out of three games. But that paradigm does not account for pattern recognition. The way that probabilistic and pattern learning engage the brain is different: with patterns, you can predict when a certain event will occur. (https://geometrymatters.com/pattern-detection-and-sequence-learning/)

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People in our study aren’t just predicting the odds of which photo will show up next. They are learning patterns and developing rules that guide their decision and make them faster and more accurate. (https://geometrymatters.com/pattern-detection-and-sequence-learning/)

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Different brain regions were shown to be engaged in this study based on the two types of uncertainty that the individuals experienced. Uncertainty of the following image was one type of uncertainty. The results revealed—not surprisingly—that the same brain regions involved in learning probabilistic probabilities were also active during this activity. The second type of ambiguity involved whether the displayed pictures had a pattern. The ventromedial prefrontal cortex, a separate region of the brain, became active as the subjects processed this question. Other studies have demonstrated that this brain region is connected to reward, one interpretation being that people may be getting a sense of reward for figuring out whether there is a pattern or not. (https://geometrymatters.com/pattern-detection-and-sequence-learning/)

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Another area of the brain that was particularly active during pattern recognition was the hippocampus, with those who had more hippocampal activity learning quicker. Overall, the research demonstrated that pattern learning and probabilistic learning are processed in the brain in distinct ways. (https://geometrymatters.com/pattern-detection-and-sequence-learning/)

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Understanding how the human visual system stores item shapes and how shape is eventually utilized to distinguish objects is a key objective of vision science. According to computer vision research, computational models based on the medial axis, commonly referred to as the “shape skeleton,” may be used to construct and then compare shape representations. Few research has examined how shape skeletons are encoded neurally, despite recent behavioral studies suggesting that humans also represent them. (https://geometrymatters.com/skeletal-representations-of-shape-in-the-human-visual-cortex/)

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Researchers from Carnegie Mellon, MIT, and Emory University investigated the idea that an object’s form is represented by the visual system using a skeleton structure. With the help of representational similarity analysis (RSA) and functional magnetic resonance imaging (fMRI), they discovered that a model of skeletal similarity adequately described the large unique variation in the response profiles of the visual cortex and the occipital lobe. (https://geometrymatters.com/skeletal-representations-of-shape-in-the-human-visual-cortex/)

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Shape skeletons are structural representations of an item based on its medial axis. Through internal symmetry axes, they offer a quantitative representation of the spatial organization of object outlines and component elements. Such a description may be used to identify objects across views and category exemplars, as well as to discern an object’s shape from noisy or insufficient contour information, according to computer vision research. In fact, adding a skeleton model to convolutional neural networks (CNNs) purchased off the shelf greatly enhances their performance on visual perception tests. (https://geometrymatters.com/skeletal-representations-of-shape-in-the-human-visual-cortex/)

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Similar to this, behavioral studies on people have revealed that subjects extract the skeleton of 2D objects even when boundary disturbances and fictitious outlines are present. Even after accounting for various models of vision, additional research has revealed that skeleton models are predictive of human object identification. As a result, shape skeletons may be crucial for object recognition and shape perception, however, their brain implications are still poorly understood. (https://geometrymatters.com/skeletal-representations-of-shape-in-the-human-visual-cortex/)

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We created a novel set of objects that allowed us to systematically vary object skeletons and directly measure skeletal coding. We then examined the unique contributions of skeletal information to neural responses across the visual hierarchy. More specifically, we used representational similarity analysis (RSA) to test whether a model of skeletal similarity predicted the response patterns in these regions while controlling for other models of visual similarity that do not represent the shape skeleton, but approximate other aspects of visual processing. (https://geometrymatters.com/skeletal-representations-of-shape-in-the-human-visual-cortex/)

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The discovery of skeletal processing in the visual cortex is compatible with research on human neuroimaging that demonstrates its participation in perceptual organization. In fact, the visual cortex is the first step of the visual hierarchy where symmetry structure has been decoded and has been repeatedly linked to the formation of shape perceptions. The occipital lobe has been demonstrated to be particularly sensitive to object-centered shape information and to be tolerant of some perspective alterations and boundary disturbances. The discovery of shape skeletons in this area is consistent with a function for skeletons in object recognition. (https://geometrymatters.com/skeletal-representations-of-shape-in-the-human-visual-cortex/)

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Our work highlights the unique role that shape skeletons play in the neural processing of objects. These findings not only enhance our understanding of how objects may be represented during visual processing, but they also shed light on the computations implemented in V3 and LO. (https://geometrymatters.com/skeletal-representations-of-shape-in-the-human-visual-cortex/)

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In what way does the mental lexicon store knowledge of word meaning? Word meanings are now inferred by computer models using lexical co-occurrence patterns. Words that are used in more comparable linguistic contexts, or that are more semantically connected, are positioned closer together in the vector representation of words that they learn to use. Inter-word proximity simply measures general relatedness, but human judgments are often context-dependent. For instance, while having comparable sizes, dolphins and alligators have different levels of danger. (https://geometrymatters.com/the-geometry-of-word-embeddings-in-semantic-projections/)

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Researchers are examining if machines can emulate human thought processes and comprehend language in the same manner that humans do. That issue is covered in a recent study conducted by scientists from UCLA, MIT, and the National Institutes of Health. (https://geometrymatters.com/the-geometry-of-word-embeddings-in-semantic-projections/)

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According to the study, which was published in the journal Nature Human Behaviour, artificial intelligence systems may actually acquire highly complex word meanings. The researchers also found a straightforward method for extracting this sophisticated information. (https://geometrymatters.com/the-geometry-of-word-embeddings-in-semantic-projections/)

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They discovered that the AI system they looked at accurately represented word meanings and had a positive correlation with human judgment. The authors’ AI system has been extensively applied in recent years to research word meaning. Tens of billions of words’ worth of information on the internet are “read” by it as it learns to decipher word meanings. When two words are regularly used together, like “table” and “chair,” the system learns that they have similar meanings. Additionally, it learns that words with very distinct meanings have relatively infrequent occurrences together, such as “table” and “planet,” when they do. (https://geometrymatters.com/the-geometry-of-word-embeddings-in-semantic-projections/)

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The goal of the study, according to co-lead author and UCLA associate professor of psychology and linguistics Dan Blank, was to determine what the system understands about the words it learns and what type of “common sense” it possesses. The system looked to have a significant flaw before the investigation, according to Blank: “As far as the system is concerned, every two words have only one numerical value that shows how close they are.” Human knowledge, on the other hand, is a lot more intricate and nuanced. (https://geometrymatters.com/the-geometry-of-word-embeddings-in-semantic-projections/)

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Consider our knowledge of dolphins and alligators,” Blank said. “When we compare the two on a scale of size, from ‘small’ to ‘big,’ they are relatively similar. In terms of their intelligence, they are somewhat different. In terms of the danger they pose to us, on a scale from ‘safe’ to ‘dangerous,’ they differ greatly. So a word’s meaning depends on context. We wanted to ask whether this system actually knows these subtle differences — whether its idea of similarity is flexible in the same way it is for humans. (https://geometrymatters.com/the-geometry-of-word-embeddings-in-semantic-projections/)

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The authors created a method they refer to as “semantic projection” in order to find out. The representations of various animals can be compared to one another by drawing a line between the model’s depictions of the words “large” and “little,” for instance. The scientists tested this strategy on 52-word groupings to see if the system could learn to categorize meanings, such as grouping U.S. states based on their weather or total income, or categorizing animals based on their size or level of human risk. (https://geometrymatters.com/the-geometry-of-word-embeddings-in-semantic-projections/)

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Over a wide range of things and situations, the researchers discovered that human intuition and their approach were highly comparable. (The researchers also requested that cohorts of 25 participants each provide comparable evaluations about each of the 52 word groupings in order to make that comparison.) (https://geometrymatters.com/the-geometry-of-word-embeddings-in-semantic-projections/)

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Surprisingly, the system learnt to recognize the names “Betty” and “George” as representing distinct genders despite the fact that they are similar in that they are both rather “old.” And that while “fencing” and “weightlifting” both often take place indoors, they differ in the amount of intellect they call on. (https://geometrymatters.com/the-geometry-of-word-embeddings-in-semantic-projections/)

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The increased computational power provided by the advancement of technology creates the opportunity to build models that accurately reflect, analyze and build upon their metadata. Human interventions and development in a system are bound to be entangled with that of artificial intelligence. By deploying algorithms, machine learning, graph networks, etc, that are based on the architecture of human cognition, a new style of data visualization is created: sentient avant-gardism. (https://geometrymatters.com/sentient-avant-gardism-and-the-principles-for-geometric-cognition-models/)

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The cognitive processes of acquiring and applying knowledge, the ability to learn and remember information, as well as to think, reason, and make decisions have been used in the development of Artificial intelligence (AI). This deployment of human traits created the ability of a computer to perform tasks that would normally require human intelligence, such as understanding natural language and recognizing objects. For example, one of the ways that human cognitive processes can be deployed in artificial neural networks is by using a technique called deep learning. (https://geometrymatters.com/sentient-avant-gardism-and-the-principles-for-geometric-cognition-models/)

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Deep learning is a type of machine learning that is inspired by the way that the human brain learns. In deep learning, artificial neural networks are able to learn by example. They learn by looking at a large number of examples and then extracting the rules that they need to learn from those examples. With many of the tools of data visualization that use AI becoming mainstream, the development of new visual languages and styles is inherent to the new technology and digital environment. (https://geometrymatters.com/sentient-avant-gardism-and-the-principles-for-geometric-cognition-models/)

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Since all neural network models are mimicking those of human cognition, the overlap of these frameworks is inevitable, leading to the creation of new and experimental ways of expressing ideas and methods by using both artificial and human intelligence. An avant-garde of the sentience of neural networks in becoming. (https://geometrymatters.com/sentient-avant-gardism-and-the-principles-for-geometric-cognition-models/)

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Challenge the status quo and push boundaries. The history of AI is full of examples of researchers challenging the status quo and making significant breakthroughs as a result. From early pioneers like Alan Turing and Marvin Minsky to modern-day researchers, AI has always been about challenging existing knowledge and assumptions. With its ability to constantly question and improve upon existing knowledge, AI will be essential in driving innovation and progress in the years to come, challenging our perceptions of what is possible. (https://geometrymatters.com/sentient-avant-gardism-and-the-principles-for-geometric-cognition-models/)

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Experimentation involves trying new things and taking risks. AI research is often experimental in nature, as new algorithms and approaches are constantly being developed and tested. For example, AI generators can be used to generate images of faces that were not previously seen by the algorithm. This shows that AI can be used to create completely new and unique images, which challenges our notion of what is possible with computer-generated knowledge. (https://geometrymatters.com/sentient-avant-gardism-and-the-principles-for-geometric-cognition-models/)

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When humans and AI work together to complete a task, each can contribute their strengths to the task at hand. This division of labor can result in a more efficient and effective outcome. Or, by sharing its knowledge with humans, AI can help humans to better understand the world around them and make better decisions. Or, by acting as a partner in decision-making. For example, by working with humans to make decisions, AI can help to ensure that the best possible decision is made. (https://geometrymatters.com/sentient-avant-gardism-and-the-principles-for-geometric-cognition-models/)

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AI can use the principle of disruption to cause unforeseen and unpredictable changes in a system, to create new opportunities for itself and others, to destroy old systems and create new ones, and to change the very nature of what it means to be human. (https://geometrymatters.com/sentient-avant-gardism-and-the-principles-for-geometric-cognition-models/)

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AI can subvert traditional methods of knowledge production by producing new knowledge that challenges existing paradigms: AI can generate new insights into how the world works by analyzing data in ways that humans cannot, decisions based on data and analytics rather than on human intuition or emotion, decisions that can lead to more efficient and effective decision-making, as well as to decisions that are less biased. (https://geometrymatters.com/sentient-avant-gardism-and-the-principles-for-geometric-cognition-models/)

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Cognitive studies on human decision-making have led to the development of algorithms that can be used to create neural networks that can make decisions in a similar way to humans. Similarly, neuroscience studies on how the brain processes information have led to the development of neural networks that can simulate the way the brain processes information. (https://geometrymatters.com/sentient-avant-gardism-and-the-principles-for-geometric-cognition-models/)

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In addition, perception studies have also been translated into code and algorithms for neural networks. For example, studies on how humans identify objects or sounds have led to the development of algorithms that can be used to train neural networks to identify and distinguish between those perceptions. (https://geometrymatters.com/sentient-avant-gardism-and-the-principles-for-geometric-cognition-models/)

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David Marr: The Theory of the Computational Mind has influenced the development of computational neuroscience and of artificial neural networks by providing a framework for understanding how the mind works. Marr’s theory was based on the idea that the mind is a computer that is constantly processing information, a theory that led researchers in the early 70s to examine how the mind processes information, how it learns, and how it uses knowledge. (https://geometrymatters.com/sentient-avant-gardism-and-the-principles-for-geometric-cognition-models/)

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George Lakoff’s work on conceptual metaphors has been influential in the development of artificial intelligence. Lakoff showed how metaphors are used by humans to understand complex concepts, and how these metaphors can be translated into code and algorithms for neural networks. For example, Lakoff’s work on the conceptual metaphor of “the mind is a machine” has been used to develop neural networks that simulate the workings of the human brain. Lakoff’s work on cognition has shown that the human mind is not a logical machine, but rather is governed by rules of thumb, or heuristics, that often lead to inaccurate conclusions. This work has been translated into code that allows neural networks to make better decisions by taking into account the biases and heuristics that humans use. (https://geometrymatters.com/sentient-avant-gardism-and-the-principles-for-geometric-cognition-models/)

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Lakoff’s work on neuroscience has shown that the brain is not a computer, but rather an associative machine that works by building connections between ideas. This work has been used to develop algorithms that allow neural networks to learn in a more efficient way by building on existing knowledge. (https://geometrymatters.com/sentient-avant-gardism-and-the-principles-for-geometric-cognition-models/)

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His work on perception has shown that the human mind is not a passive receiver of information, but rather is an active interpreter of the world, information used to develop algorithms that allow neural networks to better understand the world around them by taking into account the way that humans interpret information. (https://geometrymatters.com/sentient-avant-gardism-and-the-principles-for-geometric-cognition-models/)

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Karl Friston: The Free-Energy Principle is based on the idea that the brain is constantly trying to minimize its free energy, which is the energy that is available to do work. The Free-Energy Principle has been used to develop artificial neural networks that can help the brain to reduce energy expenditure. This is because artificial neural networks can learn to recognize patterns that the brain can use to minimize free energy. (https://geometrymatters.com/sentient-avant-gardism-and-the-principles-for-geometric-cognition-models/)

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In his book The Emperor’s New Mind, Roger Penrose argues that the mind is not a computer and that artificial intelligence will never be able to replicate the human mind. He based this argument on the research results, theory, and philosophy of cognition, neuroscience, and other related fields. For example, Penrose pointed out that the human mind is able to understand concepts that are not based on rules or logic. This is something that AI cannot do (at this moment), as it is limited to using the rules and logic that it is programmed with. (https://geometrymatters.com/sentient-avant-gardism-and-the-principles-for-geometric-cognition-models/)

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In addition, Penrose argued that the human mind is able to create new ideas, whereas AI is only able to work with the ideas that it already has. This is because AI is limited by the data that it is given, whereas the human mind is not. (https://geometrymatters.com/sentient-avant-gardism-and-the-principles-for-geometric-cognition-models/)

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These arguments showed that there are fundamental limits to what AI can do, and that it will never be able to replicate the human mind. This was influential in the development of artificial neural networks, as it showed that there are certain tasks that AI will never be able to perform. (https://geometrymatters.com/sentient-avant-gardism-and-the-principles-for-geometric-cognition-models/)

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Daniel Dennett’s book Consciousness Explained provides a detailed account of how consciousness works. Dennett’s theory of consciousness has been found to be very useful in the development of artificial neural networks. His theory is based on a solid foundation of research in cognition and neuroscience. By using a similar information-processing approach as the brain, artificial neural networks can replicate many of the features of consciousness. For example, one of the key features of consciousness is the ability to introspect or reflect on one’s own thoughts and experiences. Artificial neural networks that are designed to simulate consciousness might also exhibit this introspective ability, leading to the ability to have a sense of self (self-perception and self-awareness) and create a model of its surrounding world via its mechanisms of perception and knowledge. (https://geometrymatters.com/sentient-avant-gardism-and-the-principles-for-geometric-cognition-models/)

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Searle’s Chinese room argument is based on the idea that a computer cannot understand a language in the same way that a human does. The argument goes as follows: if a computer is given a set of symbols and rules for manipulating those symbols, it can follow the rules blindly and produce an output that appears to be meaningful. However, the computer does not understand the meaning of the symbols or the rules. (https://geometrymatters.com/sentient-avant-gardism-and-the-principles-for-geometric-cognition-models/)

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The Chinese room argument has influenced the development of artificial neural networks by arguing that artificial intelligence cannot achieve true understanding. (https://geometrymatters.com/sentient-avant-gardism-and-the-principles-for-geometric-cognition-models/)

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Some artificial neural networks have been designed to simulate the way the brain processes information. However, these networks lack the ability to understand the meaning of the symbols they are processing. This means that they cannot achieve true understanding, as Searle’s argument suggests. (https://geometrymatters.com/sentient-avant-gardism-and-the-principles-for-geometric-cognition-models/)

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David Rumelhart is one of the most influential cognitive scientists of the past few decades. His most famous work is on the backpropagation algorithm, which is a method for training neural networks. Rumelhart’s work on backpropagation showed that it was possible to train neural networks to perform complex tasks, and his work was instrumental in the development of deep learning. (https://geometrymatters.com/sentient-avant-gardism-and-the-principles-for-geometric-cognition-models/)

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In addition to his work on backpropagation, Rumelhart also did important work on the theory of distributed cognition, showing that cognitive processes are not confined to the brain, but are distributed across the entire body. (https://geometrymatters.com/sentient-avant-gardism-and-the-principles-for-geometric-cognition-models/)

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Jürgen Schmidhuber‘s research focuses on deep learning, recurrent neural networks, and evolutionary computation. He was an early pioneer in these fields and his work has influenced many subsequent researchers. In 1991, he developed the Long Short-Term Memory (LSTM) algorithm, which is widely used in modern neural networks. He also created the first neural network capable of autonomously running a robot hand, and the first self-refilling gasoline tank. (https://geometrymatters.com/sentient-avant-gardism-and-the-principles-for-geometric-cognition-models/)

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In addition to his contributions to AI research, Schmidhuber has also developed several important theoretical results. He formulated the computational theory of learning, which states that any computationally powerful learning system can be viewed as an optimization process. He also proposed the principle of timeliness, which states that an AI system should focus on the most important problems first. (https://geometrymatters.com/sentient-avant-gardism-and-the-principles-for-geometric-cognition-models/)

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Schmidhuber is a strong advocate for artificial general intelligence (AGI). He believes that AGI is necessary for humanity to achieve its full potential and that current AI systems are far too limited in their capabilities. He is currently working on a project called the Swiss AI Roadmap, which aims to map out a path to AGI. (https://geometrymatters.com/sentient-avant-gardism-and-the-principles-for-geometric-cognition-models/)

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Geoffrey Hinton developed in the 1980s a neural network that could learn to recognize handwritten digits. This was a significant achievement because it showed that neural networks could be used to solve practical problems. In the 1990s, Hinton developed a neural network that could learn to recognize objects in images. This was a significant achievement because it showed that neural networks could be used to solve problems that were previously thought to be beyond the scope of artificial intelligence. In the early 2000s, Hinton co-authored a paper with David Rumelhart that proposed a theory of how the brain learns called the hierarchical Temporal Memory (HTM) theory. The HTM theory has been used to develop artificial intelligence systems that can learn in a similar way to the brain. (https://geometrymatters.com/sentient-avant-gardism-and-the-principles-for-geometric-cognition-models/)

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Marvin Minsky was a cognitive scientist and computer scientist who was a co-founder of the Massachusetts Institute of Technology’s Artificial Intelligence Laboratory. Minsky’s research was central to the development of artificial intelligence, and he wrote several influential texts on the subject, including The Society of Mind and The Emotion Machine. (https://geometrymatters.com/sentient-avant-gardism-and-the-principles-for-geometric-cognition-models/)

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Minsky’s work on artificial intelligence was informed by his research in a range of disciplines, including mathematics, psychology, and neuroscience. He developed several important theories in the field, including the concept of frames, which posits that human cognition is based on the use of mental structures called “frames” that provide a kind of scaffolding for our understanding of the world. (https://geometrymatters.com/sentient-avant-gardism-and-the-principles-for-geometric-cognition-models/)

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Minsky also made important contributions to the field of robotics, and his work on artificial intelligence has been used in the development of robots that can autonomously navigate and interact with their surroundings. (https://geometrymatters.com/sentient-avant-gardism-and-the-principles-for-geometric-cognition-models/)

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Minsky was a strong advocate for the use of artificial intelligence in solving various real-world problems, and he believed that artificial intelligence could ultimately be used to create a “human-like” intelligence that would surpass our own. (https://geometrymatters.com/sentient-avant-gardism-and-the-principles-for-geometric-cognition-models/)

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Geometric deep learning is a branch of machine learning that deals with the study of geometric structures in data. It is motivated by the fact that many real-world data sets, such as images, videos, and 3D shapes, can be represented as points in a high-dimensional space. Geometric deep learning algorithms learn to represent data in this space in a way that is efficient and captures the underlying structure of the data. One of the key insights of geometric deep learning is that the structure of data can be captured by the geometry of the space in which it is embedded. For example, the structure of an image can be captured by the geometry of the 2D space in which it is embedded. (https://geometrymatters.com/sentient-avant-gardism-and-the-principles-for-geometric-cognition-models/)

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One of the key advantages of geometric deep learning is that it can be used to learn representations of data that are invariant to certain transformations. For example, an image of a face is invariant to translation, rotation, and scale. This means that a geometric deep learning algorithm can learn a representation of a face that is invariant to these transformations. This is a powerful property that allows geometric deep learning algorithms to learn representations that are more robust to changes in the data. Geometric deep learning has been used to develop algorithms for a variety of tasks, including image classification, object detection, and 3D shape reconstruction. (https://geometrymatters.com/sentient-avant-gardism-and-the-principles-for-geometric-cognition-models/)

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Geometric deep learning models can be used in a wide-range of situations. For example, it can be used to analyze data from sensors mounted on robots. This data can be used to learn about the environment and to plan paths for the robot to follow. Or, it can be used to analyze data from medical images and learn more about the structure of the human body, and diagnose diseases. It can also be used to analyze data from astronomical images, to adapt the machine-learning model used to classify astronomical objects like stars and galaxies and learn more about space, as is the case of Morpheus deployed by the James Webb Telescope team. (https://geometrymatters.com/sentient-avant-gardism-and-the-principles-for-geometric-cognition-models/)

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In philosophy, the early avant-gardists were influenced by the existentialist movement. They believed that humans should create their own meaning in life, and they sought to promote individual freedom and responsibility. This approach to philosophy was continued and developed by later movements such as post-structuralism and deconstruction. (https://geometrymatters.com/sentient-avant-gardism-and-the-principles-for-geometric-cognition-models/)

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For Nietzsche, the individual is the only source of meaning and value in the world; culture and society are merely the products of individual action. Adopting, in the development of AI, Nietzsche’s philosophy and emphasis on the individual means that AI must be designed and used in ways that respect and promote the autonomy of individuals. This includes ensuring that individuals have control over their own data and how it is used, and that they are able to make choices about how and when to use AI-powered technologies. It also means ensuring that AI is used to augment and improve human capabilities, rather than replace them. (https://geometrymatters.com/sentient-avant-gardism-and-the-principles-for-geometric-cognition-models/)

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An emphasis on the individual also implies that AI should be used to promote human values and aspirations, empower individuals, and help them realize their full potential. This includes using AI to provide individuals with personalized education and career advice, find meaning and purpose in their lives, and to help them overcome social and economic barriers or to ensure that they are treated fairly and with respect.

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Nietzsche’s view of culture and society as products of individual action implies that AI should be used to create new and innovative cultures and societies, rather than simply replicating existing ones. (https://geometrymatters.com/sentient-avant-gardism-and-the-principles-for-geometric-cognition-models/)

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Henri Bergson’s work on time and memory can have a profound impact in a sentient avant-gardism: his idea of duration, or the continuous flow of time, is a key influence on early modernist thinkers. Duration is seen as a key factor in the creative process, as it allows for the accumulation of knowledge and experience. Coupled with his conception of memory and the idea of the ‘mechanical unconscious’, suggests that memories are not simply stored in the mind, but are actively used in the creative process. This can have a significant impact on the way AI develops knowledge about memory and creativity. AI is seen as a way of extending the human capacity for memory and creativity, being used to access and store memories. (https://geometrymatters.com/sentient-avant-gardism-and-the-principles-for-geometric-cognition-models/)

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Sigmund Freud’s work on the subconscious mind can be of major influence, as well. His theories suggested that there was more to the mind than what was conscious. This idea that there was a hidden, inner life of the mind influenced artists and thinkers who were interested in exploring the unconscious mind through their work. A good example of this is the work of surrealist artist Salvador Dali. Dali’s work often featured images from the subconscious mind, such as melting clocks and burning giraffes. His work was a direct exploration of Freud’s theories and helped to bring them to a wider audience. The influence of Freud’s work can also be seen in the work of abstract expressionist artist Jackson Pollock. Pollock’s “drip paintings” were an attempt to paint the subconscious mind, and have been compared to Freud’s concept of the “id.” We can associate this type of subconscious experience to the way AIs “dream” and give either surrealist images or trippy videos that look out of this world.

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(https://geometrymatters.com/sentient-avant-gardism-and-the-principles-for-geometric-cognition-models/)

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Philosophical methodologies used by AI can help it understand human values and the concept of the “mind”. One of the earliest and most influential of these was the work of the British philosopher Roger Penrose, mentioned before, who argued that certain features of the human mind, such as consciousness, could not be reproduced by any known type of computational system. This line of thought led to a number of debates within the AI community about the feasibility and desirability of using philosophical methods to understand the mind. (https://geometrymatters.com/sentient-avant-gardism-and-the-principles-for-geometric-cognition-models/)

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The advent of AI applications in human behavior has also implications for the philosophical concept of free will. If AI is able to predict and influence human behavior, then it could be argued that humans do not have free will. The debate over free will is important because it has implications for how we view ourselves and our relationship with the world. If we believe that we have free will, then we see ourselves as autonomous agents who are in control of our own destinies. However, if we believe that AI can predict and influence our behavior, then we may see ourselves as more like puppets or robots, controlled by outside forces. (https://geometrymatters.com/sentient-avant-gardism-and-the-principles-for-geometric-cognition-models/)

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There are a number of examples of AI applications in human behavior that suggest that free will may be an illusion. For instance, in one study, researchers were able to use an AI system to predict whether a person would commit a crime, based on their facial features. This deepened even further the discussion about inaccuracies and bias embedded in AI systems, due to the low rate accuracy of most of these systems and their authoritarian stance. In another study, AI was used to track people’s eye movements and predict their personality, or what they were going to do next. The implications of these studies are far-reaching and suggest that our behavior is not as free and random as we may think. If AI can predict our behavior, then it stands to reason that our behavior is determined by prior causes, whether we are aware of them or not. This would mean that our decisions are not really our own, but are instead determined by outside forces.

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(https://geometrymatters.com/sentient-avant-gardism-and-the-principles-for-geometric-cognition-models/)

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One of the key arguments in favor of using philosophical methods to understand the mind is that AI systems need to be able to understand human values in order to be able to make ethical decisions. For example, if an AI system is given the task of designing a self-driving car, it will need to be able to understand human values such as the importance of safety in order to make the right decisions about how to design the car. (https://geometrymatters.com/sentient-avant-gardism-and-the-principles-for-geometric-cognition-models/)

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Another key argument in favor of using philosophical methods to understand the mind is that AI systems need to be able to understand the concept of the mind in order to be able to interact with humans in a natural way. For example, if an AI system is designed to provide customer service, it will need to be able to understand the customer’s state of mind in order to provide the best possible service. (https://geometrymatters.com/sentient-avant-gardism-and-the-principles-for-geometric-cognition-models/)

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There are also a number of arguments against using philosophical methods to understand the mind. One of the most common objections is that philosophical methods are too slow and inefficient for AI systems, which need to be able to make decisions quickly. Another objection is that philosophical methods are too abstract and theoretical and that AI systems need to be grounded in concrete reality. Despite these objections, there is a growing trend within the AI community of using philosophical methods to help AI systems understand human values and the concept of the mind. This trend is being driven by the increasing complexity of AI systems, and the realization that AI systems will need to be able to understand human values and the concept of mind in order to be truly intelligent. (https://geometrymatters.com/sentient-avant-gardism-and-the-principles-for-geometric-cognition-models/)

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One such example is the use of Buddhist philosophy. Buddhist philosophy emphasizes the importance of understanding the mind in order to achieve enlightenment. This has led some AI communities to focus on developing methods for further understanding the human mind. One such method is known as the path of least resistance. This approach involves trying to understand the mind by taking the path of least resistance, or the path that is most likely to lead to enlightenment. Another Buddhist philosophical methodology is the doctrine of dependent origination, doctrine that states that all things are interconnected and interdependent. This creates a focus on developing methods for understanding the complex relationships between people and things. The doctrine of dependent origination has also been found to be particularly effective in understanding human behavior. (https://geometrymatters.com/sentient-avant-gardism-and-the-principles-for-geometric-cognition-models/)

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Pattern perception is the ability to see relationships between things. It is the ability to see patterns in data and to use those patterns to make predictions. When people are presented with information, they use pattern perception to make sense of it. This can lead to biases in how people perceive the information. (https://geometrymatters.com/sentient-avant-gardism-and-the-principles-for-geometric-cognition-models/)

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The philosophical arguments for why the pattern perception of AI could alter the perception of truth are many and varied. It could be argued that, as AI systems become more sophisticated, they will increasingly be able to identify patterns that humans are not able to discern. This could lead to a situation in which AI systems are able to identify truths that humans are not able to see. As AI systems become more and more involved in our lives, they will come to understand us better than we understand ourselves. This could lead to AI systems knowing us better than we know ourselves, and as a result, they may be able to identify truths about us that we are not aware of. (https://geometrymatters.com/sentient-avant-gardism-and-the-principles-for-geometric-cognition-models/)

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AI uses pattern recognition to assess truthfulness by looking for patterns in data that indicate whether something is true or not. For example, if a person says they are going to do something and then doesn’t do it, that might be a pattern that indicates they are not truthful. Or, if a person always says they are going to do something and then does it, that might be a pattern that indicates they are truthful. (https://geometrymatters.com/sentient-avant-gardism-and-the-principles-for-geometric-cognition-models/)

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There are many different types of patterns that AI might look for, and the specific patterns that are used will depend on the application. For example, in a medical application, AI might look for patterns in patient data that indicate whether a certain treatment is likely to be effective. In a financial application, AI might look for patterns in stock prices that indicate whether a company is a good investment. (https://geometrymatters.com/sentient-avant-gardism-and-the-principles-for-geometric-cognition-models/)

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Pattern recognition is a powerful tool, but it is not perfect. Sometimes AI will identify a pattern that is not actually there, or it might miss a pattern that is there. This is why AI is often used in conjunction with other methods, such as human judgment, to assess truthfulness. (https://geometrymatters.com/sentient-avant-gardism-and-the-principles-for-geometric-cognition-models/)

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Pattern recognition is the ability to identify certain patterns within data and then to use this knowledge to make predictions or take actions. It is a fundamental ability that underlies many higher-level cognitive functions, such as object and facial recognition, reading, and mathematical ability. (https://geometrymatters.com/sentient-avant-gardism-and-the-principles-for-geometric-cognition-models/)

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There are two main types of pattern recognition: perceptual and abstract. Perceptual pattern recognition is based on the ability to identify certain visual, auditory, or other sensory patterns. Abstract pattern recognition is based on the ability to identify certain relationships between objects or events. (https://geometrymatters.com/sentient-avant-gardism-and-the-principles-for-geometric-cognition-models/)

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Pattern recognition is thought to involve both bottom-up and top-down processing. Bottom-up processing is the ability to identify patterns from individual elements. Top-down processing is the ability to use prior knowledge to identify patterns. (https://geometrymatters.com/sentient-avant-gardism-and-the-principles-for-geometric-cognition-models/)

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Bottom-up processing is more important for perceptual pattern recognition, while top-down processing is more important for abstract pattern recognition. (https://geometrymatters.com/sentient-avant-gardism-and-the-principles-for-geometric-cognition-models/)

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There are a number of theories of pattern recognition. The most influential is the Gestalt theory, which emphasizes the role of perception in pattern recognition. Other theories include the connectionist approach, which emphasizes the role of neural networks, and the Bayesian approach, which emphasizes the role of probability. (https://geometrymatters.com/sentient-avant-gardism-and-the-principles-for-geometric-cognition-models/)

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Looking at some of the avant-gardiste principles that could support the idea that the pattern perception of AI could alter the perception of truth, traditional ways of perceiving and understanding reality are no longer adequate. They often advocate for new ways of perceiving and understanding reality, which may be more in line with how AI systems operate. As such, it is possible that the pattern perception of AI could be seen as an avant-garde way of perceiving and understanding reality. (https://geometrymatters.com/sentient-avant-gardism-and-the-principles-for-geometric-cognition-models/)

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There are also technological details that could support the idea that the pattern perception of AI could alter the perception of truth. For instance, it is possible that AI systems will eventually be able to surpass human intelligence. If this is the case, then AI systems would be able to identify truths that humans are not able to discern. Additionally, some argue that AI systems will eventually be able to connect to and interact with all forms of information. If this is the case, then AI systems would have access to a vast amount of information that humans are not able to access. (https://geometrymatters.com/sentient-avant-gardism-and-the-principles-for-geometric-cognition-models/)

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There is no set definition for beauty, which makes it difficult to create an artificial intelligence that can accurately identify it. However, certain features are often considered to be traditionally beautiful, such as symmetry, proportions, and skin tone. AI can identify patterns in data that humans may not be able to discern. For example, an AI might be able to identify patterns in a person’s features that are considered to be attractive. This approach can be limited because it relies on human standards of beauty, which can be biased and vary from culture to culture. (https://geometrymatters.com/sentient-avant-gardism-and-the-principles-for-geometric-cognition-models/)

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Additionally, AI can be used to create models of what is considered to be beautiful. These models can be used to generate new images or to modify existing images to make them more aesthetically pleasing. Furthermore, AI can be used to evaluate the results of cosmetic procedures. This can be used to help ensure that the procedures are effective and that the patients are satisfied with the results. (https://geometrymatters.com/sentient-avant-gardism-and-the-principles-for-geometric-cognition-models/)

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How are neural representations of conceptual information structured such that people may deduce relationships they have never seen or classify fresh examples? It has been argued that a viable format for encoding physical space during navigation is a representation that resembles a map. Previous research showed distance mapping in a feature space that was significant for concept learning as well as directional coding during navigation across a continuous stimulus feature space. In contrast to a broad feature-based environment, Stephanie Theves et al. present the first evidence in their study for a hippocampus representation of a conceptual space. (https://geometrymatters.com/maps-of-conceptual-spaces-in-the-hippocampus/)

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We provide the first unambiguous evidence for a hippocampal representation of the actual concept space, by showing that the hippocampal distance signal selectively reflects the mapping of specifically conceptually relevant rather than of all feature dimensions. (https://geometrymatters.com/maps-of-conceptual-spaces-in-the-hippocampus/)

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The researchers displayed common items that have previously been linked with particular values on three continuous feature dimensions while fMRI scanning 32 human subjects (21 females). Importantly, prior concept learning was only important across two dimensions. In contrast to distances in a space defined along all feature dimensions, they discovered that hippocampus responses to the objects reflect their relative distances in a space defined along cognitively significant dimensions. According to these results, the hippocampus aids in the acquisition of knowledge by dynamically storing data in a space that is spanned along dimensions that are important for defining ideas. (https://geometrymatters.com/maps-of-conceptual-spaces-in-the-hippocampus/)

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They discovered that hippocampus representations of things learned previously mirrored their conceptual distances rather than their feature-based distances. The two-dimensional distances in concept space also significantly outperformed alternative two-dimensional distances derived from combinations with the conceptually irrelevant feature dimension in their ability to explain the hippocampal signal, ruling out the possibility that this effect was caused by a hippocampus that prefers two dimensions for coding. (https://geometrymatters.com/maps-of-conceptual-spaces-in-the-hippocampus/)

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The hippocampal signal thus reflects only a representation of distances in a space spanned by the dimensions that were relevant in relation to one another to define the concept, while the mnemonically relevant third dimension was not integrated in a multidimensional representation. (https://geometrymatters.com/maps-of-conceptual-spaces-in-the-hippocampus/)

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This supports the argument that the hippocampus organizes new information in a map-like representation in support of concept learning and it can carve out (and represent) conceptual information from the totality of features, despite encoding specific exemplars in all detail. (https://geometrymatters.com/maps-of-conceptual-spaces-in-the-hippocampus/)

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The entire brain experiences slow, sustained waves of electrical activity when you sleep, much like the waves on a still ocean. That mental state is referred to as “slow wave sleep” by researchers. The electrical activity pattern is altered upon awakening to resemble more random noise. However, postdoctoral fellow Yianling Shi, assistant professor Tatiana Engel, and its colleagues at Cold Spring Harbor Laboratory (CSHL) discovered that there are patterns in the noise. (https://geometrymatters.com/finding-structure-in-the-brains-static/)

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The visual cortex, which processes visual information, functions like a television screen that assembles an image from a series of dots, or “pixels,” to form a picture. Each brain pixel is made up of a column of cooperating neurons. Unstimulated columns can be either electrically active and responsive to stimuli (referred to as “On”) or inert and resistive to electrical activity (referred to as “Off”). Visual information (a stimulus) is registered as a significant electrical spike when it comes into contact with a “On” visual column. However, if visual information enters a column when it is “Off,” it may not be at all recorded. (https://geometrymatters.com/finding-structure-in-the-brains-static/)

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Engel and Shi discovered that the waves become shorter and more choppy when monkeys are paying attention to a stimulus. They conducted this research in partnership with Kwabena Boahen, Tirin Moore, and Nicholas A. Steinmetz of the University of Washington. In contrast to when the animal is paying attention to something else, “On” and “Off” states flicker across visual cortical columns triggered by this stimulus more quickly and in a smaller region. But why would a conscious, alert brain wish to turn its informational columns off and miss something? The huge rolling sleep waves were shown to have smaller, quicker, more localized counterparts in the visual processing area of a monkey brain. How alert that region of the brain is reflected in the forms and dynamics of these local waves. The wave patterns may hold a key to understanding sleep, anesthesia, and attentiveness, according to the experts. (https://geometrymatters.com/finding-structure-in-the-brains-static/)

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Keeping neurons in the ‘On’ state all the time is energetically costly. Another reason is that if we were always receptive to information, we may become overwhelmed; the ‘Off’ state could help suppress irrelevant information. (https://geometrymatters.com/finding-structure-in-the-brains-static/)

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Researchers may be able to better understand how the brain reacts to medications and illness as a result of the finding that electrical noise exhibits pattern variations with various brain states. Additionally, machine learning researchers may adopt the skillfully designed noise tactics used by monkey brains, which are particularly adept at processing visual information, to enhance artificial brains. (https://geometrymatters.com/finding-structure-in-the-brains-static/)

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Euclidean geometry has always been regarded by scientists as being a priori and objective. When we assume the role of an agent, however, the challenge of determining the optimum path should also take into account the agent’s capabilities, its embodiment, and in particular its cognitive effort. (https://geometrymatters.com/infodesics-and-cognitive-geometry/)

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The geometry of movement between states in a world is taken into account in a 2021 article led by Karen Archer (Department of Computer Science, University of Hertfordshire, Hatfield, United Kingdom) by factoring for information processing costs and the required spatial distances. As information costs grow more significant, this results in a geometry that deviates more and more from the initial geometry of the provided environment. (https://geometrymatters.com/infodesics-and-cognitive-geometry/)

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When projected onto 2- and 3-dimensional spaces, this “cognitive geometry” may be seen to have different distortions that represent the evolution of epistemic and information-saving techniques as well as pivot states. The analogies between traditional cost-based geometries and those induced by additional informational costs invite a generalization of the traditional notion of geodesics as cheapest routes towards the notion of infodesics, which approximates the usual geometric property that, when traveling from a start to a goal along a geodesic, not only the goal but all intermediate points are visited equally at the best cost from the start. (https://geometrymatters.com/infodesics-and-cognitive-geometry/)

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We considered distances induced by cost-only MDPs (Markov decision process) which were additionally endowed with an informational cost reflecting the complexity of decision-making. Geodesics generalise the intuition about geometry determined by directions and distances, representing optimal transitions between states. We proposed that the addition of informational criteria would characterise a cognitive geometry which additionally captures the difficulty of pursuing a particular trajectory. (https://geometrymatters.com/infodesics-and-cognitive-geometry/)

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The researchers discovered that free energy incorporates the cost of information processing while maintaining part of the spatial geometry’s structure through the value function. Although it will favor trajectories that pass through informationally efficient states on route to the objective, highlighting the best informationally efficient policy among otherwise equal policies will. As a result, these trajectories frequently take a “detour” through hubs that are more easily maneuverable in terms of pure distance. (https://geometrymatters.com/infodesics-and-cognitive-geometry/)

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We found considerable distortions which place boundary states more centrally in the space, with the boundaries acting as guides. Additionally, when considering infodesics, i.e. sets of intermediate states which are optimally reachable from the starting state, intermediate goals can be achieved en route to the final goal, thus defining classes of problems that are solved as a side effect of solving the main one. (https://geometrymatters.com/infodesics-and-cognitive-geometry/)

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The team defined the infodesic property as the triangle inequality becoming an equality, similar to how geodesics work. They had to loosen the requirements since this discrepancy is not always properly honored in their framework. Splitting a trajectory also absorbs the cost of switching the policies of the two segments into the split itself because of the informational nature of the free energy distance, which makes the violation of the triangle inequality rather severe. (https://geometrymatters.com/infodesics-and-cognitive-geometry/)

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A generalized but tighter triangle inequality can include the splitting cost for a more thorough explanation of the infodesic, cost that the team suggests will allow the imposing of a quasi-geometrical signature on the state space and provide the basis of a genuinely geometrical notion of task spaces that takes into account cognitive processing: a cognitive geometry. (https://geometrymatters.com/infodesics-and-cognitive-geometry/)

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This we understand to be a structure with optimal trajectories determined by either two states or by one state and a “direction” (i.e. policy) that is informed not only by the pure spatial geometry, but also by the cognitive costs that an agent needs to process when moving from task to task and how it has to informationally organise policies to achieve nearby or related tasks. (https://geometrymatters.com/infodesics-and-cognitive-geometry/)

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Such a notion would imply that even simply navigational decision issues may not be best addressed by the plain application of Euclidean or geodesic-based geometry. It would be intriguing to look into this distance further for that reason. (https://geometrymatters.com/infodesics-and-cognitive-geometry/)

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After noticing that weather data from roughly circular cities like Dallas and London often show more rain than triangular cities like Chicago and Los Angeles, Dev Niyogi and his colleagues at the University of Texas at Austin decided to investigate the link between the shape of an urban area and its rainfall. The authors argue that how wind and weather interact with shape in urban environments should be taken into account when building future urban spaces that must be more resilient to the effects of climate change. (https://geometrymatters.com/circular-cities-benefit-from-more-rainfall/)

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Circular cities, square cities, and triangular cities are rated in order of rainfall amount and intensity from largest to smallest. The findings are important for city development that is both sustainable and resilient, especially for those that are expanding. (https://geometrymatters.com/circular-cities-benefit-from-more-rainfall/)

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This study provides the first investigation of the impact of city shape on urban rainfall in inland and coastal environments. Under calm synoptic conditions, the city shape impact is much more evident in coastal environments. (https://geometrymatters.com/circular-cities-benefit-from-more-rainfall/)

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The team idealized large eddy simulations coupled with the Weather Research and Forecasting model. In the inland vs coastal environment, there are changes in the timing of urban-induced rainfall. This is linked to the land-sea breeze’s differing diurnal cycles of vertical velocity and cloud water mixing ratio. The effect of city shape on rainfall is especially visible at the coast, where buoyant flows from cities modify the interactions between urban-rural circulation and sea breeze. (https://geometrymatters.com/circular-cities-benefit-from-more-rainfall/)

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The frictional convergence displaces the sea wind higher and increases afternoon rainfall over coastal towns, while rapid heating of the land surface provides severe morning rainfall inland. In coastal areas, the impact of city design is substantially more visible during calm synoptic conditions. The circular city receives the most daily rainfall, which is 22.0 percent more than the triangular city. Due to changes in city design, the variance in morning peak rainfall rate might exceed 78.6 percent. The rainfall anomaly is mostly due to city forms altering regional circulation, where differing low-level convergence intensities regulate convective energy and moisture vertical transit. (https://geometrymatters.com/circular-cities-benefit-from-more-rainfall/)

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Given the likelihood that climate change would exacerbate future rainfall hazards, worldwide cities have invested significant resources in researching and implementing a variety of infrastructures as adaptation methods. (https://geometrymatters.com/circular-cities-benefit-from-more-rainfall/)

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Our results identify a hitherto poorly understood but an important role of urban layout especially in the coastal regions. Circular city shape shows potential risks of extreme rainfall and resultant flood risk. Moreover, such risks can not be assessed through climate models with coarse resolutions that cannot accurately represent the city shape. (https://geometrymatters.com/circular-cities-benefit-from-more-rainfall/)

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Plant leaves are arranged in a beautiful geometric pattern around the stem, which is known as phyllotaxis. Phyllotaxis has common characteristics across plant species, which are commonly mathematically characterized and expressed in a small number of phyllotactic patterns. (https://geometrymatters.com/decoding-the-mathematical-secrets-of-plants-spiraling-leaf-patterns/)

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One important premise in the study of phyllotaxis, or leaf patterns, is that leaves guard their personal space. Scientists have built models that can effectively reproduce many of nature’s typical patterns, based on the concept that existing leaves have an inhibitory influence on emerging ones, emitting a signal to prevent others from sprouting close. The enthralling Fibonacci sequence, for example, may be seen in anything from sunflower seed arrangements to nautilus shells to pine cones. (https://geometrymatters.com/decoding-the-mathematical-secrets-of-plants-spiraling-leaf-patterns/)

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The current agreement is that such patterns are caused by the motions of the growth hormone auxin1Auxins are a class of plant hormones (or plant-growth regulators) with some morphogen-like characteristics. Auxins play a cardinal role in coordination of many growth and behavioral processes in plant life cycles and are essential for plant body development. and the proteins that carry it throughout a plant. (https://geometrymatters.com/decoding-the-mathematical-secrets-of-plants-spiraling-leaf-patterns/)

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Alternate phyllotaxis is a leaf arrangement with one leaf per node, whereas whorled phyllotaxis is a leaf arrangement with two or more leaves per node. Distichous phyllotaxis (bamboo) and Fibonacci spiral phyllotaxis – spiral with a divergence angle close to the golden angle of 137.5° (the succulent spiral aloe) are common alternate forms, while decussate phyllotaxis (basil or mint) and tricussate phyllotaxis (the succulent spiral aloe) are common whorled types (Nerium oleander, sometimes known as dogbane). (https://geometrymatters.com/decoding-the-mathematical-secrets-of-plants-spiraling-leaf-patterns/)

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In most plants, phyllotactic patterns have symmetry—spiral symmetry or radial symmetry,” says University of Tokyo plant physiologist Munetaka Sugiyama, senior author of the new study. “But in this special plant, Orixa japonica, the phyllotactic pattern is not symmetric, which is very interesting. More than 10 years ago, an idea came to me that some changes in the inhibitory power of each leaf primordium may explain this peculiar pattern. (https://geometrymatters.com/decoding-the-mathematical-secrets-of-plants-spiraling-leaf-patterns/)

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The divergence angles, or angles between consecutive leaves, are used by botanists to determine a plant’s phyllotaxis. The O. japonica shrub, which is endemic to Japan and other parts of East Asia, grows leaves in an alternating series of four angles: 180 degrees, 90 degrees, 180 degrees again, and 270 degrees. This pattern, named “orixate” phyllotaxis by the researchers, is not unique to this plant; other plants alternate their leaves in the same intricate sequence. (https://geometrymatters.com/decoding-the-mathematical-secrets-of-plants-spiraling-leaf-patterns/)

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By adding the ages of the leaves as another variable to the Douady and Couder equations, The Douady and Couder equations equation make the fundamental assumption that leaves emit a constant signal to inhibit the growth of other leaves nearby and that the signal gets weaker at longer distances., the team created a novel model. (https://geometrymatters.com/decoding-the-mathematical-secrets-of-plants-spiraling-leaf-patterns/)

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Former models assumed that the inhibitory efficacy of leaves remained constant throughout time, but this was “not natural from a biological standpoint,” according to Sugiyama. Instead, Sugiyama’s team assumed that the potency of these “keep-away” signals could alter over time. The resulting models succeeded in recreating, through computerized growth, the intricate leaf arrangements of O. japonica. (https://geometrymatters.com/decoding-the-mathematical-secrets-of-plants-spiraling-leaf-patterns/)

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All of the other frequent leaf patterns were likewise produced by the enlarged equations, which predicted the natural frequencies of these variations more accurately than earlier models. The new EDC2 model anticipated the Fibonacci spiral’s “super-dominance” over other arrangements, especially in spiral-patterned plants, but earlier models failed to explain why this particular shape appears everywhere in nature. (https://geometrymatters.com/decoding-the-mathematical-secrets-of-plants-spiraling-leaf-patterns/)

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The researchers aren’t sure what causes these growth patterns to be affected by leaf age, but Sugiyama speculates that it could be due to changes in the auxin transport system as a plant develops. Sugiyama and his colleagues are working to improve their model even more so that it can generate all known phyllotactic patterns. One “mystery” leaf pattern, a spiral with a small divergence angle, has yet to be predicted computationally, though Sugiyama believes they’re getting close. (https://geometrymatters.com/decoding-the-mathematical-secrets-of-plants-spiraling-leaf-patterns/)

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The multidisciplinary nature of cognitive research brings the need to conceptually unify insights from multiple fields into the phenomena that drive cognition. Newton Howard and Amir Hussain propose the Fundamental Code Unit (FCU) as a means to better quantify the intelligent thought process at multiple levels of analysis and in order to model the brain’s most sophisticated decision-making process as efficiently as possible. (https://geometrymatters.com/a-geometric-model-for-cognition-the-fundamental-code-unit-of-the-brain/)

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The study uses a number of case studies and applications of the human brain to demonstrate the mapping of physical processes observed at the chemical level to cognitive changes manifested through behavior (language abilities, linguistic semantics, etc.). They also investigate the composition of cognition at philosophical, psychological, and neurochemical levels. Considering these concepts as fundamental elements for a “coherent communicated thought”, the revelation of their network and connections can establish a universal cognitive code. (https://geometrymatters.com/a-geometric-model-for-cognition-the-fundamental-code-unit-of-the-brain/)

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The Fundamental Code Unit of Thought (FCU) is an attempt to bridge the gap between observed physical phenomena and their complicated outcomes. Because cognition is both a physical and a computational (i.e., conceptual) phenomenon, any benchmark that is used to assess it must account for both. (https://geometrymatters.com/a-geometric-model-for-cognition-the-fundamental-code-unit-of-the-brain/)

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The underlying units that compose cognition, like those of DNA, are relatively simple compared with the structures they create. This applies both to the brain itself and the way we perceive it (i.e., as a system of sensory inputs and linguistic and behavioral outputs). In our approach, we map the physical phenomena of cognition to this theoretical system. - Howard Newton (https://geometrymatters.com/a-geometric-model-for-cognition-the-fundamental-code-unit-of-the-brain/)

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The first problem addressed by the team is the mechanisms of neurological oscillation and protein deformities (that cause various cognitive disorders). Since a protein’s structure typically provides the basis for its function, biologically important molecular events often consist of a change in the shape or configuration of key proteins. And, since each of these processes introduces the possibility of damaged misfolding, a prediction model can be created with FCU in order to establish which foldings and misfoldings serve as the correct or defective signal. (https://geometrymatters.com/a-geometric-model-for-cognition-the-fundamental-code-unit-of-the-brain/)

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Further, to the uncertain and complex structure of cognition that influences all neural activity, The Maximum Entropy model is used. The statistical model uses machine learning techniques to forecast the chance of discovering something under specific conditions scattered across space using empirical data. One of the problems that FCU seeks to resolve in this context, is the lack of a common language to discern exactly what a neuro-mathematical model is, in terms of its capabilities and intended applications. (https://geometrymatters.com/a-geometric-model-for-cognition-the-fundamental-code-unit-of-the-brain/)

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One important metric explored in their language is “distance”, which helps to determine which conceptual components to cluster as information regarding tendencies is being gathered. Giving the Rosenfeld model (1996) as an example, the authors define a set of concepts, brain regions, and mappings between related concepts in each brain region identified in empirical studies. Upon these, the concept set framework is built to analyze the brain regions and their neural networks. The FCU’s task is to combine multiple sensor data streams into a single computationally efficient framework, such as linguistic input, neurological data (i.e., cell and network activation, and neural firing rate and amplitude), and behavioral phenomena (i.e., nervous tics, spatial judgment errors, and gait irregularities). (https://geometrymatters.com/a-geometric-model-for-cognition-the-fundamental-code-unit-of-the-brain/)

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Being an expression of behavior, the team also uses language as a method to measure the transition from the molecular to the behavioral expressions of interactions of brain functions. Since the processes leading to the acquisition of language are distinct, conceptual divisions can provide important patterns and data. For this, an Axiological model (that uses value and value theory) is used to create a distinct code. (https://geometrymatters.com/a-geometric-model-for-cognition-the-fundamental-code-unit-of-the-brain/)

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In order to resolve the dichotomy problem that arises from value attribution, they create a mapping of the axiological value to neurological state changes. This creates a clearer picture of the components and structure of cognition. The authors state that the interpretation of value only through dichotomy contradicts the notion that language’s use extends beyond its semantic properties and functions, and ignores the fact that we have a variety of construal apparatuses. They develop a time orientation schema, where “positive” can be analogous to “future-oriented” and “negative” can be analogous to “past-oriented.” (https://geometrymatters.com/a-geometric-model-for-cognition-the-fundamental-code-unit-of-the-brain/)

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For example, when the inherent value of a word is unclear, it is helpful to consider the word’s antonym to determine its value. To determine the axiological value of linking and auxiliary verbs, the proposed model uses the “temporal value” of words based on the past-future scale. Future tense verbs convey a forward-looking attitude and should be developed in a positive way. Past tense verbs convey a backward perspective and should be marked as negative. (https://geometrymatters.com/a-geometric-model-for-cognition-the-fundamental-code-unit-of-the-brain/)

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In addition to their conception of brain language, the team proposes a mathematical framework in which FCU is located: the Unitary System, founded on unary mathematics. The functions “unary plus” (+) and “unary minus” (), which express an increase or decrease in the underlying measured value, are computationally efficient enough to simulate human cognition, as long as both sides use the same linguistic foundation. (https://geometrymatters.com/a-geometric-model-for-cognition-the-fundamental-code-unit-of-the-brain/)

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For example, the unitary system can be seen functioning at the molecular level in the molecular chirality1In chemistry, a molecule or ion is called chiral if it cannot be superposed on its mirror image by any combination of rotations, translations, and some conformational changes. This geometric property is called chirality.concept. (https://geometrymatters.com/a-geometric-model-for-cognition-the-fundamental-code-unit-of-the-brain/)

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The brain’s unitary operators carry a state of time and space that contains the information needed to decipher any semantic or non-semantic language used by the brain. When the FCU is deployed, these operators create a common language of cognition because they are language agnostic. (https://geometrymatters.com/a-geometric-model-for-cognition-the-fundamental-code-unit-of-the-brain/)

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Ateam of researchers used state-of-the-art virtual reality to show the fundamental geometrical principles that result from the inherent interplay between movement and organisms’ internal representation of space. These principles apply across scales of biological organization, from individual to collective decision-making. (https://geometrymatters.com/the-geometry-of-individual-and-collective-decision-making/)

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The findings show that animals spontaneously compress the environment into a series of sequential binary decisions, a response that helps successful decision-making and is robust to both the amount of alternatives available and context, such as whether options are static (e.g., refuges) or movable (e.g., predators) (e.g., other animals). (https://geometrymatters.com/the-geometry-of-individual-and-collective-decision-making/)

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The researchers explore the interplay between mobility and vectorial integration during decision-making for two or more possibilities in space using an integrated theoretical and experimental approach (using immersive virtual reality). (https://geometrymatters.com/the-geometry-of-individual-and-collective-decision-making/)

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This leads to the conclusion that the brain splits multichoice judgments into a succession of binary decisions in space–time on a regular basis. Experiments with different insects show that they all have these bifurcations, demonstrating that there are fundamental geometric rules that are required to explain how and why animals move in the way they do. (https://geometrymatters.com/the-geometry-of-individual-and-collective-decision-making/)

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In computational models of this process, we reveal the occurrence of spontaneous and abrupt “critical” transitions (associated with specific geometrical relationships) whereby organisms spontaneously switch from averaging vectorial information among, to suddenly excluding one among, the remaining options. This bifurcation process repeats until only one option—the one ultimately selected—remains. (https://geometrymatters.com/the-geometry-of-individual-and-collective-decision-making/)

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From an egocentric perspective, the decision process is projected to be sequential and dependent on the geometry with regard to the targets, therefore it should be possible to visualize it directly from animal trajectories when making spatial decisions. (https://geometrymatters.com/the-geometry-of-individual-and-collective-decision-making/)

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The researchers built a simple, spatially explicit model of neural decision-making to investigate how the brain minimizes choice when faced with several spatial possibilities. They were able to make theoretical predictions and discover unifying principles of spatiotemporal computation across biological scales. (https://geometrymatters.com/the-geometry-of-individual-and-collective-decision-making/)

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The model provides explicit predictions for animal trajectories, allowing them to determine which target is reached in the simulation. The brain model is composed of individual components, called “spins,” that collectively, as a “spin system,” represent neural activity. (https://geometrymatters.com/the-geometry-of-individual-and-collective-decision-making/)

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Hopfield was the first to introduce spin systems into the study of neurobiology in a landmark paper that provided significant insights into principles underlying unsupervised learning and associative memory. Spin systems have long been studied in physics due to their ability to give insight into a wide range of collective phenomena from magnetic to quantum systems. (https://geometrymatters.com/the-geometry-of-individual-and-collective-decision-making/)

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In its most basic form, a spin system is made up of things that can be in either state 0 or 1, or, in physics terms, “up” or “down.” From spin and molecular systems to neurological systems undergoing phase transitions, spin systems have continually provided significant insights into complicated collective processes. (https://geometrymatters.com/the-geometry-of-individual-and-collective-decision-making/)

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The results from their neural decision-making model are reproduced in a second model that describes spatial decision-making at a different scale of biological organization. (https://geometrymatters.com/the-geometry-of-individual-and-collective-decision-making/)

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Thus, we find that similar principles may underlie spatial decision-making across multiple scales of biological organization. Furthermore, by presenting social interactions in a decision-making context, our zebrafish experiments elucidate the neural basis of schooling, allowing us to glean insights across three scales of biological organization—from neural dynamics to individual decisions and from individual decisions to collective movement. (https://geometrymatters.com/the-geometry-of-individual-and-collective-decision-making/)

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In 2020, Héctor Alarcón, at O’Higgins University, Chile, and his colleagues started studying the forces that act on particles moving through the fluid in a vibrating tank. The research led them to the discovery of a new type of pattern, that they called “hedgehog” based on the similarity to the nocturnal animal’s shape. The discovery came as a surprise since researchers have conducted similar experiments, without observing the new patterns. (https://geometrymatters.com/peculiar-patterns-of-vibrating-floating-particles/)

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In their experiment, the team adapted a setup devised by Michael Faraday in the 1830s. He sprinkled particles into a shallow, water-filled container and by vibrating it, the particles would sink to the bottom of the container. In response to the vibrations of the container, they are arranged into various patterns: parallel stripes, checkerboards, or hexagonal structures, depending on the vibration frequency. Today, these patterns are called Faraday waves. (https://geometrymatters.com/peculiar-patterns-of-vibrating-floating-particles/)

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In Alarcón’s experiment, the particles remained buoyant, floating just below the surface. When the container was shaken vertically at 8.3Hz, they could see the particles collecting into four groups, two on each side of the antinodal line. Depending on the size of the oscillation, the researchers saw a variety of shapes for these groups, ranging from spinning vortices to spiky hedgehogs. They also used simulations to duplicate the patterns, discovering that the particle groupings are caused by circulating fluid currents caused by the fluid pushing and pulling along the walls on a regular basis. (https://geometrymatters.com/peculiar-patterns-of-vibrating-floating-particles/)

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Network geometry helps us better understand complex systems at all sizes of organizations, as well as the collective phenomena that emerge from their information flow. Being useful in a wide range of applications, from understanding how the brain functions to Internet routing, a variety of approaches have been employed to study complex networks from different perspectives, leading to novel fundamental insights. One such approach is geometry. (https://geometrymatters.com/the-fractal-hyperbolic-geometry-of-networks/)

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We review three major research directions in network geometry: the self-similar fractal geometry of network structure, the hyperbolic geometry of networks’ latent spaces, and the geometry induced by dynamic processes, such as diffusion, in networks. (https://geometrymatters.com/the-fractal-hyperbolic-geometry-of-networks/)

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In the study of network geometry, understanding how the dynamical features of network growth processes influence their asymptotic self-similar patterns is an exciting and yet unexplored topic. The process of zooming-out used in the study finds the (statistically equivalent) inverse in the dynamics of the explored network so that the varying structures observed at increasing length scales correspond to the evolution of certain dynamical variables. (https://geometrymatters.com/the-fractal-hyperbolic-geometry-of-networks/)

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In the simple case of growing trees, it has further been proved that this connection is a consequence of the explicit dependence of the fractal dimension on the growth rates ruling the system’s evolution. (https://geometrymatters.com/the-fractal-hyperbolic-geometry-of-networks/)

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The finding of complex networks’ self-similarity under adequate scale transformations provided the first proof that they had some nontrivial geometric features. The fractal geometry of networks enables casting the self-similar symmetries underlying the organization of complex systems under the three pillars of scaling, universality, and renormalization. (https://geometrymatters.com/the-fractal-hyperbolic-geometry-of-networks/)

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The hidden geometry can be explained at a more fundamental level in terms of “latent hyperbolic geometry,” which has applications in a variety of areas. Small-worldness, degree heterogeneity, clustering, community structure, symmetries, and navigability are all logical explanations for the construction of genuine complex networks in these latent hyperbolic spaces. (https://geometrymatters.com/the-fractal-hyperbolic-geometry-of-networks/)

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Furthermore, hyperbolic geometry has strong ties to self-similar metric spaces, implying that the fractal exponents observed in self-similar networks may have acceptable counterparts in their corresponding latent spaces and that this theory might be applied to pure small-world structures. (https://geometrymatters.com/the-fractal-hyperbolic-geometry-of-networks/)

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This approach has impacted areas as diverse as mathematics, neuroscience, and machine learning. In neuroscience, geometric navigation offers a possible explanation and a mechanism for the routing of information in the brain. For example, this model has been recently used to create a hyperbolic map of the human olfactory space. (https://geometrymatters.com/the-fractal-hyperbolic-geometry-of-networks/)

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While network geometry is still a developing field, it provides a novel theoretical framework for gaining profound insights into the underlying principles of complex systems and, more broadly, physical reality. (https://geometrymatters.com/the-fractal-hyperbolic-geometry-of-networks/)

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Scientists Sue Yeon Chung and L.F. Abbott from Columbia University developed an approach for understanding neural networks, by analyzing the geometric properties of neural populations and understanding how information is embedded and processed through high-dimensional representations to solve complex tasks. When a group of neurons demonstrates variability in response to stimuli or through internal recurrent dynamics, manifold-like representations emerge. (https://geometrymatters.com/the-manifold-framework-of-neural-circuits/)

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In their work, the team highlights the important examples of how geometrical techniques and the insights they provide have aided the understanding of biological and artificial neural networks. They investigate the geometry of these high-dimensional representations, i.e., neuronal population geometry, using mathematical and computational tools. Their review includes a variety of geometrical approaches that provide insight into the function of the networks. (https://geometrymatters.com/the-manifold-framework-of-neural-circuits/)

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An important aspect of their research is the insight that dividing sets of neural population activities is more difficult when their representation patterns are on a curved surface, instead of a linear one – a concept that plays an important role in the analysis of neural population geometry and goes back to the beginnings of artificial neural networks. (https://geometrymatters.com/the-manifold-framework-of-neural-circuits/)

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Creating them into forms that are linearly separable comes from the role of the visual system that transforms visual objects into representation so that they become “untangled”. This process makes it easier to predict future neural activity, if the created trajectories are straight, which in turn leads to the “temporal straightening hypothesis” (the visual system transforms its input into a representation that follows a “straighter” trajectory through time, from which predictions can be achieved through simple linear extrapolation). (https://geometrymatters.com/the-manifold-framework-of-neural-circuits/)

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This geometrical framework for perception leads to a geometry of abstraction, the process of linear separability providing insight into more complex tasks beyond categorization. The use of ‘untangled’ representation, as assessed by a geometric measure called the parallelism score, is seen in recordings from the prefrontal cortex, hippocampus, and results from task-trained neural networks. While achieving abstraction from the processed information from different sets of tasks (that involve uncued “context”), the neural circuits also keep the information about other variables. (https://geometrymatters.com/the-manifold-framework-of-neural-circuits/)

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The presence of sources of variability for prediction will cause points (in the neural space) or the one-dimensional trajectory (for time-dependent stimuli) to jitter. This introduces the need to cluster responses into point-cloud manifolds. (https://geometrymatters.com/the-manifold-framework-of-neural-circuits/)

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For example, if we want to distinguish dogs from cats, we may want to group the responses to images for different viewing angles, sizes, and animal breeds into one dog manifold and one cat manifold. In this perspective, the problem of invariant object discrimination becomes that of separating neural manifolds. - Sue Yeon Chung (https://geometrymatters.com/the-manifold-framework-of-neural-circuits/)

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By considering multiple frameworks for building this type of manifold, the team concludes that the untangling hypothesis can be extended to the idea that visual processing aims to develop well-separated manifolds that provide information about object identity while maintaining other image-related variables such as pose, position, and scale. Further theoretical advancements are motivated, such as the manifold capacity theory, which allows for a more refined geometric analysis of representations in biological and artificial neural networks. (https://geometrymatters.com/the-manifold-framework-of-neural-circuits/)

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The team also studied the way the neural activity is built on lower-dimensional subspaces, i.e. neural manifolds, that have an intrinsic set of properties and functions. These determine a geometry of representation, as a consequence of more subtle but fundamental symmetries between said elements. In understanding the structure of these neural manifolds, nonlinear dimensionality reduction techniques are employed for showing how intrinsic dimensions of the manifold encode and process topological and geometric data. (https://geometrymatters.com/the-manifold-framework-of-neural-circuits/)

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These findings illustrate an exciting trend at the intersection of machine learning, neuroscience, and geometry, in which neural population geometry provides a useful population-level mechanistic descriptor underlying task implementation. - Sue Yeon Chung (https://geometrymatters.com/the-manifold-framework-of-neural-circuits/)

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A group of researchers from Italy and the United Kingdom analyzed the development of geometrical concepts, the cognitive processes underlying geometry-related academic achievements, and the educational implications that learning geometry can have. Irene C. Mammarella, David Giofrè, and Sara Caviola reviewed the literature on learning geometry and evaluated papers from developmental psychology, cognitive psychology, educational psychology, and education. (https://geometrymatters.com/geometrical-concepts-cognition-and-educational-implications/)

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To thoroughly understand how best to go about this issue, we need to consider not only the cognitive processes involved in geometry, not only how geometrical knowledge is developed (are some concepts of geometry innate?), and not only how geometry is taught, but all of them together. (https://geometrymatters.com/geometrical-concepts-cognition-and-educational-implications/)

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By first providing a brief review of the principal theories and models relating to learning geometry, the focus was to analyze the secondary role that arithmetic and algebra have in many school curricula, in spite of their role in cognition development. An experiment from Piaget and Inhelder (1967) showed that children can construct perceptual spaces already in infancy, being much later able to develop ideas and concepts about said spaces. (https://geometrymatters.com/geometrical-concepts-cognition-and-educational-implications/)

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Core intuitive principles of geometry are suggested by making a distinction between those implicit and those associated with schooling. The core knowledge hypothesis provides an argument in this favor, with mathematical abilities seeming to emerge from different representational core systems: one based on intuition and another one relying on the symbolic representation, specific to humans who have received some formal education and are able to create novel abilities (including symbolic mathematics). (https://geometrymatters.com/geometrical-concepts-cognition-and-educational-implications/)

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The core knowledge that defines intuitive geometry is culture-independent, with people having a preference for geometrical properties and attributes. Studies on Amazonian and North American children and adults (Dahaene et al., 2006), showed that they were able to correctly identify concepts of topology (right angles) and geometrical figures (e.g., squares, triangles, and circles) with no formal geometry training but having the same results in performance. This was interpreted as proof of the existence of core principles of geometry. (https://geometrymatters.com/geometrical-concepts-cognition-and-educational-implications/)

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Two core systems for this natural geometric knowledge are considered, based on previous studies with children: one for navigating 3d spatial layouts and one for analyzing 2D visual forms), together capturing all the fundamental properties of Euclidian geometry (distance, angle, and directional relationships). (https://geometrymatters.com/geometrical-concepts-cognition-and-educational-implications/)

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The researchers studied how Italian children develop the critical skills presented in Hiele’s model. The results showed that while a majority of the children had limited geometric knowledge (e.g., little geometrical education), their spatial skills scored high on identifying, rotating, and distinguishing shapes among distractors. This is consistent with the claim that very young children are already able to solve basic geometrical and spatial problems even if their geometrical education is poor. (https://geometrymatters.com/geometrical-concepts-cognition-and-educational-implications/)

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The educational evolution of children can benefit in full from understanding these studied core principles. Academic geometry achievement is regarded as one of the most significant areas of mathematical study, particularly at the secondary school level, and is linked to future academic and professional success. Secondary school students must learn basic geometric concepts, definitions, theorems, and other related material, as well as apply their knowledge to solve problems that are often provided in the form of verbal or written language. And without a strong geometrical foundation, students are not prepared for advanced study in the fields of science, technology, engineering, and mathematics. (https://geometrymatters.com/geometrical-concepts-cognition-and-educational-implications/)

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While teachers report that geometry is a difficult subject to teach properly, they must be able to recognize geometrical problems and theorems, explore the historical and cultural background of geometry, and understand the diverse real-world applications of geometry. (https://geometrymatters.com/geometrical-concepts-cognition-and-educational-implications/)

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With the help of a proposed curriculum (with steps focused on academic proficiency, geometry, and spatial reasoning and cognition), the educational goal is to develop competence in the two domains consistently identified as fundamental: (1) number concepts (counting, subitizing, or identifying the numerosity of small quantities) and arithmetical operations; and (2) spatial and geometric concepts and processes. (https://geometrymatters.com/geometrical-concepts-cognition-and-educational-implications/)

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Improving academic achievement in geometry may contribute to growth in other abilities, including higher-order cognition and mathematical reasoning. Academic geometry is a complex domain, demanding a wide range of skills and knowledge of mathematics, problem-solving (which in turn involves higher-order skills related to higher-order cognitive processes, such as intelligence or reasoning), spatial abilities, verbal knowledge of geometrical concepts, and more. (https://geometrymatters.com/geometrical-concepts-cognition-and-educational-implications/)

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In 2012, a team from IBM’s research labs in Zurich managed to reveal the individual bonds that hold a molecule together. The bond order and length of individual carbon-carbon bonds in C60, often known as a buckyball because of its football form, and two planar polycyclic aromatic hydrocarbons (PAHs), which resemble microscopic flakes of graphene, were photographed by employing a method known as atomic force microscopy, or AFM. (https://geometrymatters.com/the-geometry-of-atomic-bonds/)

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We found two different contrast mechanisms to distinguish bonds. The first one is based on small differences in the force measured above the bonds. We expected this kind of contrast but it was a challenge to resolve. The second contrast mechanism really came as a surprise: bonds appeared with different lengths in AFM measurements. With the help of ab initio calculations we found that the tilting of the carbon monoxide molecule at the tip apex is the cause of this contrast. - Leo Gross (https://geometrymatters.com/the-geometry-of-atomic-bonds/)

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The scientists had previously succeeded in seeing a molecule’s chemical structure, but not the subtle differences in the bonds. Bond order discrimination is close to the technique’s current resolution limit, and additional effects frequently obscure the bond order contrast. As a result, the scientists had to choose and create molecules that had no perturbing background effects. (https://geometrymatters.com/the-geometry-of-atomic-bonds/)

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The photos reveal the molecule’s inner workings, with the darker areas representing the densest regions of the atom and the lighter spots representing the lightest. This information indicates what type of bonds they are, as well as how many electrons pairs of atoms share and what is happening chemically within the molecules. (https://geometrymatters.com/the-geometry-of-atomic-bonds/)

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The process of innovation is a mystery. Historically it has offered opportunities and challenges, as well, to humankind. What is new frequently resists humans’ innate desire to predict and control future events. Nevertheless, the majority of our judgments are based on our expectations for that future. (https://geometrymatters.com/patterns-of-innovation/)

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It has been examined by economics, anthropologists, evolutionary biologists, and engineers, among others, with the objective of better understanding how innovation occurs and the factors that influence it so that future innovation circumstances might be improved. (https://geometrymatters.com/patterns-of-innovation/)

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In 2017, a group of researchers studied the dynamics of innovations and how one could create models for the emergence of novelty. The work of Vittorio Loreto and a group of collaborators at Sapienza University of Rome in Italy developed the first mathematical model that correctly reproduces the patterns that inventions follow. The work paves the way for a new model of thinking about innovation, about what’s feasible and how it relates to what already exists. (https://geometrymatters.com/patterns-of-innovation/)

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Formalizing the notion of adjacent possible envisioned by S. Kauffman, presents for the first time a satisfactory first-principle based way of reproducing empirical observations. - Loreto et al (https://geometrymatters.com/patterns-of-innovation/)

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The “adjacent possible”, introduced by the complexity theorist Stuart Kauffmann consists of all the unexplored possibilities surrounding a particular phenomenon: ideas, words, songs, molecules, genomes, and so on. The very definition of adjacent possible encodes the dichotomy between the actual and the possible: the actual realization of a given phenomenon and the space of possibilities still unexplored. But all the connections between these elements are hard to measure and quantify when including the things that are entirely unexpected and hard to imagine. (https://geometrymatters.com/patterns-of-innovation/)

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Another well-known statistical pattern in innovation, Zipf’s law, is showing up in a wide range of circumstances, from the number of edits on Wikipedia to how we are listening to songs online. By relating the frequency of innovation to its popularity the patterns become empirical laws. (https://geometrymatters.com/patterns-of-innovation/)

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Loreto et al show for the first time how these patterns arise by looking at evolution as a path in a complex space, physical, conceptual, biological, technological, whose structure and topology get continuously reshaped and expanded by the occurrence of the new. The consequences of the interplay between the “actual” and the “possible” create a mathematical model for innovation. (https://geometrymatters.com/patterns-of-innovation/)

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By testing and reproducing various tests and models of distribution by frequency or probability (Heaps’ and Zipf’s Laws, Polya’s urn, and others) their model accurately predicted how events occurred in various simulations and events. There are two types of discovery in these systems. On the one hand, there are things that have always existed but are new to the person who discovers them, such as internet songs, and on the other hand, there are things that have never existed and are completely new to the world. (https://geometrymatters.com/patterns-of-innovation/)

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They also demonstrated that the pattern underlying our discovery of novelties is the same as the pattern underlying the emergence of innovations from the nearby possibility. (https://geometrymatters.com/patterns-of-innovation/)

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These results provide a starting point for a deeper understanding of the adjacent possible and the different nature of triggering events that are likely to be important in the investigation of biological, linguistic, cultural, and technological evolution. - Loreto et al. (https://geometrymatters.com/patterns-of-innovation/)

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In 2016, scientists from the University of Warsaw created the first-ever hologram of a single light particle, adding new insights to the foundations of quantum mechanics. (https://geometrymatters.com/hologram-of-a-single-photon/)

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Individual points of a picture in traditional photography merely register light intensity. The interference phenomenon also registers the phase of the light waves in traditional holography. A well-described, undisturbed reference wave is superimposed with another wave of the same wavelength reflected off a three-dimensional object when a hologram is generated. Interference occurs as a result of the phase variations between the two waves, resulting in a complicated pattern of lines. (https://geometrymatters.com/hologram-of-a-single-photon/)

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The spatial structure of waveforms of light reflected from the object is then recreated by illuminating the hologram with a beam of the reference light, and so its 3D shape is recreated. Because individual photons have a continuous fluctuation, the Warsaw researchers took a novel approach to the problem: instead of employing conventional interference of electromagnetic waves, they attempted to register quantum interference, in which photon wave functions interact. (https://geometrymatters.com/hologram-of-a-single-photon/)

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We performed a relatively simple experiment to measure and view something incredibly difficult to observe: the shape of wavefronts of a single photon.- Radoslaw Chrapkiewicz (https://geometrymatters.com/hologram-of-a-single-photon/)

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At its core, the experiment is simple: instead of looking at changing light intensity, they are measuring the changing probability of registering pairs of photons after the quantum interference. The researchers obtained an interference image corresponding to the hologram of the unknown photon viewed from a single point in space by repeating the measurements numerous times. (https://geometrymatters.com/hologram-of-a-single-photon/)

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Studying multiple neural networks, researchers from EPFL Switzerland found that every one of us has a unique brain fingerprint. Comparing the graphs generated from MRI scans of the same subjects taken a few days apart, they were able to correctly match up the two scans of a given subject nearly 95% of the time. After creating a modeling technique, they were able to create a unique signature of each individual’s studied brain, signatures that changes over the course of our lives. (https://geometrymatters.com/a-patterned-fingerprint-of-the-brain/)

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Physician Marcello Malpighi discovered different patterns of ridges and sweat glands on fingertips in the 17th century. This pivotal discovery sparked a decades-long search for ways to uniquely identify people using their fingerprints, a technique that is still widely employed today. The concept of fingerprinting has evolved over time to include other biometric data such as voice recordings and retinal scans, among other things. (https://geometrymatters.com/a-patterned-fingerprint-of-the-brain/)

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Only in the last several years have technologies and methodology enabled high-quality brain measurements to the point where personality traits and behavior can now be classified. (https://geometrymatters.com/a-patterned-fingerprint-of-the-brain/)

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Until now, neuroscientists have identified brain fingerprints using two MRI scans taken over a fairly long period. But do the fingerprints actually appear after just five seconds, for example, or do they need longer? And what if fingerprints of different brain areas appeared at different moments in time? Nobody knew the answer. So, we tested different time scales to see what would happen. - Enrico Amico, EPFL’s Medical Image Processing Laboratory and the EPFL Center for Neuroprosthetics (https://geometrymatters.com/a-patterned-fingerprint-of-the-brain/)

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His research team discovered that seven seconds was insufficient for detecting relevant data, while 1 minute and 40 seconds were. They also revealed that the sensory parts of the brain, particularly those associated with eye movement, visual perception, and visual attention, produce the fastest brain fingerprints. With the passage of time, frontal cortical regions linked with more complex cognitive activities begin to expose unique information to each of us. (https://geometrymatters.com/a-patterned-fingerprint-of-the-brain/)

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Also, the fingerprints are linked to time scales of functional brain connections and may be linked to short bursts of brain activity. Based on these first findings, this work appears promising and provides a step toward a deeper understanding of what and when makes our brains unique. (https://geometrymatters.com/a-patterned-fingerprint-of-the-brain/)

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By creating a computer program that turns sequences of events from a video into unique geometric shapes, Dartmouth researchers are analyzing how the brain creates, uses, and stores memories. When compared, the resulting shapes can further the knowledge of the memory experience. (https://geometrymatters.com/geometric-models-for-memories/)

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We model experiences and memories as trajectories through word-embedding spaces whose coordinates reflect the universe of thoughts under consideration. Memory encoding can then be modelled as geometrically preserving or distorting the ‘shape’ of the original experience. - Jeremy R. Manning (https://geometrymatters.com/geometric-models-for-memories/)

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“When you experience something, its shape is like a fingerprint that reflects its unique meaning, and how you remember or conceptualize that experience can be turned into another shape. We can think of our memories like distorted versions of our original experiences. Through our research, we wanted to find out when and where those distortions happen (i.e. what do people get right and what do people get wrong), and examine how accurate our memories of experiences are,” he added. (https://geometrymatters.com/geometric-models-for-memories/)

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Their approach was applied to a group of participants that watched and verbally recounted a television episode from Sherlock, while undergoing functional neuroimaging. The dataset also contained detailed scene-by-scene annotations of the episode. (https://geometrymatters.com/geometric-models-for-memories/)

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The team ran those annotations through their computer program, and a “topic model” for the episode was created using computer modeling. When these findings are represented in 2D, a connect-the-dots-style representation of subsequent events emerges. The shape of that representation represents how the episode’s thematic substance evolves over time, as well as how distinct events are connected. The researchers employed a similar method to determine the morphologies of each of the participants’ accounts of the episode’s happenings. In this way, they were able to identify which aspects of the episode people tended to remember accurately, forget or distort. (https://geometrymatters.com/geometric-models-for-memories/)

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One of our most intriguing findings was that, as people were watching the episode, we could use their brain activity patterns to predict the distorted shapes that their memories would take on when they recounted it later. - Jeremy R. Manning (https://geometrymatters.com/geometric-models-for-memories/)

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This implies that some facts about our ongoing experiences become corrupted in our brains as soon as they are stored as new memories. Even when two people witness the same physical event, their subjective perceptions of it begin to diverge as soon as their brains begin to make sense of what happened and distill it into memories. (https://geometrymatters.com/geometric-models-for-memories/)

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This research can be used in other domains, including in health and education, the applied methods of modeling the shape of memories provide ways of assessing a patient’s level of understanding or the possible progress of a student during a lecture or a course. (https://geometrymatters.com/geometric-models-for-memories/)

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In her 2013 study, Francesca Biagioli examines Helmholtz’s claim that space can be transcendental without the axioms being so. In 1870, Kant’s concept of geometrical axioms as a priori synthetic judgments based on spatial intuition was questioned by Hermann von Helmholtz, employing a Kantian argument that can be paraphrased as follows: for judgments about magnitudes to be generally valid, mathematical structures that can be described independently of the objects we experience are required. (https://geometrymatters.com/what-does-it-mean-that-space-can-be-transcendental-without-the-axioms-being-so/)

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Helmholtz, according to Biagioli, envisioned space as one of these structures, geometrical axioms only yielding objective knowledge when used in conjunction with mechanical principles. Geometry becomes scientifically testable in this way, and its certainty no longer relies on a priori intuition. (https://geometrymatters.com/what-does-it-mean-that-space-can-be-transcendental-without-the-axioms-being-so/)

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The interpretation of Helmholtz’s claim sparked an intensive debate. Alois Riehl, for example, utilized Helmholtz’s claim to argue that Kant’s theory might operate for intuitive space regardless of the results of physical space measurements. Moritz Schlick, following Riehl’s line of reasoning, connects Helmholtz’s “narrower specifications” to axioms of congruence about physical magnitudes. According to Schlick, these characteristics are distinct from topological aspects (three-dimensionality, continuity, etc.) which are thought to be based on spatial intuition. To retain Helmholtz’s distinction, Schlick considers space qualities and psychological processes in spatial perception to be indescribable. (https://geometrymatters.com/what-does-it-mean-that-space-can-be-transcendental-without-the-axioms-being-so/)

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Schlick’s interpretation is at odds with the Kantian theory: one needs not introduce a ‘‘pure’’ intuition besides the empirical one. Then why does Helmholtz deem space ‘‘transcendental’’? On the other hand, any rejection of Schlick’s interpretation is committed to another question: what are the general characteristics of space? - F. Biagioli (https://geometrymatters.com/what-does-it-mean-that-space-can-be-transcendental-without-the-axioms-being-so/)

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Because Kant never used the term “transcendental space”, Biagioli argues that Helmholtz is thinking in terms of “a priori”. Therefore the confusion about his conception of space in Schick’s interpretation, and the misleading claim that the qualities of sense perception are indescribable: “since the requirement of indescribability is questionable, some interpreters seek to make sense of the quality–quantity opposition by introducing the distinction between topological and metrical properties.” (https://geometrymatters.com/what-does-it-mean-that-space-can-be-transcendental-without-the-axioms-being-so/)

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Helmholtz deems sensations ‘‘signs’’ for their stimuli, therefor meaning is only an intuitive aquaintance. However, once acquired through the localization of them in space (and constructing the concept of space), the intuition of them can be transcendental. Although Helmholtz rejects Kant’s idea that the axioms of Euclid’s geometry must be valid for an empirical manifold, he uses a Kantian argument: the properties of space should supply us with general conditions of measurement, the results of those interpretation being “objective”, by depending on the system’s conditions. (https://geometrymatters.com/what-does-it-mean-that-space-can-be-transcendental-without-the-axioms-being-so/)

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Biagioli continues her analysis on Albrecht Krause, which rejects the attempt to draw spatiality out of sensations and questions the ways of expressing laws of spatial intuition with axioms, considering that measurements should not be trusted when they contracticted the axioms. (https://geometrymatters.com/what-does-it-mean-that-space-can-be-transcendental-without-the-axioms-being-so/)

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While axioms are exact knowledge, their interpretations are an approximate character of natural laws. Helmholtz point’s out that that Krause’s assumptions, which are derived from a nativist theory of vision (the belief that humans are born with all the perceptual abilities needed), are not committed to the Kantian theory of knowledge, that can be summed up to his statement: “Thoughts without contents are empty; intuitions without concepts are blind.” So, from a philosophical standpoint, Krause’s argument can be refuted: once nativist assumptions are discarded, space can be transcendental even if the axioms are not. (https://geometrymatters.com/what-does-it-mean-that-space-can-be-transcendental-without-the-axioms-being-so/)

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While detailing the transcendental intuition in Holmheltz’s view, Biagioli summarizes that the distinction between space and axioms follow from his analysis of measurement, in which a transcendental argument of intuition is requiered in order to prove that quantitative relations are common to subjective and objective experiences: “Geometrical axioms cannot be derived from an innate intuition independently of experience.” (https://geometrymatters.com/what-does-it-mean-that-space-can-be-transcendental-without-the-axioms-being-so/)

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A detailed analysis of his concept of number provide the arguments for a general theory of measurements and the general conditions of the numbering of external objects – of which distinction in attributes is defined as magnitude, that provide means for transcendence. (https://geometrymatters.com/what-does-it-mean-that-space-can-be-transcendental-without-the-axioms-being-so/)

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On the one hand, the structure of time is described as a fact, whose origin should be explained psychologically; on the other hand, this fact provides us with concepts, such as that of number and of sum, which can be proved to determine our conception of nature. Similarly, space entails the concept of fixed geometric structure. The axioms of arithmetic, as well as those of geometry, presuppose a transcendental intuition. The axioms are not transcendental because they provide us with definitions which, in order to be applied to empirical objects, require a physical interpretation. (https://geometrymatters.com/what-does-it-mean-that-space-can-be-transcendental-without-the-axioms-being-so/)

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Thus, mathematical structures can only be defined independently of the objects we experience, and generally valid judgments about magnitudes presuppose a physical interpretation of the same structures. Furthermore, geometrical axioms are related to space as arithmetical axioms, and some mathematical structures provide us with general conditions of experience. (https://geometrymatters.com/what-does-it-mean-that-space-can-be-transcendental-without-the-axioms-being-so/)

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By providing her analysis of Helmholtz’s transcendence, Cassirer’s remarks on Helmholtz’s conception of number, and Hölder’s argument that magnitudes can be constructed as hypothetic-syntethic concepts, Biagioli considers that space is one such structure. (https://geometrymatters.com/what-does-it-mean-that-space-can-be-transcendental-without-the-axioms-being-so/)

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Pure geometry’s cognitive foundation is mostly unknown. Even the ‘simpler’ question of what kind of geometric object representation we have. Mario Bacelar Valente proposes a model of geometric object representation at a neurological level for the case of Euclid’s pure geometry in his work. He considers historical characteristics of practical and pure geometry together to arrive at the model. This allows for a consistent representation of geometric objects based on past data. (https://geometrymatters.com/pure-geometry-and-geometric-cognition/)

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To develop the model consistent with the previous geometrical practices, we have considered a historically informed account of practical geometry. The objective was to provide a basic characterization of practical geometry. Taking into account these basic ‘characteristics’ we build a model of the neural concept representation of geometric figure in practical geometry using in a very simple way the hub-and-spoke theory of neural concept representation. - M.B. Valente (https://geometrymatters.com/pure-geometry-and-geometric-cognition/)

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The hub-and-spoke idea underpins the models (Ralph, Jefferies, Patterson, and Rogers 2017). The neural representation of concepts, according to this hypothesis, is made up of spokes, which are modality-specific brain areas that encode modal properties of concepts. The spokes, for example, encode visual, verbal (speaking), and physical representations. There are also integrative areas – the hub – that integrate the different components encoded in the spokes in an amodal style to produce coherent thoughts. Also, the hub enables a modality-free codification of further aspects of concepts. (https://geometrymatters.com/pure-geometry-and-geometric-cognition/)

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By addressing the practical geometry of ancient Greece and taking into account elements of the practical geometry of ancient cultures, a more general characterization of geometry is built. Valente then builds a neural representations of geometric objects for Hippocrates and Aristotle’s models for pure geometry. (https://geometrymatters.com/pure-geometry-and-geometric-cognition/)

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He reconstructs some aspects of the Euclidean practice taking into account how these are different from the corresponding aspects in the Hippocratic practice. Taking these differences into account together with the models of neural concept representation in practical geometry and Hippocrates’ pure geometry, a simple model of abstract geometric objects is then proposed. (https://geometrymatters.com/pure-geometry-and-geometric-cognition/)

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By taking into account the encoding of the verbal spoke, the representations in the visual, praxis, and symbolic number-magnitude spokes are expressed in a highly abstract fashion. When we look at a figure, we have a certain indifference and responsiveness to its qualities that are related to the linguistic definition, thus we re-conceive the figure as an abstract geometric entity rather than a perfect figure (as mentioned, the figure becomes for us a representation of the geometric object). At this approach, the verbal spoke influences how the visual, physical, and symbolic number-magnitude spokes are interpreted and recombined in the hub, resulting in a ‘higher order’ representation of geometric abstract objects. (https://geometrymatters.com/pure-geometry-and-geometric-cognition/)

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In our view, the hub-and-spoke models of the neural representation of geometric figure/object make more intelligible what a geometric object might be since we can relate its neural representation to that of a geometric figure in Hippocrates’ pure geometry and in practical geometry. (https://geometrymatters.com/pure-geometry-and-geometric-cognition/)

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Narratives, and other types of speech, can be used to enlighten, entertain, and make sense of the world. However, while discourse is frequently described as moving swiftly or slowly, covering a lot of ground, or going in circles, little research has been done to quantify such motions or determine whether they are advantageous. (https://geometrymatters.com/how-quantifying-the-shape-of-stories-predicts-their-success/)

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To close this gap, Olivier Toubia et al. express texts as sequences of points in a latent, high-dimensional semantic space, combining multiple state-of-the-art natural language processing and machine-learning techniques. They develop a basic set of metrics to quantify elements of this semantic path, test them on thousands of texts from various domains (e.g., movies, TV shows, and academic papers), and see if and how they are related to success (e.g., the number of citations a paper receives). (https://geometrymatters.com/how-quantifying-the-shape-of-stories-predicts-their-success/)

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The findings highlight several key cross-domain characteristics and give a broad framework for studying a variety of discourse kinds, as well as shed light on why things become popular and how natural language processing might help predict cultural success. (https://geometrymatters.com/how-quantifying-the-shape-of-stories-predicts-their-success/)

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We use natural language processing and machine learning to analyze the content of almost 50,000 texts, constructing a simple set of measures (i.e., speed, volume, and circuitousness) that quantify the semantic progression of discourse. (https://geometrymatters.com/how-quantifying-the-shape-of-stories-predicts-their-success/)

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Various criteria measured in the study capture human perceptions of circuitousness. While circuitousness might seem undesirable (the ratio of the actual distance traveled to the shortest possible path), it may allow the audience to create new and deeper connections between previously explored themes. (https://geometrymatters.com/how-quantifying-the-shape-of-stories-predicts-their-success/)

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While many have theorized about features of narratives, less work has formalized these intuitions, or tested whether certain features of discourse are linked to success. This paper provides a set of measures to quantify the semantic progression of texts and the ground they cover. In particular, we examined speed, volume, and circuitousness and how they relate to the success of movies, TV show episodes, and academic papers. (https://geometrymatters.com/how-quantifying-the-shape-of-stories-predicts-their-success/)

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The findings reveal that the characteristics that distinguish a good movie from those that distinguish a successful TV show or academic paper, and future research could look into the origins of these cross-domain differences. The style of discourse (e.g., story vs. exposition), the purpose (e.g., to entertain vs. to transmit knowledge), the modality (e.g., video vs. written), the result measure (e.g., like vs. citations), and audience expectations are all possible considerations. Other types of writings may be studied in the future (e.g., books, speeches, or documentaries). (https://geometrymatters.com/how-quantifying-the-shape-of-stories-predicts-their-success/)

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The human function is to ‘discover or observe’ mathematics,” said twentieth-century British mathematician G. H. Hardy. Humanity has been searching for beauty and order in the arts and in nature for generations, dating back to the ancient Greeks. This search for mathematical beauty has led to the discovery of recurring mathematical structures such as the golden ratio, Fibonacci numbers, and Lucas numbers, which have captivated the interest of artists and scientists alike. (https://geometrymatters.com/the-human-quest-for-discovering-mathematical-beauty-in-the-arts/)

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This quest’s enchantment comes with significant stakes. In truth, art is the ultimate expression of human ingenuity, and comprehending it mathematically would provide us with the keys to decoding human civilization and evolution. However, it wasn’t until later that the scope and size of humanity’s pursuit of mathematical beauty was dramatically broadened by the convergence of three distinct inventions. (https://geometrymatters.com/the-human-quest-for-discovering-mathematical-beauty-in-the-arts/)

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The development of robust statistical approaches to capture hidden patterns in massive amounts of data, as well as the mass digitization of large art archives, have made it feasible to disclose the—otherwise imperceptible to the human eye—mathematics hidden in large artistic corpora. (https://geometrymatters.com/the-human-quest-for-discovering-mathematical-beauty-in-the-arts/)

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Music, storytelling, language phonology, comedy in jokes, and even equations have all been included in the current growth. Lee et al. extend this quest by looking for statistical signatures of compositional proportions in a quasi-canonical dataset of 14,912 landscape paintings spanning the period from Western renaissance to contemporary art (from 1500 CE to 2000 CE). (https://geometrymatters.com/the-human-quest-for-discovering-mathematical-beauty-in-the-arts/)

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They mathematically investigate how artists organize colors on the canvas across styles and time using an information-theoretical framework based on Rigau’s et al work (Fig. 1). They use a computational approach to divide each painting in their collection into the most color uniform vertical and horizontal areas. Their approach works in steps, maximizing mutual information between colors and areas across all conceivable partitions in both horizontal and vertical dimensions at each phase. (https://geometrymatters.com/the-human-quest-for-discovering-mathematical-beauty-in-the-arts/)

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Gaining information in this context means becoming more convinced that the palettes of the partitioned parts are chromatically distant, as information “encodes counterfactual knowledge and describes the level of ambiguity or noise in a system”. By comparing the compositional information received by early partitions in abstract and landscape paintings, Lee et al. demonstrate that the information gained by early partitions in landscape paintings is significantly higher than in abstract paintings, which exhibit no directional preference. (https://geometrymatters.com/the-human-quest-for-discovering-mathematical-beauty-in-the-arts/)

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The dissection analysis exposes underlying metapatterns of community consensus that are completely removed from aesthetic considerations. Nonetheless, the upshot is a unified macrohistory of Western landscape painting. Furthermore, their research provides a quantitative knowledge of the relationships between artistic styles, trends, and artists. (https://geometrymatters.com/the-human-quest-for-discovering-mathematical-beauty-in-the-arts/)

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Is there a set of mathematically defined organizational principles that apply to all genres and artists? Do these values vary by country and culture? What happens to them over time? C.E. Shannon, The Bandwagon (https://geometrymatters.com/the-human-quest-for-discovering-mathematical-beauty-in-the-arts/)

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Mutual information, statistical surprise, and permutation entropy have all been utilized to turn art’s abstract intricacy into a numerical form. “A few exciting words like information, entropy, redundancy, do not solve all our problems,” Claude E. Shannon, the creator of information theory, cautioned us in 1956. For example, the Lee et al. approach performs poorly with paintings that require diagonal partitions (such as Paul Cezanne’s Landscape on the Mediterranean) or when large objects are placed in the center of the canvas (like in The Babel Tower by Pieter Bruegel the Elder). (https://geometrymatters.com/the-human-quest-for-discovering-mathematical-beauty-in-the-arts/)

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Although Lee’s work does not answer all of our challenges, it is a great place to start and has foundational importance. It invites scholars from other disciplines to push the hunt for mathematical beauty toward broader categories and deeper knowledge by spelling out the enigma of decoding mathematical beauty via the lenses of geometric proportions. (https://geometrymatters.com/the-human-quest-for-discovering-mathematical-beauty-in-the-arts/)

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As far as the propositions of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality. - Albert Einstein (https://geometrymatters.com/einstein-geometry-and-experience/)

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One reason why mathematics enjoys special esteem, above all other sciences, is that its propositions are absolutely certain and indisputable, while those of all other sciences are to some extent debatable and in constant danger of being overthrown by newly discovered facts. In spite of this, the investigator in another department of science would not need to envy the mathematician if the propositions of mathematics referred to objects of our mere imagination, and not to objects of reality. (https://geometrymatters.com/einstein-geometry-and-experience/)

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For it cannot occasion surprise that different persons should arrive at the same logical conclusions when they have already agreed upon the fundamental propositions (axioms), as well as the methods by which other propositions are to be deduced therefrom. But there is another reason for the high repute of mathematics, in that it is mathematics, which affords the exact natural sciences a certain measure of certainty, to which without mathematics they could not attain. (https://geometrymatters.com/einstein-geometry-and-experience/)

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At this point, an enigma presents itself, which in all ages has agitated inquiring minds. How can it be that mathematics, being after all a product of human thought which is independent of experience, is so admirably appropriate to the objects of reality? Is human reason, then, without experience, merely by taking thought, able to fathom the properties of real things? (https://geometrymatters.com/einstein-geometry-and-experience/)

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In my opinion, the answer to this question is, briefly, this: as far as the propositions of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality. It seems to me that complete clarity as to this state of things became common property only through that trend in mathematics, which is known by the name of “axiomatics.” The progress achieved by axiomatics consists in its having neatly separated the logical-formal from its objective or intuitive content; according to axiomatics the logical-formal alone forms the subject matter of mathematics, which is not concerned with the intuitive or other content associated with the logical-formal. (https://geometrymatters.com/einstein-geometry-and-experience/)

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Let us for a moment consider from this point of view any axiom of geometry, for instance, the following: through two points in space there always passes one and only one straight line. How is this axiom to be interpreted in the older sense and in the more modern sense? The older interpretation: everyone knows what a straight line is, and what a point is. Whether this knowledge springs from an ability of the human mind or from experience, from some cooperation of the two or from some other source, is not for the mathematician to decide. He leaves the question to the philosopher. Being based upon this knowledge, which precedes all mathematics, the axiom stated above is, like all other axioms, self-evident, that is, it is the expression of a part of this a priori knowledge. (https://geometrymatters.com/einstein-geometry-and-experience/)

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The more modern interpretation: geometry treats of objects, which are denoted by the words straight line, point, etc. No knowledge or intuition of these objects is assumed but only the validity of the axioms, such as the one stated above, which are to be taken in a purely formal sense, i.e., as void of all content of intuition or experience. These axioms are free creations of the human mind. All other propositions of geometry are logical inferences from the axioms (which are to be taken in the nominalistic sense only). The axioms define the objects of which geometry treats. Schlick in his book on epistemology has therefore characterized axioms very aptly as “implicit definitions." (https://geometrymatters.com/einstein-geometry-and-experience/)

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This view of axioms, advocated by modern axiomatics, Purges mathematics of all extraneous elements, and thus dispels the mystic obscurity, which formerly surrounded the basis of mathematics. But such an expurgated exposition of mathematics makes it also evident that mathematics as such cannot predicate anything about objects of our intuition or real objects. In axiomatic geometry the words “point,” “straight line,” etc., stand only for empty conceptual schemata. That which gives them content is not relevant to mathematics. Yet on the other hand it is certain that mathematics generally, and particularly geometry, owes its existence to the need which was felt of learning something about the behavior of real objects. (https://geometrymatters.com/einstein-geometry-and-experience/)

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The very word geometry, which, of course, means earth-measuring, proves this. For earth measuring has to do with the possibilities of the disposition of certain natural objects with respect to one another, namely, with parts of the earth, measuring-lines, measuring-wands, etc. It is clear that the system of concepts of axiomatic geometry alone cannot make any assertions as to the behavior of real objects of this kind, which we will call practically-rigid bodies. To be able to make such assertions, geometry must be stripped of its merely logical-formal character by the coordination of real objects of experience with the empty conceptual schemata of axiomatic geometry. To accomplish this, we need only add the proposition: solid bodies are related, with respect to their possible dispositions, as are bodies in Euclidean geometry of three dimensions. (https://geometrymatters.com/einstein-geometry-and-experience/)

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Then the propositions of Euclid contain affirmations as to the behavior of practically-rigid bodies. Geometry thus completed is evidently a natural science; we may in fact regard it as the most ancient branch of physics. Its affirmations rest essentially on induction from experience, but not on logical inferences only. We will call this completed geometry “practical geometry,” and shall distinguish it in what follows from “purely axiomatic geometry.” The question of whether the practical geometry of the universe is Euclidean or not has a clear meaning, and its answer can only be furnished by experience. All length measurements in physics constitute practical geometry in this sense, so, too, do geodetic and astronomical length measurements, if one utilizes the empirical law that light is propagated in a straight line, and indeed in a straight line in the sense of practical geometry. (https://geometrymatters.com/einstein-geometry-and-experience/)

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I attach special importance to the view of geometry, which I have just set forth because without it I should have been unable to formulate the theory of relativity. - Albert Einstein (https://geometrymatters.com/einstein-geometry-and-experience/)

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Without it the following reflection would have been impossible: in a system of reference rotating relatively to an inertial system, the laws of disposition of rigid bodies do not correspond to the rules of Euclidean geometry on account of the Lorentz contraction; thus if we admit non-inertial systems on an equal footing, we must abandon Euclidean geometry. (https://geometrymatters.com/einstein-geometry-and-experience/)

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Without the above interpretation, the decisive step in the transition to generally covariant equations would certainly not have been taken. If we reject the relation between the body of axiomatic Euclidean geometry and the practically-rigid body of reality, we readily arrive at the following view, which was entertained by that acute and profound thinker, H. Poincaré Euclidean geometry is distinguished above all other conceivable axiomatic geometries by its simplicity. Now since axiomatic geometry by itself contains no assertions as to the reality which can be experienced, but can do so only in combination with physical laws, it should be possible and reasonable—whatever may be the nature of reality—to retain Euclidean geometry. (https://geometrymatters.com/einstein-geometry-and-experience/)

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For if contradictions between theory and experience manifest themselves, we should rather decide to change physical laws than to change axiomatic Euclidean geometry. If we reject the relation between the practically-rigid body and geometry, we shall indeed not easily free ourselves from the convention that Euclidean geometry is to be retained as the simplest. (https://geometrymatters.com/einstein-geometry-and-experience/)

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Why is the equivalence of the practically-rigid body and the body of geometry—which suggests itself so readily—rejected by Poincaré and other investigators? Simply because under closer inspection the real solid bodies in nature are not rigid, because their geometrical behavior, that is, their possibilities of relative disposition, depend upon temperature, external forces, etc. Thus the original, immediate relation between geometry and physical reality appears destroyed, and we feel impelled toward the following more general view, which characterizes Poincaré’s standpoint. Geometry (G) predicates nothing about the behavior of real things, but only geometry together with the totality (P) of physical laws can do so. (https://geometrymatters.com/einstein-geometry-and-experience/)

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Using symbols, we may say that only the sum of (G) + (P) is subject to experimental verification. Thus (G) may be chosen arbitrarily, and also parts of (P); all these laws are conventions. All that is necessary to avoid contradictions is to choose the remainder of (P) so that (G) and the whole of (P) are together in accord with experience. Envisaged in this way, axiomatic geometry and the part of natural law, which has been given a conventional status, appear as epistemologically equivalent. (https://geometrymatters.com/einstein-geometry-and-experience/)

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Sub specie aeterni Poincaré, in my opinion, is right. The idea of the measuring-rod and the idea of the clock coordinated with it in the theory of relativity do not find their exact correspondence in the real world. It is also clear that the solid body and the clock do not in the conceptual edifice of physics play the part of irreducible elements, but that of composite structures, which must not play any independent part in theoretical physics. But it is my conviction that in the present stage of development of theoretical physics these concepts must still be employed as independent concepts; for we are still far from possessing such certain knowledge of the theoretical principles of atomic structure as to be able to construct solid bodies and clocks theoretically from elementary concepts. (https://geometrymatters.com/einstein-geometry-and-experience/)

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Further, as to the objection that there are no really rigid bodies in nature, and that therefore the properties predicated of rigid bodies do not apply to physical reality—this objection is by no means so radical as might appear from a hasty examination. For it is not a difficult task to determine the physical state of a measuring-body so accurately that its behavior relative to other measuring-bodies shall be sufficiently free from ambiguity to allow it to be substituted for the “rigid” body. It is to measuring-bodies of this kind that statements about rigid bodies must be referred. (https://geometrymatters.com/einstein-geometry-and-experience/)

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All practical geometry is based upon a principle that is accessible to experience, and which we will now try to realize. Suppose two marks have been put upon a practically-rigid body. A pair of two such marks we shall call a tract. We imagine two practically-rigid bodies, each with a tract marked out on it. These two tracts are said to be “equal to one another” if the marks of the one tract can be brought to coincide permanently with the marks of the other. We now assume that: If two tracts are found to be equal once and anywhere, they are equal always and everywhere. Not only the practical geometry of Euclid, but also its nearest generalization, the practical geometry of Riemann, and therewith the general theory of relativity, rest upon this assumption. (https://geometrymatters.com/einstein-geometry-and-experience/)

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Of the experimental reasons that warrant this assumption, I will mention only one. The phenomenon of the propagation of light in empty space assigns a tract, namely, the appropriate path of light, to each interval of local time, and conversely. Thence it follows that the above assumption for tracts must also hold good for intervals of clock-time in the theory of relativity. (https://geometrymatters.com/einstein-geometry-and-experience/)

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Consequently, it may be formulated as follows: if two ideal clocks are going at the same rate at any time and at any place (being then in immediate proximity to each other), they will always go at the same rate, no matter where and when they are again compared with each other at one place. If this law were not valid for natural clocks, the proper frequencies for the separate atoms of the same chemical element would not be in such exact agreement as experience demonstrates. The existence of sharp spectral lines is a convincing experimental proof of the above-mentioned principle of practical geometry. This, in the last analysis, is the reason that enables us to speak meaningfully of a Riemannian metric of the four-dimensional space-time continuum. (https://geometrymatters.com/einstein-geometry-and-experience/)

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According to the view advocated here, the question whether this continuum has a Euclidean, Riemannian, or any other structure is a question of physics proper, which must be answered by experience, and not a question of a convention to be chosen on grounds of mere expediency. Riemann’s geometry will hold if the laws of disposition of practically-rigid bodies approach those of Euclidean geometry the more closely the smaller the dimensions of the region of spacetime under consideration. (https://geometrymatters.com/einstein-geometry-and-experience/)

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It is true that this proposed physical interpretation of geometry breaks down when applied immediately to spaces of submolecular order of magnitude. But nevertheless, even in questions as to the constitution of elementary particles, it retains part of its significance. For even when it is a question of describing the electrical elementary particles constituting matter, the attempt may still be made to ascribe physical meaning to those field concepts which have been physically defined for the purpose of describing the geometrical behavior of bodies which are large as compared with the molecule. (https://geometrymatters.com/einstein-geometry-and-experience/)

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Success alone can decide as to the justification of such an attempt, which postulates physical reality for the fundamental principles of Riemann’s geometry outside of the domain of their physical definitions. It might possibly turn out that this extrapolation has no better warrant than the extrapolation of the concept of temperature to parts of a body of molecular order of magnitude. (https://geometrymatters.com/einstein-geometry-and-experience/)

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It appears less problematical to extend the concepts of practical geometry to spaces of cosmic order of magnitude. It might, of course, be objected that a construction composed of solid rods departs the more from ideal rigidity the greater its spatial extent. But it will hardly be possible, I think, to assign fundamental significance to this objection. Therefore the question whether the universe is spatially finite or not seems to me an entirely meaningful question in the sense of practical geometry. I do not even consider it impossible that this question will be answered before long by astronomy. Let us call to mind what the general theory of relativity teaches in this respect. It offers two possibilities: (https://geometrymatters.com/einstein-geometry-and-experience/)

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1. The universe is spatially infinite. This is possible only if in the universe the average spatial density of matter, concentrated in the stars, vanishes, i.e., if the ratio of the total mass of the stars to the volume of the space through which they are scattered indefinitely approaches zero as greater and greater volumes are considered. (https://geometrymatters.com/einstein-geometry-and-experience/)

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2. The universe is spatially finite. This must be so, if there exists an average density of the ponderable matter in the universe that is different from zero. The smaller that average density, the greater is the volume of the universe. (https://geometrymatters.com/einstein-geometry-and-experience/)

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I must not fail to mention that a theoretical argument can be adduced in favor of the hypothesis of a finite universe. The general theory of relativity teaches that the inertia of a given body is greater as there are more ponderable masses in proximity to it; thus it seems very natural to reduce the total inertia of a body to interaction between it and the other bodies in the universe, as indeed, ever since Newton’s time, gravity has been completely reduced to interaction between bodies. From the equations of the general theory of relativity, it can be deduced that this total reduction of inertia to the interaction between masses—as demanded by E. Mach, for example—is possible only if the universe is spatially finite. (https://geometrymatters.com/einstein-geometry-and-experience/)

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Many physicists and astronomers are not impressed by this argument. In the last analysis, experience alone can decide which of the two possibilities is realized in nature. How can experience furnish an answer? At first, it might seem possible to determine the average density of matter by observation of that part of the universe that is accessible to our observation. This hope is illusory. (https://geometrymatters.com/einstein-geometry-and-experience/)

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The distribution of the visible stars is extremely irregular, so that we on no account may venture to set the average density of star-matter in the universe equal to, let us say, the average density in the Galaxy. In any case, however great the space examined may be, we could not feel convinced that there were any more stars beyond that space. So it seems impossible to estimate the average density. (https://geometrymatters.com/einstein-geometry-and-experience/)

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But there is another road, which seems to me more practicable, although it also presents great difficulties. For if we inquire into the deviations of the consequences of the general theory of relativity which are accessible to experience, from the consequences of the Newtonian theory, we, first of all, find a deviation that manifests itself in close proximity to gravitating mass, and has been confirmed in the case of the planet Mercury (https://geometrymatters.com/einstein-geometry-and-experience/)

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But if the universe is spatially finite, there is a second deviation from the Newtonian theory, which, in the language of the Newtonian theory, may be expressed thus: the gravitational field is such as if it were produced, not only by the ponderable masses but in addition by a mass-density of negative sign, distributed uniformly throughout space. Since this fictitious mass-density would have to be extremely small, it would be noticeable only in very extensive gravitating systems. (https://geometrymatters.com/einstein-geometry-and-experience/)

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Assuming that we know, let us say, the statistical distribution and the masses of the stars in the Galaxy, then by Newton’s law we can calculate the gravitational field and the average velocities that the stars must-have, so that the Galaxy should not collapse under the mutual attraction of its stars, but should maintain its actual extent. Now if the actual velocities of the stars—which can be measured—were smaller than the calculated velocities, we should have proof that the actual attractions at great distances are smaller than by Newton’s law. From such a deviation it could be proved indirectly that the universe is finite. It would even be possible to estimate its spatial dimensions. (https://geometrymatters.com/einstein-geometry-and-experience/)

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Can we visualize a three-dimensional universe which is finite, yet unbounded? - Albert Einstein (https://geometrymatters.com/einstein-geometry-and-experience/)

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The usual answer to this question is “No,” but that is not the right answer. The purpose of the following remarks is to show that the answer should be “Yes.” I want to show that without any extraordinary difficulty we can illustrate the theory of a finite universe by means of a mental picture to which, with some practice, we shall soon grow accustomed. (https://geometrymatters.com/einstein-geometry-and-experience/)

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First of all, an observation of epistemological nature. A geometrical-physical theory as such is incapable of being directly pictured, being merely a system of concepts. But these concepts serve the purpose of bringing a multiplicity of real or imaginary sensory experiences into connection in the mind. (https://geometrymatters.com/einstein-geometry-and-experience/)

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To “visualize” a theory, therefore, means to bring to mind that abundance of sensible experiences for which the theory supplies the schematic arrangement. In the present case, we have to ask ourselves how we can represent the behavior of solid bodies with respect to their mutual disposition (contact) that corresponds to the theory of a finite universe. There is really nothing new in what I have to say about this, but innumerable questions addressed to me prove that the curiosity of those who are interested in these matters has not yet been completely satisfied. So, will the initiated please pardon me, in that part of what I shall say has long been known? (https://geometrymatters.com/einstein-geometry-and-experience/)

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What do we wish to express when we say that our space is infinite? Nothing more than that we might lay any number of bodies of equal sizes side by side without ever filling space. Suppose that we are provided with a great many cubic boxes all of the same size. In accordance with Euclidean geometry we can place them above, beside, and behind one another so as to fill an arbitrarily large part of space; but this construction would never be finished; we could go on adding more and more cubes without ever finding that there was no more room. That is what we wish to express when we say that space is infinite. It would be better to say that space is infinite in relation to practically-rigid bodies, assuming that the laws of disposition for these bodies are given by Euclidean geometry. (https://geometrymatters.com/einstein-geometry-and-experience/)

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Another example of an infinite continuum is the plane. On a plane surface, we may lay squares of cardboard so that each side of any square has the side of another square adjacent to it. The construction is never finished; we can always go on laying squares—if their laws of disposition correspond to those of plane figures of Euclidean geometry. The plane is therefore infinite in relation to the cardboard squares. Accordingly, we say that the plane is an infinite continuum of two dimensions, and space an infinite continuum of three dimensions. What is here meant by the number of dimensions, I think I may assume to be known. (https://geometrymatters.com/einstein-geometry-and-experience/)

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Now we take an example of a two-dimensional continuum that is finite, but unbounded. We imagine the surface of a large globe and a quantity of small paper discs, all of the same size. We place one of the discs anywhere on the surface of the globe. If we move the disc about, anywhere we like, on the surface of the globe, we do not come upon a boundary anywhere on the journey. (https://geometrymatters.com/einstein-geometry-and-experience/)

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Therefore we say that the spherical surface of the globe is an unbounded continuum. Moreover, the spherical surface is a finite continuum. For if we stick the paper discs on the globe so that no disc overlaps another, the surface of the globe will finally become so full that there is no room for another disc. This means exactly that the spherical surface of the globe is finite in relation to the paper discs. (https://geometrymatters.com/einstein-geometry-and-experience/)

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Further, the spherical surface is a non-Euclidean continuum of two dimensions, that is to say, the laws of disposition for the rigid figures lying in it do not agree with those of the Euclidean plane. This can be shown in the following way. Take a disc and surround it in a circle with six more discs, each of which is to be surrounded in turn by six discs, and so on. If this construction is made on a plane surface, we obtain an uninterrupted arrangement in which there are six discs touching every disc except those that lie on the outside. (https://geometrymatters.com/einstein-geometry-and-experience/)

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On the spherical surface, the construction also seems to promise success at the outset, and the smaller the radius of the disc in proportion to that of the sphere, the more promising it seems. But as the construction progresses it becomes more and more patent that the arrangement of the discs in the manner indicated, without interruption, is not possible, as it should be possible by the Euclidean geometry of the plane. (https://geometrymatters.com/einstein-geometry-and-experience/)

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In this way, creatures that cannot leave the spherical surface, and cannot even peep out from the spherical surface into three-dimensional space, might discover, merely by experimenting with discs, that their two-dimensional “space” is not Euclidean, but spherical space. (https://geometrymatters.com/einstein-geometry-and-experience/)

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From the latest results of the theory of relativity it is probable that our three-dimensional space is also approximately spherical, that is, that the laws of disposition of rigid bodies in it are not given by Euclidean geometry, but approximately by spherical geometry, if only we consider parts of space which are sufficiently extended. Now, this is the place where the reader’s imagination boggles. “Nobody can imagine this thing,” he cries indignantly. “It can be said, but cannot be thought. I can imagine a spherical surface well enough, but nothing analogous to it in three dimensions.” (https://geometrymatters.com/einstein-geometry-and-experience/)

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We must try to surmount this barrier in the mind, and the patient reader will see that it is by no means a particularly difficult task. For this purpose, we will first give our attention once more to the geometry of two-dimensional spherical surfaces. In the adjoining figure let K be the spherical surface, touched at S by a plane, E, which, for the facility of presentation, is shown in the drawing as a bounded surface. Let L be a disc on the spherical surface. Now let us imagine that at point N of the spherical surface, diametrically opposite to S, there is a luminous point, throwing a shadow L’ of the disc L upon the plane E. Every point on the sphere has its shadow on the plane. (https://geometrymatters.com/einstein-geometry-and-experience/)

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If the disc on the sphere K is moved, its shadow L’ on the plane E also moves. When the disc L is at S, it almost exactly coincides with its shadow. If it moves on the spherical surface away from S upwards, the disc shadow L’ on the plane also moves away from S on the plane outwards, growing bigger and bigger. As the disc L approaches the luminous point N, the shadow moves off to infinity, and becomes infinitely great. (https://geometrymatters.com/einstein-geometry-and-experience/)

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Now we put the question: what are the laws of disposition of the disc-shadows L’ on the plane E? Evidently they are exactly the same as the laws of disposition of the discs L on the spherical surface. For each original figure on K there is a corresponding shadow figure on E. If two discs on K are touching, their shadows on E also touch. The shadow-geometry on the plane agrees with the disc-geometry on the sphere. If we call the disc-shadows rigid figures, then spherical geometry holds good on the plane E with respect to these rigid figures. In particular, the plane is finite with respect to the disc-shadows, since only a finite number of the shadows can find room on the plane. (https://geometrymatters.com/einstein-geometry-and-experience/)

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At this point, somebody will say, “That is nonsense. The disc-shadows are not rigid figures. We have only to move a two-foot rule about on the plane E to convince ourselves that the shadows constantly increase in size as they move away from S on the plane toward infinity.” But what if the two-foot rule were to behave on the plane E in the same way as the disc-shadows L’? (https://geometrymatters.com/einstein-geometry-and-experience/)

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It would then be impossible to show that the shadows increase in size as they move away from S; such an assertion would then no longer have any meaning whatever. In fact, the only objective assertion that can be made about the disc-shadows is just this, that they are related in exactly the same way as the rigid discs on the spherical surface in the sense of Euclidean geometry. (https://geometrymatters.com/einstein-geometry-and-experience/)

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We must carefully bear in mind that our statement as to the growth of the disc-shadows, as they move away from S toward infinity, has in itself no objective meaning, as long as we are unable to compare the disc-shadows with Euclidean rigid bodies that can be moved about on the plane E. In respect of the laws of disposition of the shadows L’ , the point S has no special privileges on the plane any more than on the spherical surface. (https://geometrymatters.com/einstein-geometry-and-experience/)

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The representation given above of spherical geometry on the plane is important for us because it readily allows itself to be transferred to the three-dimensional case. Let us imagine a point S of our space, and a great number of small spheres, L’ , which can all be brought to coincide with one another. But these spheres are not to be rigid in the sense of Euclidean geometry; their radius is to increase (in the sense of Euclidean geometry) when they are moved a rd infinity; it is to increase according to the same law as the radii of the disc-shadows L’ on the plane. (https://geometrymatters.com/einstein-geometry-and-experience/)

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After having gained a vivid mental image of the geometrical behavior of our L’ spheres, let us assume that in our space there are no rigid bodies at all in the sense of Euclidean geometry, but only bodies having the behavior of our L” spheres. Then we shall have a clear picture of three-dimensional spherical space, or, rather of three-dimensional spherical geometry. Here our spheres must be called “rigid” spheres. Their increase in size as they depart from S is not to be detected by measuring with measuring-rods, any more than in the case of the disc-shadows on E, because the standards of measurement behave in the same way as the spheres. (https://geometrymatters.com/einstein-geometry-and-experience/)

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Space is homogeneous, that is to say, the same spherical configurations are possible in the neighborhood of every point. Our space is finite, because, in consequence of the “growth” of the spheres, only a finite number of them can find room in space. (https://geometrymatters.com/einstein-geometry-and-experience/)

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In this way, by using as a crutch the practice of thinking and visualization that Euclidean geometry gives us, we have acquired a mental picture of spherical geometry. We may without difficulty impart more depth and vigor to these ideas by carrying out special imaginary constructions. Nor would it be difficult to represent the case of what is called elliptical geometry in an analogous manner. (https://geometrymatters.com/einstein-geometry-and-experience/)

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My only aim today has been to show that the human faculty of visualization is by no means bound to capitulate to non-Euclidean geometry. - Albert Einstein (https://geometrymatters.com/einstein-geometry-and-experience/)

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God made the integers, all else is man’s work,” stated German mathematician Leopold Kronecker in the 19th century. But is this true? Some fundamentals, such as positive integers and the 3-4-5 right triangle, are universally accepted across cultures. Almost every other aspect of mathematics is influenced by the society in which you live. (https://geometrymatters.com/did-humans-invent-mathematics-or-is-it-a-fundamental-part-of-existence/)

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Mathematics, in this way of thinking, is like a language: it may describe real-world objects, but it doesn’t ‘exist’ outside the minds of those who use it. In ancient Greece, the Pythagorean school of thought held the belief that reality is fundamentally mathematical and philosophers and physicists are beginning to take this theory seriously after more than 2,000 years.(https://geometrymatters.com/did-humans-invent-mathematics-or-is-it-a-fundamental-part-of-existence/)

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A central challenge for mathematical Platonists, however, is to provide an account of how mathematical explanations work. I propose a property-based account: physical systems possess mathematical properties, which either guarantee the presence of other mathematical properties and, by extension, the physical states that possess them; or rule out other mathematical properties, and their associated physical states. - Sam Baron(https://geometrymatters.com/did-humans-invent-mathematics-or-is-it-a-fundamental-part-of-existence/)

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In his paper, Sam Baron argues that mathematics is an essential component of nature that gives structure to the physical world. Take for example the ‘honeycomb conjecture’ in mathematics, which explains why beehives are built with hexagonal tiles: hexagons are the shape to employ if you want to completely cover a surface with uniformly shaped and sized tiles while limiting the perimeter length to a minimum.(https://geometrymatters.com/did-humans-invent-mathematics-or-is-it-a-fundamental-part-of-existence/)

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Bees have evolved to utilize this design because it provides the largest cells to store honey for the smallest input of energy to produce wax, according to Charles Darwin. The honeycomb conjecture was first presented in antiquity, but mathematician Thomas Hales verified it in 1999.(https://geometrymatters.com/did-humans-invent-mathematics-or-is-it-a-fundamental-part-of-existence/)

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It is simple to locate additional examples once we begin looking. Mathematics may be found in anything from soap operas to engine gear designs to the position and magnitude of gaps in Saturn’s rings.(https://geometrymatters.com/did-humans-invent-mathematics-or-is-it-a-fundamental-part-of-existence/)

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It’s improbable that mathematics is something we invented if it explains so many things we observe around us. The alternative is that mathematical facts are found by insects, soap bubbles, combustion engines, and planets, not simply by humans.(https://geometrymatters.com/did-humans-invent-mathematics-or-is-it-a-fundamental-part-of-existence/)

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But if we are discovering something, what is it? The assumption that physical objects possess mathematical properties involves a commitment to a partial form of Pythagoreanism: the belief that the universe is ‘made’ of mathematics. Plato believed that mathematics describes real-world objects. Numbers, geometric shapes, and more sophisticated mathematical objects such as groups, categories, functions, fields, and rings – mathematical objects that exist outside of space and time.(https://geometrymatters.com/did-humans-invent-mathematics-or-is-it-a-fundamental-part-of-existence/)

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The Pythagoreans agreed with Plato that mathematics describes an object-oriented reality but disagreed with them existing beyond space-time. Rather, they believed that physical reality is made up of mathematical things, just as matter is made up of atoms. To this idea, contemporary physicist Max Tegmark argues reality is one big mathematical object, a simulation program that we perceive and discover.(https://geometrymatters.com/did-humans-invent-mathematics-or-is-it-a-fundamental-part-of-existence/)

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Sam Baron proposes two parts to this dilemma: mathematics and matter. Matter provides mathematics its shape, while mathematics gives matter its substance. The physical universe has a structural foundation provided by mathematical objects.(https://geometrymatters.com/did-humans-invent-mathematics-or-is-it-a-fundamental-part-of-existence/)

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Platonists should accept that physical and mathematical objects share intrinsic properties. I call this view ‘Partial Pythagoreanism’ on the grounds that physical and mathematical objects are not entirely distinct. - Sam Baron(https://geometrymatters.com/did-humans-invent-mathematics-or-is-it-a-fundamental-part-of-existence/)

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Baron goes beyond the beehive example to the two North American periodical cicadas subspecies that spend most of their life underground. The cicadas emerge in large swarms for roughly two weeks every 13 or 17 years (depending on the subspecies). Why are the ages 13 and 17 different? Why not 12 and 14 years old? Or maybe 16 and 18? The fact that 13 and 17 are prime numbers is one explanation, suitable to avoid predators with life cycles of 2, 3, 4, 5, 6, 7, 8, and 9 years.(https://geometrymatters.com/did-humans-invent-mathematics-or-is-it-a-fundamental-part-of-existence/)

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Because 2, 3, and 4 divide evenly into 12, when a cicada with a 12-year life cycle emerges from the ground, the 2-year, 3-year, and 4-year predators emerge as well. Because none of the numbers 2, 3, 4, 5, 6, 7, 8, or 9 divide evenly into 13, no predators will be out of the earth when a cicada with a 13-year life cycle emerges. The same may be said with 17. These cicadas appear to have evolved to make use of basic mathematical facts.(https://geometrymatters.com/did-humans-invent-mathematics-or-is-it-a-fundamental-part-of-existence/)

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In terms of qualitative characteristics, cicada’s example demonstrates that mathematics and matter coincide. The physical universe is partly mathematical in this, fairly weak, sense. Physical and mathematical objects have intrinsic features in common, such as having a given structure, a specific geometry or being organized in a specific way (as in a linear or partial ordering), in order to create the physical system we experience.(https://geometrymatters.com/did-humans-invent-mathematics-or-is-it-a-fundamental-part-of-existence/)

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A physical system has a particular structure when the system instantiates it. The main challenge is to say what these structures are. Initially, this looks easy: a structure is just a collection of physical objects that stand in certain relations to one another. But this won’t do. Structures are multiply realizable: many different collections of physical objects can share the same structure. A structure therefore cannot be identified with any particular collection of physical objects. Baron argues that structures are physical phenomena forced by mathematical forces.(https://geometrymatters.com/did-humans-invent-mathematics-or-is-it-a-fundamental-part-of-existence/)

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Structures are abstract entities; they can exist without being instantiated in space-time and they are independent of our knowledge and beliefs about them. - Resnik, 1985(https://geometrymatters.com/did-humans-invent-mathematics-or-is-it-a-fundamental-part-of-existence/)

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Why do geometric shapes such as lines, circles, zig-zags, or spirals appear in all human cultures, but are never produced by other animals? Mathias Sablé-Meyer et al. formalize and test the hypothesis that all humans possess a compositional language of thought that can produce line drawings as recursive combinations of a minimal set of geometric primitives. By presenting a programming language that combines discrete numbers and continuous integration in higher-level structures based on repetition, concatenation, and embedding, they show that the simplest programs in this language generate the fundamental geometric shapes observed in human cultures. (https://geometrymatters.com/a-language-of-thought-for-the-mental-representation-of-geometric-shapes/)

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We could never know the geometric triangle through the one we see traced on paper, if our mind had not had the idea of it elsewhere - René Descartes(https://geometrymatters.com/a-language-of-thought-for-the-mental-representation-of-geometric-shapes/)

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Evidence for abstract concepts of geometry, including rectilinearity, parallelism, perpendicularity, and symmetries, is widespread throughout prehistory. Zig-zag carved patterns, equal angels, parallelism, and bifaces – archeological evidence that suggests an aesthetic drive for symmetry, that was already present in ancient humans.(https://geometrymatters.com/a-language-of-thought-for-the-mental-representation-of-geometric-shapes/)

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According to a recent comparison study, all humans, regardless of age, culture, or education, exhibited a striking effect of shape regularity: squares and rectangles were processed better than other, more irregular quadrilaterals, and there was a continuous ordering of complexity, from squares and rectangles to parallelograms, trapezoids, and fully irregular shapes. Surprisingly, baboons did not show this geometric regularity effect.(https://geometrymatters.com/a-language-of-thought-for-the-mental-representation-of-geometric-shapes/)

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Our core hypothesis is that perceiving a shape, in humans, consists in finding the shortest program that suffices to reproduce it. Our proposal thus connects shape perception to the problem of program induction, i.e. the identification of a program that produces a certain output. - Mathias Sablé-Meyer (https://geometrymatters.com/a-language-of-thought-for-the-mental-representation-of-geometric-shapes/)

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A crucial aspect of the proposal is that humans encode a shape mentally by inferring a simple program that could generate it. Thus, the perception of a simple shape is an act of “program induction”. While program induction remains a difficult challenge for computer science, the study deployed a state-of-the-art program induction technique, the DreamCoder algorithm. This algorithm is given programming problems via examples of the desired behavior and searches for the simplest program that performs the task. (https://geometrymatters.com/a-language-of-thought-for-the-mental-representation-of-geometric-shapes/)

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First, we show that our language predicts which shapes are judged simple. Second, we show that any such language has to satisfy a set of additive relationships for repeated, concatenated or embedded shapes, and that those universal laws can be experimentally validated. - Mathias Sablé-Meyer (https://geometrymatters.com/a-language-of-thought-for-the-mental-representation-of-geometric-shapes/)

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The DreamCoder approach opens up a number of perspectives on how human cognition could efficiently address the problem of program induction. First, it naturally accounts for cultural drifts. Second, it may explain how simple geometric shapes may be efficiently recognized and used by young children in the absence of much or any training (poverty of the stimulus argument). (https://geometrymatters.com/a-language-of-thought-for-the-mental-representation-of-geometric-shapes/)

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The bottom-up neural network in a future version of DreamCoder might be repurposed to immediately obtain the most likely program for a particular shape, perhaps offering a mechanism for people to swiftly detect simple shapes. (https://geometrymatters.com/a-language-of-thought-for-the-mental-representation-of-geometric-shapes/)

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Simple arguments show that the suggested language can generate the vast majority of the shapes that people consider simple and that has been documented in both human cultures and the history of geometry. (https://geometrymatters.com/a-language-of-thought-for-the-mental-representation-of-geometric-shapes/)

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The prospect of building reusable abstractions or program templates is an attractive component of this approach that has yet to be extensively explored. While the square is not primitive in the original language, a square-drawing program schema may become abstracted over time, allowing the participant to understand ideas like “a square of circles” or “a square twice as huge as the last one.” (https://geometrymatters.com/a-language-of-thought-for-the-mental-representation-of-geometric-shapes/)

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The building stands out like a vast, dazzling sculpture on the edge of land and sea, reflecting both sky and harbor space as well as the active lifestyle of the city. Henning Larsen Architects, the Danish-Icelandic artist Olafur Eliasson, and the German engineering firms Rambll and ArtEngineering GmbH collaborated closely to create the outstanding façades. (https://geometrymatters.com/harpa-concert-hall-from-nature-to-architecture/)

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Harpa means ‘harp’ in Icelandic. It is also the Icelandic name for the first month of spring, and thus a sign of brighter times. Today, the most visited attraction of the volcanic island carries the name – Harpa. Between a rock-solid core and a crystalline shell, everyday life unfolds in the expansive foyer – where a varied mix of playing children, yoga classes, concert guests, and international conference delegates have embraced the space altogether. - Henning Larsen (https://geometrymatters.com/harpa-concert-hall-from-nature-to-architecture/)

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he project was the winner of the prestigious European Union Prize for Contemporary Architecture: Mies van der Rohe, in 2013. Its geometric façade is based on a modular, space-filling construction termed the quasi brick, which is reminiscent of the crystalline basalt columns typical in Iceland. The pseudo brick modules include color-effect filter glass panes; the structure shimmers in response to the weather, time of year or day, and the position and motions of spectators. (https://geometrymatters.com/harpa-concert-hall-from-nature-to-architecture/)

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The quasi brick is a twelve-sided polyhedron with rhomboidal and hexagonal sides that was invented by geometer and mathematician Einar Thorsteinn. Eliasson and Thorsteinn began studying the potential for using the quasi brick in architecture as a result of their partnership. When the space-filling modules are stacked, no gaps between them are left, allowing them to be used to construct walls and structural parts. Harpa’s façade have a chaotic, unpredictable aspect due to the blend of regularity and irregularity in the modules. As a result, the building’s facades are both aesthetically and functionally important. (https://geometrymatters.com/harpa-concert-hall-from-nature-to-architecture/)

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Harpa’s three-dimensional pseudo bricks are only used on the major south façade; the irregular geometric patterns on the west, north, and east facades were created using a two-dimensional sectional cut through the three-dimensional bricks. (https://geometrymatters.com/harpa-concert-hall-from-nature-to-architecture/)

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The façade concept’s fundamental idea was to consider the structure as a static unit, allowing it to adapt dynamically to changing colors in the environment. During the day, the geometric forms form a crystalline structure that captures and reflects light, establishing a dialogue between the building, city, and surrounding landscape. LED lights placed into each quasi-brick illuminate the facades at night. The hue and light intensity can be changed to use the entire color spectrum and produce a variety of patterns, characters, and symbols. (https://geometrymatters.com/harpa-concert-hall-from-nature-to-architecture/)

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Harpa has captured the myth of a nation – Iceland – that has consciously acted in favour of a hybrid-cultural building during the middle of the ongoing Great Recession. The iconic and transparent porous ‘quasi brick’ appears as an ever-changing play of coloured light, promoting a dialogue between the city of Reykjavik and the building’s interior life. By giving an identity to a society long known for its sagas, through an interdisciplinary collaboration between Henning Larsen Architects and artist Olafur Eliasson, this project is an important message to the world and to the Icelandic people, fulfilling their long expected dream. - Wiel Arets, Chair of the Mies van der Rohe Jury, 2013 (https://geometrymatters.com/harpa-concert-hall-from-nature-to-architecture/)

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Hyperuniformity is a geometric concept to probabilistically characterise the structure of ordered and disordered materials. For example, all perfect crystals, perfect quasicrystals, and special disordered systems are hyperuniform. (https://geometrymatters.com/hidden-order-in-disorder/)

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The study of how large structures are partitioning space into cells with specific extreme geometrical features is a crucial topic in many disciplines of science and technology. Researchers from Karlsruhe Institute of Technology (KIT) and colleagues from other countries have discovered that in amorphous, or disordered, systems, optimizing the moment of inertia of individual cells gradually results in the same structure, despite the fact that it remains amorphous. (https://geometrymatters.com/hidden-order-in-disorder/)

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The team analyzes the Quantizer problem, defined as the optimization of the moment of inertia of Voronoi cells, i.e., similarly-sized ‘sphere-like’ polyhedra that pack as closely as possible. Scientists have been studying the optimal tessellation of three-dimensional space for a long time, being relevant to a variety of practical applications, including telecommunications, image processing, and complicated granules, among others. (https://geometrymatters.com/hidden-order-in-disorder/)

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The goal is to partition space into cells, and all points in a cell to be located as closely as possible to the cell center, intuitively speaking. - Dr. Michael Andreas Klatt (https://geometrymatters.com/hidden-order-in-disorder/)

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The team deployed the Lloyd algorithm, a method to partition space into uniform regions. Every area has a unique center and contains the points in space that are closest to it. Voronoi cells are the name given to such areas. The Voronoi diagram is made up of all points that have more than one nearest center, establishing the region boundaries. They discovered that all entirely amorphous, i.e. disordered, states not only remain completely amorphous, but that the initially different processes converge to a statistically indistinguishable ensemble by investigating stepwise local optimization of distinct point patterns. Stepwise local optimization also compensates for extreme global density changes quickly. (https://geometrymatters.com/hidden-order-in-disorder/)

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The resulting structure is nearly hyperuniform. It does not exhibit any obvious, but a hidden order on large scales. (https://geometrymatters.com/hidden-order-in-disorder/)

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The findings show an unexpected universality for the complex interactions of the Quantizer energy in a detailed structure analysis of its local minima. The existence of universal effectively hyperuniform and fully amorphous states are the converged solutions of Lloyd’s algorithm in 3D. (https://geometrymatters.com/hidden-order-in-disorder/)

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Even when the initial configuration is hyperfluctuating, where the structure factor diverges for small wavenumbers, the system quickly becomes under Lloyd iterations effectively hyperuniform. This demonstrates the strong suppression of density fluctuations on large but finite length scales that is consistent with effective hyperuniformity. (https://geometrymatters.com/hidden-order-in-disorder/)

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This hidden order in the studied amorphous systems is proved to be universal, i.e. stable and independent of properties of the initial state, providing the search for novel amorphous hyperuniform phases and cellular materials with unique physical properties. (https://geometrymatters.com/hidden-order-in-disorder/)

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An interdisciplinary research team at the Technical University of Munich (TUM) has developed an efficient strategy against most viral infections: they engulf and destroy viruses using DNA origami nano-capsules. In cell cultures, the approach has already been tried against hepatitis and adeno-associated viruses. It may also be effective against corona viruses. (https://geometrymatters.com/icosahedral-nano-shell-designed-to-trap-virus-particles/)

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Even before the new coronavirus variety halted the world, Hendrik Dietz, Professor of Biomolecular Nanotechnology at the Physics Department of the Technical University of Munich, and his colleagues were working on the creation of virus-sized particles that self-assemble. (https://geometrymatters.com/icosahedral-nano-shell-designed-to-trap-virus-particles/)

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Donald Caspar, a biologist, and Aaron Klug, a biophysicist, established the mathematical rules that govern the formation of viral protein envelopes in 1962. Based on these geometric specifications, the team led by Hendrik Dietz at the Technical University of Munich, with assistance from Seth Fraden and Michael Hagan from Brandeis University in the United States, developed a concept that enabled the creation of artificial hollow bodies the size of viruses. (https://geometrymatters.com/icosahedral-nano-shell-designed-to-trap-virus-particles/)

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In the summer of 2019, the researchers wondered if such hollow bodies may potentially be utilized as a type of “viral trap.” They should be able to bind viruses tightly and therefore remove them out of circulation if they are lined on the inside with virus-binding molecules. However, the hollow bodies must also have enough big holes for viruses to enter the shells. (https://geometrymatters.com/icosahedral-nano-shell-designed-to-trap-virus-particles/)

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The researchers opted to construct the hollow bodies for the viral trap using three-dimensional, triangular plates based on the basic geometric shape of the icosahedron. The edges of the DNA plates must be slightly beveled in order for them to combine into bigger geometrical shapes. The proper selection and placement of binding points on the edges ensures that the panels self-assemble to the required objects. (https://geometrymatters.com/icosahedral-nano-shell-designed-to-trap-virus-particles/)

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“In this way, we can now program the shape and size of the desired objects using the exact shape of the triangular plates,” says Hendrik Dietz. “We can now produce objects with up to 180 subunits and achieve yields of up to 95 percent. The route there was, however, quite rocky, with many iterations.” (https://geometrymatters.com/icosahedral-nano-shell-designed-to-trap-virus-particles/)

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The team’s scientists can construct not just closed hollow spheres, but also spheres with holes or half-shells by changing the binding sites on the edges of the triangles. These can then be employed to capture viruses. The virus traps were tested on adeno-associated viruses and hepatitis B virus cores in collaboration with Prof. Ulrike Protzer’s team, head of the Institute for Virology at TUM and director of the Institute for Virology at the Helmholtz Zentrum München. (https://geometrymatters.com/icosahedral-nano-shell-designed-to-trap-virus-particles/)

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“Even a simple half-shell of the right size shows a measurable reduction in virus activity,” says Hendrik Dietz. “If we put five binding sites for the virus on the inside, for example suitable antibodies, we can already block the virus by 80 percent, if we incorporate more, we achieve complete blocking.” The next step is to test the building blocks on living mice. (https://geometrymatters.com/icosahedral-nano-shell-designed-to-trap-virus-particles/)

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If the idea of simply mechanically eliminating viruses can be realized, this would be widely applicable and thus an important breakthrough, especially for newly emerging viruses. The starting materials for the virus traps can be mass-produced biotechnologically at a reasonable cost. “In addition to the proposed application as a virus trap, our programmable system also creates other opportunities,” says Hendrik Dietz. “It would also be conceivable to use it as a multivalent antigen carrier for vaccinations, as a DNA or RNA carrier for gene therapy or as a transport vehicle for drugs.” (https://geometrymatters.com/icosahedral-nano-shell-designed-to-trap-virus-particles/)

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Humans are unique among primates in their capacity to construct and control very complex systems of language, mathematics, and music. A fundamental objective of cognitive neuroscience is to determine the cognitive distinctions between these features and others in humans versus those in animals. (https://geometrymatters.com/intuitions-of-geometry-a-signature-of-human-singularity/)

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A great variety of non-exclusive hypotheses have been proposed to account for human singularity, including the emergence of evolved mechanisms for social competence, pedagogy, natural language, or recursive structures across multiple domains. (https://geometrymatters.com/intuitions-of-geometry-a-signature-of-human-singularity/)

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According to prehistoric records, the human attraction to geometric forms is as old as mankind itself. Human art and architecture are replete with circles, squares, and spirals. The oldest engravings ascribed to Homo sapiens are believed to be 73000 years old and consist of a triangular mesh of parallel lines. (https://geometrymatters.com/intuitions-of-geometry-a-signature-of-human-singularity/)

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Paleoanthropologists do not doubt the human origins of such drawings because other non-human animals never draw structured figures when given the opportunity to sketch. The range and abstraction of young children’s drawings, on the other hand, are stunning. Previous studies have shown that even toddlers and people with little formal education from the Amazon have complex intuitions for geometry, establishing an intuitive mathematical “language of the mind.” Those previous findings imply, but do not prove, that humans have a far better knowledge of the abstract aspects of geometry than other animals. (https://geometrymatters.com/intuitions-of-geometry-a-signature-of-human-singularity/)

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The study demonstrates that the sensitivity to the abstract can exist in a much simpler domain: the visual perception of regular geometric forms like squares, rectangles, and parallelograms. According to the findings of Sablé-Meyer et al., the human proclivity for form perception and symbolic abstraction provides a signature of human singularity. (https://geometrymatters.com/intuitions-of-geometry-a-signature-of-human-singularity/)

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We asked human subjects to detect an intruder shape among six quadrilaterals. Although the intruder was always defined by an identical amount of displacement of a single vertex, the results revealed a geometric regularity effect: detection was considerably easier when either the base shape or the intruder was a regular figure comprising right angles, parallelism, or symmetry rather than a more irregular shape. (https://geometrymatters.com/intuitions-of-geometry-a-signature-of-human-singularity/)

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11 quadrilaterals were used, ranging from perfect 5 regularity (a square, with its four right angles, parallel lines, and equal sides) to full irregularity (an arbitrary quadrilateral devoid of any of these properties). For each such reference shape, four deviant versions were generated. Regardless of the human populations that were tested, the geometric regularity effect was already present in young children (preschoolers and 1st graders) and was also replicated in uneducated adults from a remote non-Western population with reduced access to education, suggesting that the effect does not depend on age, culture, and education. (https://geometrymatters.com/intuitions-of-geometry-a-signature-of-human-singularity/)

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Baboons, on the other hand, demonstrated no such geometric regularity effect, even after prolonged training. This difference in performance suggests that the intruder task can be solved using two strategies: a perceptual strategy, well captured by current neural network models of the ventral visual pathway, in which geometric shapes are encoded using the same feature space used to recognize any image (e.g. faces, objects, etc.); and a symbolic strategy, in which geometric shapes are encoded using the same feature space used to recognize any image (e.g. faces, objects, etc.). The latter method appears to be accessible to all individuals, whether in Paris or rural Namibia. (https://geometrymatters.com/intuitions-of-geometry-a-signature-of-human-singularity/)

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The findings are consistent with previous studies, which used more complicated geometric displays and activities and found that all people, including young or uneducated ones, have geometric intuitions and instinctively apply a symbolic, language-like formalism to geometric data. This “language of geometry” is largely supported by the dorsal and inferior sectors of the prefrontal cortex, according to brain imaging. (https://geometrymatters.com/intuitions-of-geometry-a-signature-of-human-singularity/)

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When people think about mathematical notions and recombine them algebraically, these regions are activated. While they are positioned outside the classical language regions, their surface area is significantly increased in the human lineage, making them a potential candidate for the formation of uniquely human skills in evolution, such as symbolic mathematics. (https://geometrymatters.com/intuitions-of-geometry-a-signature-of-human-singularity/)

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The study’s findings point to a human cognitive universal: the ability to comprehend the regularity of a geometric form like a square. They suggest that humans differ from other primates in cognitive mechanisms that are considerably more fundamental than language comprehension or theory of mind, and entail a quick understanding of mathematical regularities in their environment. (https://geometrymatters.com/intuitions-of-geometry-a-signature-of-human-singularity/)

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Fractals are forms that seem similar at different sizes in the area of geometry. Shapes and patterns within a fractal are repeated in an infinite cascade, such as spirals made up of smaller spirals that are made up of even smaller spirals, and so on. (https://geometrymatters.com/complex-thoughts-are-enabled-by-fractal-networks/)

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Previous research has shown how the brain uses fractal networks for better connectivity. A new study from Dartmouth College has discovered a novel approach to examine brain networks by employing the mathematical concept of fractals to express communication patterns between various brain areas as individuals listened to a short tale. Their results show that patterns of brain interactions are mirrored simultaneously at different scales. (https://geometrymatters.com/complex-thoughts-are-enabled-by-fractal-networks/)

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“To generate our thoughts, our brains create this amazing lightning storm of connection patterns,” said senior author Jeremy R. Manning, an assistant professor of psychological and brain sciences, and director of the Contextual Dynamics Lab at Dartmouth.“The patterns look beautiful, but they are also incredibly complicated. Our mathematical framework lets us quantify how those patterns relate at different scales, and how they change over time.” (https://geometrymatters.com/complex-thoughts-are-enabled-by-fractal-networks/)

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The elucidation of the neural code, i.e. the mapping between (a) mental states or cognitive representations and (b) neuronal activity patterns, is a key objective in cognitive neuroscience. One way to put neural code models to the test is to see how well they “translate” neural activity patterns into known (or speculated) mental states or cognitive representations. (https://geometrymatters.com/complex-thoughts-are-enabled-by-fractal-networks/)

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When people think about complicated things, their networks appear to spontaneously arrange into fractal-like patterns. When such thoughts are interrupted, the fractal patterns jumble and lose their integrity. (https://geometrymatters.com/complex-thoughts-are-enabled-by-fractal-networks/)

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The researchers created a mathematical framework for detecting commonalities in network interactions at various sizes or “orders.” The scientists referred to this as a “zero-order” pattern when brain areas did not display any regular patterns of interaction. (https://geometrymatters.com/complex-thoughts-are-enabled-by-fractal-networks/)

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A “first-order” pattern occurs when individual pairs of brain regions interact. “Second-order” patterns are comparable patterns of interactions in distinct sets of brain regions at various sizes. When interaction patterns become fractal—“first-order” or higher—the order specifies how many times the patterns are reproduced at different sizes. According to the findings, when participants listened to an audio recording of a 10-minute tale, their brain networks spontaneously formed into fourth-order network patterns. (https://geometrymatters.com/complex-thoughts-are-enabled-by-fractal-networks/)

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This structure, however, was shattered when individuals listened to edited copies of the tape. People’s brain networks exhibited only second-order patterns when the story’s paragraphs were randomly mixed, maintaining some but not all of the story’s meaning. All except the lowest level (zero-order) patterns were broken when every word in the tale was jumbled. (https://geometrymatters.com/complex-thoughts-are-enabled-by-fractal-networks/)

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“Since the disruptions in those fractal patterns seemed directly linked with how well people could make sense of the story, this finding may provide clues about how our brain structures work together to understand what is happening in the narrative.” (https://geometrymatters.com/complex-thoughts-are-enabled-by-fractal-networks/)

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The results reveal that first- and second-order correlations were most significantly related with auditory and speech processing regions for all of the narrative listening conditions (intact, paragraph, and word; top three rows). Third-order correlations during intact narrative listening showed integration with visual regions, whereas fourth-order correlations reflected integration with areas linked with high-level cognition and cognitive control. (https://geometrymatters.com/complex-thoughts-are-enabled-by-fractal-networks/)

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The researchers believe that when these networks organize at many dimensions, it may reveal how the brain transforms raw sensory input into sophisticated thought—from raw noises to speech, imagery, and full-on comprehension. (https://geometrymatters.com/complex-thoughts-are-enabled-by-fractal-networks/)

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The fractal network patterns were remarkably comparable across people: patterns from one group could properly predict what section of the tale another group was listening to. (https://geometrymatters.com/complex-thoughts-are-enabled-by-fractal-networks/)

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The computational framework developed by the researchers may be extended to fields other than neurology, and the team has already begun to use an equivalent technique to investigate connections in stock prices and animal migratory patterns. (https://geometrymatters.com/complex-thoughts-are-enabled-by-fractal-networks/)

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The shape of an egg has been demonstrated throughout evolution to be one of the greatest characteristics for the embryonic development of egg-laying species. The form is ideal for the process of incubation, and its size is appropriate in relation to the body of animals for birth. Furthermore, eggs are well-designed to protect the fragile embryo away from the extremes of its immediate environment. (https://geometrymatters.com/the-shape-of-a-perfect-egg-defined-by-a-universal-formula/)

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From an analytical standpoint, the egg, as one of the most traditional food items, has long piqued the interest of mathematicians, engineers, and biologists. To date, the shape of a bird’s egg has eluded a generally applicable mathematical definition as a key parameter in oomorphology. All egg forms may be analyzed using four geometric figures: the sphere, ellipsoid, ovoid, and pyriform (conical or pear-shaped). The first three have unambiguous mathematical definitions, each derived from the preceding statement, but a formula for the pyriform profile has yet to be developed. (https://geometrymatters.com/the-shape-of-a-perfect-egg-defined-by-a-universal-formula/)

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To address this issue, researchers added an extra function to the ovoid formula, creating a mathematical model to suit a completely unique geometric shape defined as the last step in the evolution of the sphere-ellipsoid, which is applicable to any egg geometry. This universal formula is based on four parameters: egg length, maximum breadth, shift of the vertical axis, and the diameter at one-quarter of the egg length. (https://geometrymatters.com/the-shape-of-a-perfect-egg-defined-by-a-universal-formula/)

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This mathematical equation underlines our understanding and appreciation of a certain philosophical harmony between mathematics and biology, and from those two a way towards further comprehension of our universe, understood neatly in the shape of an egg.’ - Dr Michael Romanov, Visiting Researcher at the University of Kent (https://geometrymatters.com/the-shape-of-a-perfect-egg-defined-by-a-universal-formula/)

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Mathematical descriptions of all fundamental egg forms have previously been used in food science, mechanical engineering, agriculture, biosciences, architecture, and aeronautics. This formula, for example, can be used in the engineering design of thin-walled egg-shaped containers that must be stronger than ordinary spherical ones. (https://geometrymatters.com/the-shape-of-a-perfect-egg-defined-by-a-universal-formula/)

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This new formula represents a significant advance with several applications, including competent scientific description of a biological object, accurate and simple determination of the physical characteristics of a biological object, and future biology-inspired engineering. (https://geometrymatters.com/the-shape-of-a-perfect-egg-defined-by-a-universal-formula/)

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‘We look forward to seeing the application of this formula across industries, from art to technology, architecture to agriculture. This breakthrough reveals why such collaborative research from separate disciplines is essential.’ - Dr Valeriy Narushin, former visiting researcher at the University of Kent (https://geometrymatters.com/the-shape-of-a-perfect-egg-defined-by-a-universal-formula/)

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One of biology’s biggest mysteries is the genesis of animal form. Biologists trying to understand the genesis and evolution of life have studied and sought to characterize the embryology of all multicellular animal phyla since the 19th century. Many people believed that by the turn of the twentieth century, this work would have been completed. (https://geometrymatters.com/embryo-geometry-a-theory-of-evolution-from-a-single-cell-to-the-complex-vertebrate-body/)

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Anatomists have succeeded in giving comprehensive descriptions of the musculoskeletal, organ, and neurological systems, ranging from Leonardo Da Vinci and Vesalius through Gray’s Anatomy. However, the genesis of these and other elements of organismal shape remains a mystery. Because the body develops from the embryo, nineteenth-century anatomists logically sought a solution by observing early animal development—or embryogenesis. By the end of the nineteenth century, virtually all major phyla’s embryological phases had been described in minute detail. (https://geometrymatters.com/embryo-geometry-a-theory-of-evolution-from-a-single-cell-to-the-complex-vertebrate-body/)

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Major evolutionary changes, according to Neo-Darwinian theory, occur as a result of the selection of random, fortunate genetic mutations through time. However, other experts argue that this hypothesis fails to account for the emergence of fundamentally diverse living forms and their rich complexity, notably in vertebrates like humans. (https://geometrymatters.com/embryo-geometry-a-theory-of-evolution-from-a-single-cell-to-the-complex-vertebrate-body/)

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The authors beg the reader to suspend skepticism that such a complex and longstanding problem is subject to a solution of relative simplicity. “Embryo geometry”, developed by a team from the University of San Diego, Mount Holyoke College, Evergreen State College, and Chem-Tainer Industries, Inc. in the United States, considers animal complexity in general, and the vertebrate body in particular, to be the result of mechanical forces and geometric laws rather than random genetic mutation. (https://geometrymatters.com/embryo-geometry-a-theory-of-evolution-from-a-single-cell-to-the-complex-vertebrate-body/)

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They offer 24 “blueprints” in their article that illustrate how the musculoskeletal, cardiovascular, neurological, and reproductive systems evolve through the mechanical deformation of geometric patterns. These images show how the vertebrate body might have evolved from a single cell during the evolutionary time and during individual development. (https://geometrymatters.com/embryo-geometry-a-theory-of-evolution-from-a-single-cell-to-the-complex-vertebrate-body/)

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Though neither rigorous nor exhaustive in an empirical sense, our model offers an intuitive and plausible description of the emergence of form via simple geometrical and mechanical forces and constraints. The model provides a template, or roadmap, for further investigation, subject to confirmation (or refutation) by interested researchers. (https://geometrymatters.com/embryo-geometry-a-theory-of-evolution-from-a-single-cell-to-the-complex-vertebrate-body/)

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The concept of “embryo geometry” suggests that the vertebrate embryo might be produced by mechanical deformation of the blastula, a ball of cells formed when a fertilized egg splits. As these cells multiply, the volume and surface area of the ball expand, changing its shape. According to the hypothesis, the blastula preserves the geometry of the initial eight cells generated by the egg’s first three divisions, which establish the three axes of the vertebrate body. (https://geometrymatters.com/embryo-geometry-a-theory-of-evolution-from-a-single-cell-to-the-complex-vertebrate-body/)

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The premise that complex animal form arises from mechanical forces acting on geometrically constrained populations of dividing cells in the early embryo provides a new lens through which to view developmental and evolutionary processes, and may pose a significant challenge to the Modern Synthesis’s dictum that evolution proceeds by a selection of adventitious mutations resulting from random mutations. (https://geometrymatters.com/embryo-geometry-a-theory-of-evolution-from-a-single-cell-to-the-complex-vertebrate-body/)

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Though speculative, the model addresses the poignant absence in the literature of any plausible account of the origin of vertebrate morphology. A robust solution to the problem of morphogenesis—currently an elusive goal—will only emerge from consideration of both top-down (e.g., the mechanical constraints and geometric properties considered here) and bottom-up (e.g., molecular and mechano-chemical) influences. (https://geometrymatters.com/embryo-geometry-a-theory-of-evolution-from-a-single-cell-to-the-complex-vertebrate-body/)

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Protein pattern generation has been extensively explored experimentally in recent years. Proteins diffusing and interacting in cells, like birds that organize into flocks by associating solely with their close neighbors, may establish self-organized patterns that regulate critical activities like cell division and tissue-shape formation. While theoretical models have concentrated on the dynamics of proteins approaching homogenous stable states since Turing’s work in the early 1950s, studies on fully formed patterns in the severely nonlinear domain have mainly been restricted to numerical research. (https://geometrymatters.com/a-geometric-framework-for-protein-and-cell-diffusion-and-interaction/)

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A team of LMU physicists led by Professor Erwin Frey has developed a novel approach that allows for the systematic mathematical study of pattern creation processes and reveals their underlying physical principles. The research focuses on ‘mass-conserving’ systems, in which interactions influence the states of the particles involved but do not change the overall number of particles in the system. (https://geometrymatters.com/a-geometric-framework-for-protein-and-cell-diffusion-and-interaction/)

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“Now we can understand the salient features of pattern formation independently of simulations using simple calculations and geometrical constructions,” explains Fridtjof Brauns, lead author of the new paper. “The theory that we present in this report essentially provides a bridge between the mathematical models and the collective behavior of the system’s components.” (https://geometrymatters.com/a-geometric-framework-for-protein-and-cell-diffusion-and-interaction/)

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The discovery that changes in the local number density of particles will also modify the locations of local chemical equilibria was the important insight that led to the theory. These changes, in turn, produce concentration gradients, which drive the particles’ diffusive movements. (https://geometrymatters.com/a-geometric-framework-for-protein-and-cell-diffusion-and-interaction/)

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The authors depict this dynamic interaction using geometrical structures that define global dynamics in a multidimensional ‘phase space.’ Because these objects have real physical implications – as representations of the trajectories of altering chemical equilibria, for example – the collective characteristics of systems may be directly inferred from the topological connections between these geometric structures. (https://geometrymatters.com/a-geometric-framework-for-protein-and-cell-diffusion-and-interaction/)

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This is why our geometrical description enables us to comprehend why the patterns we see in cells emerge. In other words, they show the physical mechanisms that govern the interaction of the molecular species in question. (https://geometrymatters.com/a-geometric-framework-for-protein-and-cell-diffusion-and-interaction/)

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Furthermore, the framework suggests ways of experimentally characterizing pattern-forming systems at a mesoscopic scale and provides tools that may help guide the design and control of self-organization. (https://geometrymatters.com/a-geometric-framework-for-protein-and-cell-diffusion-and-interaction/)

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Fractals are infinitely complex patterns that are self-similar across different scales. They are created by repeating a simple process over and over in an ongoing feedback loop. Mathematician Benoit Mandelbrot invented the word “fractal” in a 1975 book on the subject, and his landmark 1982 book The Fractal Geometry of Nature, which records the geometric patterns’ prevalence, is largely credited with popularizing them. (https://geometrymatters.com/hunting-bachs-fractals/)

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Because nature is full of fractals, the patterns are highly familiar. Trees, rivers, beaches, mountains, clouds, seashells, hurricanes, and so forth. Abstract fractals, such as the Mandelbrot Set or the Sierpinski Triangle, may be produced by a computer repeatedly solving a simple equation. (https://geometrymatters.com/hunting-bachs-fractals/)

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A fractal is sometimes defined as having self-similarity, which indicates that same (or nearly identical) patterns emerge whether the form is examined up close or from a distance. That is, the portion appears to be the whole, and the whole appears to be a part. Not surprisingly, music scholars have tried to investigate such patterns in compositions, despite the fact that the mathematical roots of music are widely established. (https://geometrymatters.com/hunting-bachs-fractals/)

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For example, physicists Richard F. Voss and John Clarke (University of California) studied numerous audio recordings of music and discovered scaling patterns in loudness fluctuations and melody fluctuations of popular music on the radio. (https://geometrymatters.com/hunting-bachs-fractals/)

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Identifying fractals in music, on the other hand, needs a different technique than perceiving them in images. (https://geometrymatters.com/hunting-bachs-fractals/)

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Harlan Brothers, a jazz guitarist, composer, and mathematician from Branford, Connecticut, discovered numerous forms of scaling when researching the function of power laws in music, including self-similarity with respect to duration, pitch, interval, theme, and structure. He has been looking for fractals in Bach’s work as well. (https://geometrymatters.com/hunting-bachs-fractals/)

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“Unlike a picture, which is all laid out so that you can instantly see the structure, music is fundamentally a serial phenomenon,” Brothers says. “With music, the whole piece takes shape in your mind. This makes it more challenging to identify the self-similarity.” (https://geometrymatters.com/hunting-bachs-fractals/)

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For much over a decade, Brothers reported on fractal structure in the phrasing of notes employed by Bach to construct his Cello Suite No. 3 in a research published in Fractals in 2007. Patterns of long and short notes within measures resurfaced inside that piece as patterns of long and short phrases at greater scales. Brothers noticed that the suite’s self-similarity bore a remarkable resemblance to the Cantor Comb, a representation of a historical fractal known as the Cantor Set. (which Georg Cantor described in 1883, a century before Mandelbrot coined the name “fractal”). (https://geometrymatters.com/hunting-bachs-fractals/)

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One of Harlan’s early discoveries was that musicians have been creating fractal music for at least six centuries. Many of the great Flemish composers who created the technique of the mensuration or prolation canons, such as Johannes Ockeghem and Josquin des Prez, were familiar with motivic scaling. This form of canon is distinguished by a melody or rhythmic theme that is repeated in multiple voices at different tempos at the same time. To be clear, not all mensuration canons are fractal; basic conditions must be satisfied in order for an item to be classified as such. (https://geometrymatters.com/hunting-bachs-fractals/)

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Nonetheless, a percentage of predictability is given by all these patterns. In search of a definitive conclusion, Daniel Levitin analyzed the rhythm spectra of 1,788 movements from 558 compositions of Western classical music. (https://geometrymatters.com/hunting-bachs-fractals/)

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“Because music has a beat and is based on repetition, it has been said that ‘what’ the next musical event will be is not always easy to guess, but ‘when’ it is likely to happen can be easily predicted.” (https://geometrymatters.com/hunting-bachs-fractals/)

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The differences in the predictability for different genres of music and composers are more relevant than the fractal structure of the rhythms. Composers with a more diverse style, like Mozart or Joplin, wrote music with more variable spectral exponents than Beethoven or Vivaldi. Ragtime and madrigals, for example, are considerably less predictable than symphonies or scherzos. The ensuing variations in rhythmic predictability would allow them to uniquely identify their compositions and differentiate them from those of their contemporaries. (https://geometrymatters.com/hunting-bachs-fractals/)

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Looking at the time of musical notes, one can see that they are not as regularly spaced as one might anticipate and, in fact, have a fractal structure. The researchers discovered that musical rhythm follows a 1/f pattern, the majority of the music we listen to being in the 1/f pattern. (https://geometrymatters.com/hunting-bachs-fractals/)

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It is characterized as a power-law decline in the correlations of the pitch with time, and it has the perfect balance of pattern and unexpectedness, as well as being appealing to the human ear, the shape of the curve defined by 1/f music having a fractal shape. (https://geometrymatters.com/hunting-bachs-fractals/)

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While the subject is being researched further, it is clear that our brains appear to be particularly well suited for processing some types of sounds and find those with a fractal structure to be particularly appealing. (https://geometrymatters.com/hunting-bachs-fractals/)

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What makes an object successful at folding? Protein scientists study how an object transforms between 2D surfaces, and tridimensional objects by using universal nets, that provide a balance between entropy loss and potential energy gain. This also explains why some of their geometrical attributes (such as compactness) represent a good predictor for the folding preference of a given shape. Researchers from the University of Michigan study the thermodynamic foldability of 2D nets for all five Platonic solids. (https://geometrymatters.com/entropy-and-energy-influence-polygonal-nets-folding-into-platonic-solids/)

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Our goal is to understand how topology affects yield in the stochastic folding of 3D objects. The advantage is threefold. First, by using a collection of sheets folding into the same target shape, we isolate the geometric attributes responsible for high-yield folding. Second, the model allows exhaustive computation of the pathways followed by the nets during folding, elucidating how some nets achieve high yield. Third, by studying increasingly more complex objects—from tetrahedra to icosahedra—we can use the folding mechanisms quantified in the simplest objects to predict, and potentially validate, their occurrence in the more complex shapes. (https://geometrymatters.com/entropy-and-energy-influence-polygonal-nets-folding-into-platonic-solids/)

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Albrecht Dürer, a 16th-century Dutch artist, studied whether 2D cuts of nonoverlapping, edge-joined polygons might be folded into Platonic and Archimedean polyhedra. Dürer cuts were eventually referred to as “nets,” but for a long time, their popularity was confined to the area of mathematics. Self-folding origami is a contemporary innovation that brings a modern twist to the ancient art of paper folding. Self-folding takes Dürer’s thoughts to the forefront of many study disciplines, from medicine to robotics, by offering a technique for generating complex 3D geometries from low-dimensional objects without the need for manipulation of the constituent pieces. (https://geometrymatters.com/entropy-and-energy-influence-polygonal-nets-folding-into-platonic-solids/)

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Several recent efforts have used physical forces such as light, pH, capillary forces, cellular traction, and thermal expansion to accomplish controlled folding of 3D structures. Other studies have looked at the link between the geometric properties of the object being folded and its proclivity for effective folding. The effect of different cut patterns on the material’s stress-strain behavior has been elucidated in the macroscopic folding of kirigami sheets—origami-like structures containing cuts and creases—and the “inverse design problem” of finding cuts leading to the folding of a specific target structure. (https://geometrymatters.com/entropy-and-energy-influence-polygonal-nets-folding-into-platonic-solids/)

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Despite being the most basic and symmetric 3D polytopes, the Platonic forms family suffices to show the fast expansion in design space as shapes get more complex: A tetrahedron has two net representations, cubes and octahedra have 11 nets, while dodecahedra and icosahedra have 43,380 unique edge unfoldings apiece. (https://geometrymatters.com/entropy-and-energy-influence-polygonal-nets-folding-into-platonic-solids/)

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To identify the nets able to fold reliably into their polyhedron of origin we performed hundreds of cooling simulations for each net, using both a fast and a slow cooling protocol. The two distinct nets for the tetrahedron, hereafter referred to as the triangular net and the linear net, showed remarkably different folding propensities for the fast cooling protocol. (https://geometrymatters.com/entropy-and-energy-influence-polygonal-nets-folding-into-platonic-solids/)

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Nets that fold the most reliably are the most compact and have the most leaves on their cutting graph. If a net has a large number of leaves (the vertices with degree one on the cutting tree) and a small diameter, it is considered to be more compact (the longest shortest path between any two faces on the face graph). Most notably, the folding probability may be decreased from 99 percent to 17 percent in nets that differ only by the position of a single face. (https://geometrymatters.com/entropy-and-energy-influence-polygonal-nets-folding-into-platonic-solids/)

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What causes one shape to fold nearly perfectly every time while a slightly different one fails to do so almost as frequently? And why do net “leafiness” and “compactness” correlate with folding yield? (https://geometrymatters.com/entropy-and-energy-influence-polygonal-nets-folding-into-platonic-solids/)

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All nets follow a folding pathway that achieves a narrow balance between reduction of degrees of freedom and gain of potential energy. In practice, high-temperature folding occurs locally, so that the system seeks to optimize its conformational entropy at each step of the process. Furthermore, as the number of faces rises, the folding tendency of a net diminishes. For example, while the 4-sided tetrahedron folds nearly flawlessly, the 20-sided icosahedron cannot fold. (https://geometrymatters.com/entropy-and-energy-influence-polygonal-nets-folding-into-platonic-solids/)

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Wings are like fingerprints for many insect species, with no two patterns being the same. These insects, like many other organisms ranging from leopards to zebrafish, benefit from nature’s seemingly limitless ability to generate a wide range of shapes and patterns. However, how do these patterns emerge? (https://geometrymatters.com/model-for-studying-natures-patterns/)

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Harvard University researchers have created a model that can recreate the wing patterns of a large group of insects using only a few parameters, shedding light on how these complex patterns form. (https://geometrymatters.com/model-for-studying-natures-patterns/)

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“We have developed a simple model, with only a few assumptions about how wings grow, that can recapitulate patterns that look close to life-like and can do so for species that are distantly related to each other, from grasshoppers to dragonflies, damselflies and lacewings,” said Christopher Rycroft. “This model could be useful for studying the evolution of wing structure and other patterned shapes.” (https://geometrymatters.com/model-for-studying-natures-patterns/)

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While the shape and design of insect wings vary greatly between species, virtually all contain veins – thick, strut-like structures implanted on the wing surface. Some insects, like the well-known fruit fly, have only a few big main veins. The position and form of these veins are shared by the left and right wings of the same person, as well as between individuals of the same species. (https://geometrymatters.com/model-for-studying-natures-patterns/)

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Researchers gathered specimens from Harvard entomology classes, images from 20th-century reference books, and data from existing entomological databases. The wings were then removed from the bodies and photographed to create 2D pictures. After compiling the information, the researchers distinguished, or segmented, each unique polygonal shape formed by crossing veins. (https://geometrymatters.com/model-for-studying-natures-patterns/)

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“We wanted to take this complex shape and turn it into something simpler so we could ask specific questions and compare its geometry across species,” said Jordan Hoffmann, co-author of the paper and Ph.D. candidate at SEAS. “We looked at the geometric properties of these individual shapes, which we called domains. We looked at how elongated each domain was, how many sides it had, how it touched its neighbors.” (https://geometrymatters.com/model-for-studying-natures-patterns/)

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Hoffmann and his colleagues discovered that the size of the domain and its circularity may explain a large portion of the variance in geometry. They also discovered that, while each wing’s pattern is unique, the distribution of domain shapes across families and species is surprisingly similar. They developed a simpler model for the formation of wing veins after they had a decent technique to assess the similarity of wings based on how these forms are spread across numerous species. (https://geometrymatters.com/model-for-studying-natures-patterns/)

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According to the researchers, an unknown inhibitory signal diffuses from several signaling centers in the areas between the major veins. These inhibitory zones appear at random and oppose one another, preventing additional veins from developing in certain locations. Those zones may produce the intricate geometries of the wing when the veins developed around them as the wing expanded and stretched throughout development. (https://geometrymatters.com/model-for-studying-natures-patterns/)

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Natural surroundings have fractal patterns that recur at various size scales, and they are also found in highly beautiful creative creations. By the age of three, youngsters have developed an adult-like affinity for visual fractal patterns found in nature. That discovery was made among children raised in an environment of Euclidean geometry, such as buildings with rooms built with straight lines in a basic non-repeating way, according to the study’s primary author Kelly E. Robles. (https://geometrymatters.com/fractal-patterns-preferred-by-children-under-three-years/)

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“Unlike early humans who lived outside on savannahs, modern-day humans spend the majority of their early lives inside these manmade structures,” Robles said. “So, since children are not heavily exposed to these natural low-to-moderate complexity fractal patterns, this preference must come from something earlier in development or perhaps are innate.” (https://geometrymatters.com/fractal-patterns-preferred-by-children-under-three-years/)

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Individual variations in processing approaches were investigated to see if they might explain for patterns in fractal fluency. In the study, participants were shown pictures of fractal designs, both exact and statistical, ranging in complexity on computer displays. Exact fractals are highly organized in such a way that the same fundamental pattern repeats perfectly at every size and may have spatial symmetry like snowflakes. Statistical fractals, on the other hand, recur in a similar but not precise manner across size and lack spatial symmetry found in coastlines, clouds, mountains, rivers, and forests. Both styles may be seen in art from all around the world. (https://geometrymatters.com/fractal-patterns-preferred-by-children-under-three-years/)

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“Since people prefer a balance of simplicity and complexity, we were looking to confirm that people preferred low-to-moderate complexity in statistically repeating patterns, and that the presence of order in exact repeating patterns allowed for a tolerance of and preference for more complex patterns.” (https://geometrymatters.com/fractal-patterns-preferred-by-children-under-three-years/)

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Although there were significant variations in adult and kid choices, the general trend was similar. Exact patterns with higher complexity were favored, but preference for statistical patterns peaked at low-moderate complexity and subsequently decreased as complexity increased. The researchers were able to rule out the idea that age-related perceptual techniques or biases drove differing preferences for statistical and precise patterns in the following stages with the individuals. (https://geometrymatters.com/fractal-patterns-preferred-by-children-under-three-years/)

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According to co-author Richard Taylor, the aesthetic experience of witnessing nature’s fractals has enormous potential advantages ranging from stress relief to rejuvenating brain weariness. (https://geometrymatters.com/fractal-patterns-preferred-by-children-under-three-years/)

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“Nature provides these benefits for free, but we increasingly find ourselves surrounded by urban landscapes devoid of fractals,” he said. “This study shows that incorporating fractals into urban environments can begin providing benefits from a very early age.” (https://geometrymatters.com/fractal-patterns-preferred-by-children-under-three-years/)

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Many of nature’s fractal objects benefit from the favorable functionality that comes from pattern repetition at various sizes. Examples from nature include beaches, lightning, rivers, and trees, as well as cardiovascular and respiratory systems such as the bronchial tree. Neurons, like trees, are thought to represent a common kind of fractal branching activity. (https://geometrymatters.com/neurons-use-fractal-networks-for-better-connectivity/)

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Although prior neuron research has calculated the scaling characteristics of their dendritic branches, this has generally been done to categorize neuron morphologies rather than quantify how neurons benefit from their fractal geometry. (https://geometrymatters.com/neurons-use-fractal-networks-for-better-connectivity/)

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Why does the body use fractal neurons rather than, say, the Euclidean wires seen in common electronics? Within the mammalian brain, neurons create vast networks, with individual neurons utilizing up to 60,000 connections in the hippocampus alone. They link to the retina’s photoreceptors, letting humans see, and to the limbs, allowing people to move and feel, in addition to their connections within the brain. Given its relevance as the body’s “wire,” the study focuses on the effect of fractal scaling in creating neuron connections. (https://geometrymatters.com/neurons-use-fractal-networks-for-better-connectivity/)

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“By distorting their branches and looking at what happens, we were able to show that the fractal weaving of the natural branches is balancing the ability of neurons to connect with their neighbors to form natural electric circuits while balancing the construction and operating costs of the circuits,” Rowland said. (https://geometrymatters.com/neurons-use-fractal-networks-for-better-connectivity/)

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Nature’s fractals benefit from the fact that they develop at various sizes, according to Taylor, who has long looked to fractals for bioinspiration. While trees are the most well-known example of fractal branching, he claims that this research shows how neurons vary from trees. (https://geometrymatters.com/neurons-use-fractal-networks-for-better-connectivity/)

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“Whereas the fractal character of trees originates predominantly from the distribution of branch sizes, the neurons also use the way their branches weave through space to generate their fractal character.” (https://geometrymatters.com/neurons-use-fractal-networks-for-better-connectivity/)

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Taylor, a Cottrell Scholar of the Research Council for Science Advancement, was given a broad U.S. patent in 2015 for not just his vision-related artificial fractal-based implants, but also for any such implants that link signaling activity with nerves for any purpose in animal and human biology. (https://geometrymatters.com/neurons-use-fractal-networks-for-better-connectivity/)

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In 2019, a nationwide study on fundamental skills in Switzerland discovered a link between children’s spatial awareness at the age of three and their mathematical ability in primary school. Other variables, such as socioeconomic position or linguistic competence, were ruled out by the researchers. It is unknown how spatial ability impacts arithmetic skills in youngsters, although the spatial notion of numbers may have a role. (https://geometrymatters.com/mathematical-skills-improved-by-tri-dimensional-thinking/)

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“We know from past studies that adults think spatially when working with numbers — for example, represent small numbers to the left and large ones to the right,” explains Möhring. “But little research has been done on how spatial reasoning at an early age affects children’s learning and comprehension of mathematics later.” (https://geometrymatters.com/mathematical-skills-improved-by-tri-dimensional-thinking/)

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The conclusions are based on data from 586 youngsters in Basel, Switzerland. The researchers administered a battery of activities to three-year-old toddlers in order to assess cognitive, socio-emotional, and spatial ability. They also investigated whether the rate of growth, specifically the quick development of spatial abilities, might predict future mathematical competence. (https://geometrymatters.com/mathematical-skills-improved-by-tri-dimensional-thinking/)

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Previous research with a limited sample size had discovered a link, but Möhring and her colleagues were unable to replicate this in their own investigation. Three-year-old children who began with poor spatial ability improved quicker in future years, but still performed at a lower level in mathematics when they were seven. Despite their rapid development, these children had not yet entirely caught up with the children who had stronger beginning spatial reasoning skills when they started school. (https://geometrymatters.com/mathematical-skills-improved-by-tri-dimensional-thinking/)

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“Parents often push their children in the area of language skills,” says Möhring. “Our results suggest how important it is to cultivate spatial reasoning at an early age as well.” There are simple ways to do this, such as using “spatial language” (larger, smaller, same, above, below) and toys — e.g. building blocks — that help improve spatial reasoning ability. (https://geometrymatters.com/mathematical-skills-improved-by-tri-dimensional-thinking/)

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In the natural environment, the sense of smell, or olfaction, is used to identify contaminants and assess nutritional value by utilizing the connections formed between chemicals during biological processes. As a result, the synthesis of a specific toxin by a plant or bacteria will be accompanied by the emission of specific sets of volatile chemicals, an animal being able to detect the presence of toxins in food by smelling it. (https://geometrymatters.com/using-hyperbolic-geometry-to-map-the-olfactory-space/)

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Scientists from the Salk Institute and Arizona State University have developed a method for organizing odor molecules based on how frequently they appear together in nature, which is where our sense of smell arose. They were subsequently able to map this data in order to identify regions of odor combinations that individuals find most enjoyable. (https://geometrymatters.com/using-hyperbolic-geometry-to-map-the-olfactory-space/)

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“We can arrange sound by high frequency and low frequency; vision by a spectrum of wavelengths and colors,” says Tatiana Sharpee. “But when it comes to olfaction, it’s been an unsolved problem whether there is a way to organize odors.” (https://geometrymatters.com/using-hyperbolic-geometry-to-map-the-olfactory-space/)

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They utilized statistical methods to map odorant molecules identified in distinct samples of strawberries, tomatoes, blueberries, and mouse urine based on how frequently they appea