non-Euclidean Geometry Research Articles

Число и кантианская исследовательская программа в современной нейронауке

The article presents the attempt to consider the Kantian research program in modern neuroscience in its part, which relates to the representation of the “number” and the mechanisms for processing numerical information by neu­ral structures. We claim that Kantian ideas about the a priori nature of certain mathematical categories related to the status of space and time [geometry and arithmetic], which were subjected to doubt as a result of the discovery of non-Euclidean geometries, proved to be highly demanded and reassessed as a result of the intensive progress of modern cognitive and cultural neuroscience. The dis­covery of the subitizing phenomena, “sense of number” and “sense of place” (analogous to the navigation system of the brain) push us to recall the old Kan­tian judgments concerning certain a priori constructions of mathematics. The on­togenetic foundations of such phenomena, their conditionality by the features of the functioning of the brain, reveal not the metaphorical, but the strategic nature of the Kantian research program in modern neuroscience. In the context of these studies, it also turned out that in the case of living systems, one can speak about their proto-arithmetic traits, and in the case of humans, mathematical abilities that are largely independent of the language, and their systematic development from an early age significantly increases the likelihood of successful mathemati­cal activity in future. Attention drawn to the interdependence of the activity of the developing brain, social and cultural contexts, which intersects and expressed in the process of acculturation of the brain and vice versa – neural determination of culture. This kind of interaction support the idea of the possibility of expand­ing original Kantian idea and introducing the idea related to the transcendental­ism of the activity type

On Some Properties of Distance in TO-Space

The aim of this work is to investigate some properties of the truncated octahedron metric introduced in the space in further studies on metric geometry. With this metric, the 3-dimensional analytical space is a Minkowski geometry which is a non-Euclidean geometry in a finite number of dimensions. In a Minkowski geometry, the unit ball is a certain symmetric closed convex set instead of the usual sphere in Euclidean space. The unit ball of the truncated octahedron geometry is a truncated octahedron which is an Archimedean solid. In this study, first, metric properties of truncated octahedron distance, d_TO, in R^2 has been examined by metric approach. Then, by using synthetic approach some distance formulae in R_TO^3, 3-dimensional analytical space furnished with the truncated octahedron metric has been found.

Learning to dance: A graph convolutional adversarial network to generate realistic dance motions from audio

Synthesizing human motion through learning techniques is becoming an increasingly popular approach to alleviating the requirement of new data capture to produce animations. Learning to move naturally from music, i.e., to dance, is one of the more complex motions humans often perform effortlessly. Each dance movement is unique, yet such movements maintain the core characteristics of the dance style. Most approaches addressing this problem with classical convolutional and recursive neural models undergo training and variability issues due to the non-Euclidean geometry of the motion manifold structure. In this paper, we design a novel method based on graph convolutional networks to tackle the problem of automatic dance generation from audio information. Our method uses an adversarial learning scheme conditioned on the input music audios to create natural motions preserving the key movements of different music styles. We evaluate our method with three quantitative metrics of generative methods and a user study. The results suggest that the proposed GCN model outperforms the state-of-the-art dance generation method conditioned on music in different experiments. Moreover, our graph-convolutional approach is simpler, easier to be trained, and capable of generating more realistic motion styles regarding qualitative and different quantitative metrics. It also presented a visual movement perceptual quality comparable to real motion data. The dataset and project are publicly available at: https://www.verlab.dcc.ufmg.br/motion-analysis/cag2020.

Demonstration in geometry: historical and philosophical perspectives

In this article, we weave historical-philosophical reflections about demonstration in mathematics, based on works of researchers that discuss the different philosophical perspectives on the topic, more specifically on geometry. We focus first on demonstration and its relationship with intuition and figural representations. Second, we criticize Poincaré’s conception of mathematical demonstration. Third, we reflect, in a non-exhaustive way, on the philosophy of demonstration in geometry, confronting Kant’s conceptions with the axiomatizations of the non-Euclidean geometries. In this text, we do not adopt a single definition that would cover all modes of scientific validation, since we admit the possibility of an evolution of ideas about the validity of a proposition. Not to fall into the symmetrical flaws of the glorification of the Ancients or even being ungrateful to them, we must start from the naive idea that the demonstration has a historical origin and, therefore, maintains a historical character, but we should be more attentive to what characterizes, in its particularity or even its uniqueness, the productions of past and present centuries. Keywords: Philosophy of demonstration; Axiomatization; Induction; Intuition; Representation.

Asymptotic normality for random polytopes in non-Euclidean geometries

Asymptotic normality for the natural volume measure of random polytopes generated by random points distributed uniformly in a convex body in spherical or hyperbolic spaces is proved. Also the case of Hilbert geometries is treated and central limit theorems in Lutwak’s dual Brunn–Minkowski theory are established. The results follow from a central limit theorem for weighted random polytopes in Euclidean spaces. In the background are Stein’s method for normal approximation and geometric properties of weighted floating bodies.

Learnable Pooling in Graph Convolutional Networks for Brain Surface Analysis.

Brain surface analysis is essential to neuroscience, however, the complex geometry of the brain cortex hinders computational methods for this task. The difficulty arises from a discrepancy between 3D imaging data, which is represented in Euclidean space, and the non-Euclidean geometry of the highly-convoluted brain surface. Recent advances in machine learning have enabled the use of neural networks for non-Euclidean spaces. These facilitate the learning of surface data, yet pooling strategies often remain constrained to a single fixed-graph. This paper proposes a new learnable graph pooling method for processing multiple surface-valued data to output subject-based information. The proposed method innovates by learning an intrinsic aggregation of graph nodes based on graph spectral embedding. We illustrate the advantages of our approach with in-depth experiments on two large-scale benchmark datasets. The ablation study in the paper illustrates the impact of various factors affecting our learnable pooling method. The flexibility of the pooling strategy is evaluated on four different prediction tasks, namely, subject-sex classification, regression of cortical region sizes, classification of Alzheimer's disease stages, and brain age regression. Our experiments demonstrate the superiority of our learnable pooling approach compared to other pooling techniques for graph convolutional networks, with results improving the state-of-the-art in brain surface analysis.

Material Affine Connections for Growing Solids

The present paper aims to develop geometrical approach for finite incompatible deformations arising in growing solids. The phenomena of incompatibility is modeled by specific affine connection on material manifold, referred to as material connection. It provides complete description of local incompatible deformations for simple materials. Meanwhile, the differential-geometric representation of such connection is not unique. It means that one can choose different ways for analytical definition of connection for single given physical problem. This shows that, in general, affine connection formalism provides greater potential than is required to the theory of simple materials (first gradient theory). For better understanding of this inconsistence it is advisable to study different ways for material connection formalization in details. It is the subject of present paper. Affine connection endows manifolds with geometric properties, in particular, with parallel transport on them. For simple materials the parallel transport is elegant mathematical formalization of the concept of a materially uniform (in particular, a stress-free) non-Euclidean reference shape. In fact, one can obtain a connection of physical space by determining the parallel transport as a transformation of the tangent vector, which corresponds to the structure of the physical space containing shapes of the body. One can alternatively construct affine connection of material manifold by defining parallel transport as the transformation of the tangent vector, in which its inverse image with respect to locally uniform embeddings does not change. Utilizing of the conception of material connections and the corresponding methods of non-Euclidean geometry may significantly simplify formulation of the initial-boundary value problems of the theory of incompatible deformations. Connection on the physical manifold is compatible with metric and Levi-Civita relations holds for it. Connection on the material manifold is considered in three alternative variants. The first leads to Weitzenbock space (the space of absolute parallelism or teleparallelism, i.e., space with zero curvature and nonmetricity, but with non-zero torsion) and gives a clear interpretation of the material connection in terms of the local linear transformations which transform an elementary volume of simple material into uniform state. The second one allows to choose the Riemannian space structure (with zero torsion and nonmetricity, but nonzero curvature) in material manifold and it is the most convenient way for deriving of field equations. The third variant is based on Weyl manifold with specified volume form and non-vanishing nonmetricity.

Liquid crystal director fields in three-dimensional non-Euclidean geometries

This paper investigates nematic liquid crystals in three-dimensional curved space, and determines which director deformation modes are compatible with each possible type of non-Euclidean geometry. Previous work by Sethna et al showed that double twist is frustrated in flat space , but can fit perfectly in the hypersphere . Here, we extend that work to all four deformation modes (splay, twist, bend, and biaxial splay) and all eight Thurston geometries. Each pure mode of director deformation can fill space perfectly, for at least one type of geometry. This analysis shows the ideal structure of each deformation mode in curved space, which is frustrated by the requirements of flat space.

Topological Hyperbolic Lattices.

Non-Euclidean geometry, discovered by negating Euclid's parallel postulate, has been of considerable interest in mathematics and related fields for the description of geographical coordinates, Internet infrastructures, and the general theory of relativity. Notably, an infinite number of regular tessellations in hyperbolic geometry-hyperbolic lattices-are expected to extend Euclidean Bravais lattices and the consequent wave phenomena to non-Euclidean geometry. However, topological states of matter in hyperbolic lattices have yet to be reported. Here we investigate topological phenomena in hyperbolic geometry, exploring how the quantized curvature and edge dominance of the geometry affect topological phases. We report a recipe for the construction of a Euclidean photonic platform that inherits the topological band properties of a hyperbolic lattice under a uniform, pseudospin-dependent magnetic field, realizing a non-Euclidean analog of the quantum spin Hall effect. For hyperbolic lattices with different quantized curvatures, we examine the topological protection of helical edge states and generalize Hofstadter's butterfly, by employing two empirical parameters that measure the edge confinement and defect immunity. We demonstrate that the proposed platforms exhibit the unique spectral-magnetic sensitivity of topological immunity in highly curved hyperbolic planes. Our approach is applicable to general non-Euclidean geometry and enables the exploitation of infinite lattice degrees of freedom for band theory.

Probabilistic collaborative representation on Grassmann manifold for image set classification

For image-set based classification, sparse coding and collaborative representation have gained a lot of attention due to their robustness and effectiveness. However, most existing methods focus on collaborative representation in Euclidean space. It still remains a research gap to handle this problem from Geometry-Aware perspective and interpret the mechanism of collaborative representation on nonlinear manifold. In this paper, we propose a novel method named probabilistic collaborative representation on Grassmann manifold for image set classification, which is interpreted from a probabilistic viewpoint. Specifically, we regard each image set as a point on Grassmann manifold inspired by its non-Euclidean geometry and then perform collaborative representation on the space of symmetric matrices, which enables us to explain the internal mechanism of classification and derive a closed form solution. Moreover, classification criterion is designed to further improve the performance of the proposed method. Experimental results on four databases (i.e. Honda/UCSD, YaleB, Youtube Celebrities and ETH-80) for face recognition task and object recognition task demonstrate the robustness and effectiveness of our proposed method.

Solid angles in perspective

The specialised uses of solid angles mean that they are quite unfamiliar quantities. This article, apart from making solid angles a little more familiar, brings out several topics of general interest, such as how units are interrelated and how equations depend on the choice of units. Although the steradian is commonly used as the unit for solid angle, another unit, the square degree, is used in astronomy, and a unit introduced here, the solid degree (with 360 solid degrees in a hemisphere) could be used with benefits that are similar to those of the degree when it is used as the unit for plane angle. The article, which is suitable for students at A-level and introductory undergraduate level, also shows how solid angles can provide a gentle introduction to crystal structure, spherical trigonometry and non-Euclidean geometry.

Counting Form: Gender and the Geometries of Address, in Frances Presley and Carol Watts

This essay treats two innovative site-specific sequences produced by women in the first decade of the twenty first century. Both are explicitly interested in the relationship between geometry, writing (as material and political practice) and geo-cultural space, a relationship each finds inflected to some extent by gender emphases. Starting from the premise that any piece of writing is itself a place, the essay considers the self-conscious textualities of its primary texts—one concerned with Exmoor; the other with a sheep-farm in rural mid-Wales—in the light of their different, if similarly rural and relatively remote, contexts. Presley’s ‘Stone Settings’ explores the relationship between some of the quasi-geometrical Neolithic stone arrangements dotted across Exmoor, and the mediation of their apparently Euclidean sometimes barely visible forms in/as text. Watts’ work-in-progress Zeta Landscape mobilises in the ‘analytical’ or ‘projective’ (ie non-Euclidean) geometry of its title the complex weave of routine care-giving and accountancy charging the contemporary (Michel Foucault’s ‘distributive’) pastoral. Both sequences wryly suggest that poetic form can finally no more adequately figure place than the abstractions of mathematical discourse can utter the cultural ecology of any environment, however concrete-seeming. Aided by Jacques Derrida’s powerful essay ‘White Mythology’, the account comes to rest on the equally equivocal recognition of the in/effectuality of metaphor in any kind of address, critical or creative.

3D Laser Forming of Metal Foam Sandwich Panels

Abstract Metal foam sandwich panels have been the subject of many concept studies, due to their exceptional stiffness, light weight, and crash absorption capacity. Yet, the industrial production of the material has been hampered by the fact that it is challenging to bend the material into practical engineering shapes. Only recently, it has been shown that bending of metal foam sandwich panels is possible using lasers. It was also shown that the material can be bent into Euclidean (2D) geometries, and the governing laser-induced bending mechanisms were analyzed. This study was focused on laser forming of metal foam sandwich panels into non-Euclidean (3D) geometries. It was investigated whether the bending mechanisms and process parameters identified for 2D laser forming translate to 3D deformation. Additionally, the impact of the laser scan length was determined by comparing different scan patterns that achieve the same 3D geometries. It was shown that laser forming could induce 3D deformation necessary for both bowl and saddle shapes, the two fundamental non-Euclidean geometries. The amount of laser-induced bending and in-plane strains vary depending on process conditions and the governing bending mechanisms. Lastly, the laser scan length was shown to become more important for metal foam sandwich panels, where the panel thickness tends to be large.

Koch Fractal in Non-Euclidean Geometries

Koch Fractal in Non-Euclidean Geometries

ШИРОКОСМУГОВІ АНТЕНИ НА ОСНОВІ КІЛЬЦЕВОЇ ГЕОМЕТРІЇ

В роботі розглянуто варіанти моделей квазіфрактальних кільцевих структур. Для їх синтезу запропонований векторний опис фрактальної трансформації окремих сегментів початкової геометричної форми. Поєднання декомпозиції геометричної форми та багатовимірного фрактального підходу дозволяє спростити синтез фрактальних 3D-структур та/або використовувати кілька видів фракталів, у тому числі, з різною кількістю ітерацій. Такий підхід дозволяє досягти широкосмуговості та багатодіапазонності антенних систем. При цьому досліджено вплив рівня сегментації та варіантів схеми живлення на просторово-частотні характеристики антени. Через складність опису взаємодії антен неевклідової геометрії з радіохвилями для їх синтезу та аналізу обрано методи чисельного моделювання. Для аналізу просторово-частотних характеристик проектованих антенних рішень використані такі показники, як зворотні втрати, діаграма спрямованості та коефіцієнт стоячої хвилі. Отримані оцінки відносної смуги пропускання δf = 0,9 свідчать про перспективність запропонованого підходу для реалізації антенних систем з високим рівнем широкосмуговості

JUSTIFICATION OF INDIRECT INSOLUBLE ACHILLES APORIA BY VISUAL REPRESENTATION OF THE NON-EUCLIDEAN GEOMETRY

To justify the insoluble aporia of the Achilles, the author uses Klein's model of visual rep-resentation of non-Euclidean geometry, according to which unequal and at the same time diminishing segments in the Euclidean space correspond to equal segments in the non-Euclidean one. However, this justification, by necessity, is only the nature of appearance, the reverse of which is the ambiguity of the implicitly used principle of Parmenides on the identity of thinking and being.

Non-Euclidean Geometry in Russian Art History: On a Little-Known Application of a Scientific Theory

The author has previously proposed that there are at least six different definitions of “reverse” or “inverse” perspective, i.e. the principle of organizing pictorial space in the icon. Reverse perspective is still a largely unresolved art historical problem. The author focuses on one of the six defi nitions, the one least familiar to Western scholars—namely, the view, common in Russian art-historical writing at the beginning of the twentieth century, that space in the icon is a visual analogue of non-Euclidean geometry. Russian mathematician-turned-theologian and priest Pavel Florensky claimed that the space of the icon is that of non-Euclidean geometry and truer to the way human vision functions. The author considers the scientifi c validity of Florensky's claim.

Image Set Classification Using a Distance-Based Kernel Over Affine Grassmann Manifold.

Modeling image sets or videos as linear subspaces is quite popular for classification problems in machine learning. However, affine subspace modeling has not been explored much. In this article, we address the image sets classification problem by modeling them as affine subspaces. Affine subspaces are linear subspaces shifted from origin by an offset. The collection of the same dimensional affine subspaces of [Formula: see text] is known as affine Grassmann manifold (AGM) or affine Grassmannian that is a smooth and noncompact manifold. The non-Euclidean geometry of AGM and the nonunique representation of an affine subspace in AGM make the classification task in AGM difficult. In this article, we propose a novel affine subspace-based kernel that maps the points in AGM to a finite-dimensional Hilbert space. For this, we embed the AGM in a higher dimensional Grassmann manifold (GM) by embedding the offset vector in the Stiefel coordinates. The projection distance between two points in AGM is the measure of similarity obtained by the kernel function. The obtained kernel-gram matrix is further diagonalized to generate low-dimensional features in the Euclidean space corresponding to the points in AGM. Distance-preserving constraint along with sparsity constraint is used for minimum residual error classification by keeping the locally Euclidean structure of AGM in mind. Experimentation performed over four data sets for gait, object, hand, and body gesture recognition shows promising results compared with state-of-the-art techniques.

Entre palavras e imagens de um mundo surreal: pensamentos matemáticos em uma experiência com crianças e Salvador Dalí

Inaugurada por Flores (2013), a “perspectiva da visualidade em educação matemática” surge como uma estratégia de análise e alargamento das pesquisas sobre o visual, discutindo como nosso olhar foi constituído historicamente e formas outras de pensar a relação da arte com a matemática. O presente artigo é a escrita de uma experiência com crianças no encontro com a arte de Salvador Dalí, com o objetivo de experimentar e problematizar uma geometria não-euclidiana junto à sua obra. Usando a ideia de oficina-dispositivo, criou-se uma oficina a partir de alguns aspectos da obra do pintor que tinha a intenção de provocar visualidades, fazendo emergir pensamentos e saberes matemáticos. A partir da escuta atenta a uma história que narrava características de um mundo surreal, crianças do 5º ano do Ensino Fundamental do Colégio de Aplicação da Universidade Federal de Santa Catarina foram convidadas a imaginar e construir o seu próprio mundo, fazendo colagens de figuras em caixas de ovo. A colagem, na superfície ondulada da caixa, causou inquietação e fez com que pensamentos matemáticos emergissem do estranhamento com uma geometria não-euclidiana. Durante a oficina, puderam ser observadas narrativas que ligavam a deformação das figuras a algo errado ou feio, e também tentativas de não-deformação das figuras através de estratégias diversas. Além disso, de outro modo, a oficina efetivou um espaço em que o ensino e a aprendizagem da matemática elaboram-se a partir dos sentidos da experiência, problematizando conexões com o real e o surreal.

The ‘Cardinality of the Continuum’ Is False in Non-Euclidean Geometries

According Georg Cantor's Cardinality of the Continuum, as proved in Euclidean geometry, any segment (not inclusive of its two end points) has as many points as an entire line. This theorem is an underlying assumption of the Hypothesis, that conjectures that every infinite subset of the continuum of real numbers $\mathbb{R}$ is either equivalent to the set of natural numbers $\mathbb{N}$, or to $\mathbb{R}$ itself. However, the proof in Euclidean geometry uses parallel lines, and therefore assumes that Euclid's Parallel Postulate is true. But in Non-Euclidean geometries, Euclid's Parallel Postulate is false. In Hyperbolic geometry, at any point off of a given line, there are a plurality of lines parallel to the given line. In Elliptic geometry (which includes Spherical geometry), no lines are parallel, so two lines always intersect. Absolute geometry has neither Euclid's parallel postulate nor either of its alternatives. We provide examples in Spherical geometry and in Hyperbolic geometry where the Cardinality of the Continuum is false. Therefore it is also false in Absolute geometry. So it is a false proposition, as is the Hypothesis, that falsely assumes that the Cardinality of the Continuum is true. Concepts in theory, such as the number line of real numbers, cannot be limited to Euclidean geometry.