Geometry Matters

The shape of neural representations might be essential for understanding how the brain supports diverse types of flexible behavior:

The geometry of abstractions - Geometry Matters Many forms of cognitive, emotional, and behavioral flexibility place diverse demands on brain representations, ranging from generalization in unfamiliar settings to the capacity to create a wide variety of responses to complicated patterns of inputs. According to research done by a team at Columbia....

Nikos Salingaros' architectural language is presented as a series of forms that can be used to guide the design of buildings and spaces, with a specific focus on the geometry and form of the buildings and spaces to create a sense of relaxation, calm, and well-being:

The Geometry of Control in Architectural Design - Geometry Matters Nikos Salingaros is a mathematician and architectural theorist who has developed a theory of architecture that emphasizes the importance of socially-organized housing and the geometry of control. Salingaros argues that traditional architecture, which is based on symmetrical and hierarchical designs,...

Tirthabir Biswas and James E. Fitzgerald introduce a geometric framework that can be applied to neural activity data to make rigorous anatomical predictions that follow generally from the model architecture.

Geometric framework predicts structure from neural functions - Geometry Matters Tirthabir Biswas and James E. Fitzgerald introduce a geometric framework for identifying synaptic connections required by steady-state responses in recurrent networks of threshold-linear neurons.

The effect of the shapes we perceive is well-known and utilized in various domains. For example, the Bouba/Kiki effect is an effect that allows cartoon or animation characters to be designed with specific emotional attributes. The question of whether individuals consistently ascribed emotions to static forms is based on the appearance of dynamic forces between these forms. Irena Pavlova, Arseny Sokolov, and Alexander Sokolov conducted research on emotions and dynamics in single forms in 2005.

Perceived dynamics of static images enable emotional attribution - Geometry Matters The effect of the shapes we perceive is well-known and utilized in various domains. For example, the Bouba/Kiki effect is an effect that allows cartoon or animation characters to be designed with specific emotional attributes. The question of whether individuals consistently ascribed emotions to sta...

Although consciousness is a multifaceted phenomena, important aspects like awareness and alertness have only been theoretically rather than neurobiologically defined. According to a novel theory, different neurofunctional dimensions of the brain contain aspects of consciousness that may be detected and measured by variations in blood flow across time.

Geometry of Brain, Dimensions of Mind - Geometry Matters Although consciousness is a multifaceted phenomena, important aspects like awareness and alertness have only been theoretically rather than neurobiologically defined. According to a novel theory, different neurofunctional dimensions of the brain contain aspects of consciousness that may be detected....

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.

A multidisciplinary team examines the most important theoretical and practical breakthroughs in network geometry techniques, as well as future research directions and problems in this frontier of complexity research.

The fractal-hyperbolic geometry of networks - Geometry Matters 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 approache...

In a 2020 study, Lee et al. searched 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).

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. 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.

The human quest for discovering mathematical beauty in the arts - Geometry Matters Lee et al. 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. They use a computational approach to divide each painting in their collection into the most color uniform vertical and horizo...

Albert Einstein in his lecture before the Prussian Academy of Sciences, 27 January 1921.

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.

"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

Pure geometry and geometric cognition - Geometry Matters 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. H...

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.

Geometrical concepts, cognition, and educational implications - Geometry Matters 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.

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 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.

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.

Patterns of innovation - Geometry Matters 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 patt...

"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."

Einstein, geometry, and experience 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.

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.

Geometric models for memories - Geometry Matters

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.

Hologram of a single photon - Geometry Matters

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.

A language of thought for the mental representation of geometric shapes - Geometry Matters A team of researchers formalizes and tests 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....

It turns out, even larger systems that seem un-organized, have a hidden order.

Hidden order in disorder - Geometry Matters 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.

Earlier this year, 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 coronaviruses.

Icosahedral nano-shell designed to trap virus particles - Geometry Matters 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-associa...

Humans are unique among primates in their capacity to construct and control very complex systems of language, mathematics, and music. According to the findings of Sablé-Meyer et al., the human proclivity for form perception and symbolic abstraction provides a signature of human singularity.

Intuitions of geometry: a signature of human singularity - Geometry Matters 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...

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.

Complex thoughts are enabled by fractal networks - Geometry Matters 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...

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.

The shape of a perfect egg, defined by a universal formula - Geometry Matters 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 a...

A geometric model that offers an intuitive and plausible description of the emergence of form via simple geometrical and mechanical forces and constraints.

Embryo geometry: a theory of evolution from a single cell to the complex vertebrate body - Geometry Matters "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 geo...

Unlike a picture, which is all laid out so that you can instantly see the structure, music is fundamentally a serial phenomenon. With music, the whole piece takes shape in your mind. This makes it more challenging to identify self-similarity.

Hunting Bach's fractals - Geometry Matters 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....

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.

Mathematical skills improved by tri-dimensional thinking - Geometry Matters % 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. I...

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.

Entropy and energy influence polygonal nets folding into Platonic solids - Geometry Matters What makes an object successful at folding? Protein scientists study how an object transforms between 2D surfaces, called nets, 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....

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.

Using hyperbolic geometry to map the olfactory space - Geometry Matters 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. Because biological activities produce numerous by-products, the sense of smell can be utilize...

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.

Fractal patterns preferred by children under three years - Geometry Matters 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 youngsters re...

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.

Neurons use fractal networks for better connectivity - Geometry Matters 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 th...

By studying their shapes and fragmentation patterns, a team of researchers discovered that the average of all the shapes and fragmentation patterns of various rocks is a cube.

Plato's cube is nature's preferred choice for fragmentation. - Geometry Matters A group of researchers from the University of Pennsylvania, Budapest University of Technology and Economics, and the University of Debrecen combine arithmetic, geology, and physics to prove that the average of all the shapes and fragmentation patterns of various rocks is a cube.

Hans Jenny was a natural scientist and physician who coined the term cymatics to explain the acoustic impacts of sound wave phenomena.

"If you spear a little of your imagination, as you watch this film as it runs, you will see many things that answers many questions, you will see living forms, living amoeba, almost animal like creatures. You will see continents being formed, the Earth itself coming to existence, explosions, erruptions, atomic explosions and bombs. You can see all this and watch it before your eyes. Everything owes it existence solely and completely to sound.

Hans Jenny and the science of sound: cymatics. - Geometry Matters 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, tha...