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Geometrical concepts, cognition, and educational implications

A group of researchers from Italy and the United Kingdom analyzed the development of geometrical…

Patterns of innovation

The process of innovation is a mystery. Historically it has offered opportunities and challenges, as…

Geometric models for memories

By creating a computer program that turns sequences of events from a video into unique…

What Does It Mean That “Space Can Be Transcendental Without the Axioms Being So”?

In her 2013 study, Francesca Biagioli examines Helmholtz’s claim that space can be transcendental without…

Pure geometry and geometric cognition

Pure geometry’s cognitive foundation is mostly unknown. Even the ‘simpler’ question of what kind of…

How quantifying the shape of stories predicts their success

Narratives, and other types of speech, can be used to enlighten, entertain, and make sense…

The human quest for discovering mathematical beauty in the arts

“The human function is to ‘discover or observe’ mathematics,” said twentieth-century British mathematician G. H.…

Einstein, geometry, and experience

As far as the propositions of mathematics refer to reality, they are not certain; and…

Did Humans Invent Mathematics, or Is It a Fundamental Part of Existence?

“God made the integers, all else is man’s work,” stated German mathematician Leopold Kronecker in…

A language of thought for the mental representation of geometric shapes

Why do geometric shapes such as lines, circles, zig-zags, or spirals appear in all human…

Hans Jenny and the science of sound: cymatics.

This is not unregulated chaos; it is a dynamic but ordered pattern. Hans Jenny (16 August 1904, Basel – 23 June 1972) was a natural scientist and physician who coined the term cymatics to explain the acoustic impacts of sound wave phenomena. To this field have contributed a number of scholars, that believed sound plays a…

The fractal-hyperbolic geometry of networks

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…

The manifold framework of neural circuits

Manifold: a topological space that locally resembles a Euclidean space in mathematics. Perceptual manifold: the population structures of sensory neurons that emerge as a result of identity-preserving variations in the input stimulus space. Neural manifold: low-dimensional subspaces underlying population activities embedded in high-dimensional neural state space Point-cloud manifold: a set of data points with an…

Geometrical concepts, cognition, and educational implications

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…

The geometry of atomic bonds

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,…

Patterns of innovation

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. It has been examined by economics, anthropologists, evolutionary biologists, and…

Hologram of a single photon

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. 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…

A patterned fingerprint of the brain

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…

Geometric models for memories

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. We model experiences and memories as trajectories through word-embedding spaces whose coordinates reflect…

What Does It Mean That “Space Can Be Transcendental Without the Axioms Being So”?

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…

Pure geometry and geometric cognition

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…

How quantifying the shape of stories predicts their success

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. To close…

The human quest for discovering mathematical beauty in the arts

“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…

Einstein, geometry, and experience

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. Lecture before the Prussian Academy of SciencesPresented: 27 January 1921Published: 1921 by Julius Springer (Berlin)Translation: from Einstein, Ideas and Opinionstrans. Sonja Bargmann (New York: Crown, 1982)© The Collected…

Did Humans Invent Mathematics, or Is It a Fundamental Part of Existence?

“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. Mathematics, in this…

A language of thought for the mental representation of geometric shapes

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…

Harpa Concert Hall: from nature to architecture

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. Harpa…

Hidden order in disorder

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. 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.…

Icosahedral nano-shell designed to trap virus particles

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. Even before the new coronavirus…

Intuitions of geometry: a signature of human singularity

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. A great variety of non-exclusive hypotheses have been proposed to account for…

Complex thoughts are enabled by fractal networks

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. Previous research has shown how the brain uses fractal networks…

The shape of a perfect egg, defined by a universal formula

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 geometry: a theory of evolution from a single cell to the complex vertebrate body

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.…

A geometric framework for protein and cell diffusion and interaction

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…

Hunting Bach’s fractals

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…

Entropy and energy influence polygonal nets folding into Platonic solids

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…

Model for studying nature’s patterns.

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? Harvard University researchers have created a model that…

Fractal patterns preferred by children under three years

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…

Neurons use fractal networks for better connectivity

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. Although prior neuron research…

Mathematical skills improved by tri-dimensional thinking

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…

Using hyperbolic geometry to map the olfactory space

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,…

Geometry of the stem cell’s micro-environment can influence organ aging and susceptibility to cancer.

Stem cells are the raw materials of the body, the cells that give rise to all other cells with specific tasks. Stem cells divide to create new cells in the body or in a laboratory under the correct conditions. Referred to as daughter cells, these cells either become new stem cells (self-renewal) or specialized cells…

Plato’s cube is nature’s preferred choice for fragmentation.

Plato is widely recognized as the first to create the concept of an atom, or the idea that matter is made up of indivisible components at the smallest scale. The Greek philosopher proposed the primordial forms of the universe’s structure, arguing that the universe was made up of five different forms of matter: earth, air, fire,…

Number sense: emergence from the recognition of visible objects

Humans and animals have a “number sense,” or the capacity to intuitively estimate the numerosity of visual elements in a collection. This capacity suggests that processes for extracting numerosity are inside the brain’s visual system, which is largely concerned with visual object recognition. Researchers have long questioned if these number neurons are created in the…

Hans Jenny and the science of sound: cymatics.

This is not unregulated chaos; it is a dynamic but ordered pattern. Hans Jenny (16 August 1904, Basel – 23 June 1972) was a natural scientist and physician who coined the term cymatics to explain the acoustic impacts of sound wave phenomena. To this field have contributed a number of scholars, that believed sound plays a…

Are patterns real or imagined?

For neuroscientists studying complex systems, patterns exhibit valuable data that may or may not correspond to higher levels of cognitive processes. Tyler Millhouse proposes a criterion evaluating just how real a pattern is likely to be, improving a SFI External Professor Daniel Dennett’s 1991 explanation, which utilized ‘compressibility’ to determine how genuine a pattern is…

World’s oldest example of applied geometry revealed by Australian mathematician.

Pythagoras of Samos is credited with several of the world’s most important mathematical theories. The Pythagorean theorem, the Theory of Proportions, and the sphericity of the Earth are only a few examples of Pythagora’s bright mind. At least, that’s what we believed. As our grasp of history grows, a few gaps in these attributions become…

Sounds, shapes, speech and body movements convey emotion through one shared property

People communicate their emotions using their voice, face, and movement, as well as through abstract forms such as art, architecture, and music. The structure of these expressions is frequently intuitively related to their meaning: flowery curlicues are used in romantic poetry, while a spiky script is used in death metal band logos. A Dartmouth study published in…

Mathematical Beauty Activates Same Brain Region as Great Art or Music

The beauty of mathematical formulations resides in abstracting, in simple equations, universal truths. Many people, including mathematicians Bertrand Russell (1919) and Hermann Weyl (Dyson, 1956; Atiyah, 2002), physicist Paul Dirac (1939), and art critic Clive Bell (1914), have written about the importance of beauty in mathematical formulations and compared the experience of mathematical beauty to…

Deciphering the brain’s color and shape coding

A human can identify hundreds of thousands of unique colors and forms visually, but how does the brain process all of this information? Previously, scientists assumed that the visual system records shape and color separately with discrete groups of neurons and then integrates them much later. According to the Salk researchers, there are neurons that…

Depth and coordinates processing in neural networks

The MonoDepth model, which calculates depth from a single picture, was studied by researchers from the Technical University of Delft in the Netherlands. A single image necessitates Deep Neural Networks’ relying on visual cues, which necessitates awareness of the surroundings, a fundamentally assumption-laden process. The current focus of monocular depth estimation research has been on…

Honeycomb style pattern used by the brain to code one’s position in space

A European-American research team has used electrophysiological data to prove the presence of grid-like activity in the human brain. Researchers used various methods to visualize grid cell activity while subjects explored images of everyday scenes under the direction of Prof. Christian Doeller of the Max Planck Institute for Human Cognitive and Brain Sciences (MPI CBS)…

The Scutoid: additional solution for three-dimensional packing

Tissues flex into intricate three-dimensional forms that lead to organs as an embryo grows. This process’s building blocks are epithelial cells, which create the outer layer of skin, for example. They also line all animals’ blood arteries and organs. These cells are closely packed together. It has been suggested that epithelial cells acquire either columnar…

Parametric semiology: creating meaningful architectural spaces via human patterns of behaviour

In his Parametric Semiology – The Design of Information Rich Environments (2013), Patrik Schumacher from Zaha Hadid Architects, creates a new type of value for human spaces. He states that designed spaces are spatial communications that frame and organize subsequent conversations. They organize the participants into precise constellations that correspond to the anticipated conversation circumstances. A…

Geometry, physics, and the chameleon’s change of colour

For many years, scientists thought that the chameleon’s ability to change colors was due to its capacity to transfer pigments around inside its cells; however, it appears that this is not the case. Instead of pigments, the little animals rely heavily on geometry and physics. To understand this, you must first understand how light and…

‘Multi-dimensional universe’ in brain networks

Researchers were able to identify architectural patterns that arise when the brain had to comprehend information before disintegrating into nothing by utilizing a sophisticated mathematical method. The discovery showed that the brain is full of multi-dimensional geometrical structures operating in as many as 11 dimensions. A team of researchers from the Blue Brain Project, a…

Tom Noddy’s Bubble Magic

“Tom Noddy introduced America and the world to Bubble Magic via television in the early 80’s. Before that, he spent a decade inventing and developing this astounding art.” Stage name of Tom McAllister, Tom Noddy spent almost a decade constructing a new type of performance piece before initially bringing it to television in the early…

Creating a self-assembling quasicrystal

A quasiperiodic crystal, also known as a quasicrystal, is an organized but nonperiodic structure. A quasicrystalline design may occupy all available space indefinitely, although it lacks translational symmetry. While crystals can only have two-, three-, four-, and six-fold rotational symmetries, the Bragg diffraction pattern of quasicrystals reveals distinct peaks with additional symmetry orders, such as…

Measuring devices from the time of Stonehenge: The chalk drums from Folkton and Lavant

Investigation on mathematical understanding and the usage of conventional units of measurement in prehistoric civilizations show that there is a clear relationship between the design of Stonehenge and the chalk artifacts known as the Folkton and Lavant Drums, where the Drums reflect measuring standards that were required for precise and reproducible monument building. This has…

The geometry of an electron

University of Basel physicists have demonstrated for the first time how a single electron appears in an artificial atom. They can now indicate the likelihood of an electron being present in a space using a newly discovered technique. This enables better control of electron spins, which might be the smallest information unit in a future…

The math behind musical harmony: phase transitions and balance between order and disorder

Music, while allowing nearly unlimited creative expression, almost always conforms to a set of rigid rules at a fundamental level. Jesse Berezovsky, an associate professor of physics at Case Western Reserve University, defines the “emergent structures of musical harmony” inherent in the art, just as order comes from disorder in the physical world. I present a…