non-Euclidean Geometry Research Articles

Distortions in Classical Mechanics as a Consequence of Euclidean Geometry Utilization

We live on a huge sphere. When the sum of interior angles in a triangle is greater than [Formula: see text] and each point has a different local vertical, we must use spherical geometry. Euclidean and non-Euclidean geometries are non-equivalent systems, with their own premises and completely different foundations. The use of any logic and tool from orthogonal into non-orthogonal realities (and vice versa), leads to big errors. Newtonian concepts are treated as absolutes in classical mechanics. Content analysis shows that even recent physics publications have inadmissible distortions in the description of natural phenomena in the domain of classical mechanics, inherited through the utilization of the Euclidean model. The whole objective reality has been presented using the tools of Euclidean geometry. When we use the tools of spherical geometry to examine the same phenomena, we get very different findings. The Law of conservation of energy appears as the most efficient tool for the evaluation of these significantly different findings. It confirms the validity of the new discoveries presented in this paper and gives legitimacy to the whole project! Classical mechanics becomes more consistent with the conservation laws of Physics, and the subject’s coherence is lifted to a significantly higher level.

Cognitive Maps for a Non-Euclidean Environment: Path Integration and Spatial Memory on a Sphere

Humans build mental models of the world and utilize them for various cognitive tasks. The exact form of cognitive maps is not fully understood, especially for novel and complex environments beyond the flat Euclidean environment. To address this gap, we investigated path integration—a critical process underlying cognitive mapping—and spatial-memory capacity on the spherical (non-Euclidean) and planar (Euclidean) environments in young healthy adults (N = 20) using immersive virtual reality. We observed a strong Euclidean bias during the path-integration task on the spherical surface, even among participants who possessed knowledge of non-Euclidean geometry. Notably, despite this bias, participants demonstrated reasonable navigation ability on the sphere. This observation and simulation suggest that humans navigate nonflat surfaces by constructing locally confined Euclidean maps and flexibly combining them. This insight sheds light on potential neural mechanisms and behavioral strategies for solving complex cognitive tasks.

On the Preeminence of Euclidean Geometry: Nash’s Embedding Theorems

AbstractAccording to Kant’s philosophy of geometry, Euclidean geometry is synthetic a priori. The advent of non-Euclidean geometries proved this position at least problematic, if not obsolete. However, based on Nash’s embedding theorems we show that a weaker notion of preeminence supports the view that Euclidean geometry, even though not strictly a priori, enjoys a more fundamental status than non-Euclidean geometries.

On Geodesic Triangles in Non-Euclidean Geometry

On Geodesic Triangles in Non-Euclidean Geometry

Hyperbolic prototype rectification for few-shot 3D point cloud classification

Few-shot point cloud classification is a challenging task in 3D computer vision and has received widespread attention from researchers. Most of the works on deep learning models rely heavily on Euclidean spatial metrics. However, point cloud objects often have complex non-Euclidean geometric structures, with underlying inter/intra-class hierarchical structures, which are difficult to capture by current Euclidean-based deep learning models. Moreover, due to the lack of training samples, many few-shot learning methods often suffer from the overfitting problem. Given the Hyperbolic metric of non-Euclidean geometry offering hierarchical structural prior, as we assume to be able to assist FSL task, we propose Hyperbolic Prototype Rectification (HPR) for few-shot point cloud classification, without requiring extra learnable parameter. Firstly, point clouds are embedded into hyperbolic space to better describe hierarchical similarity relationships in data. Secondly, the HPR utilizes hyperbolic spatial and distributional information to enhance the feature representation and improve the generalization capability, with more appropriate hyperbolic prototypes. The few-shot classification experiments and further ablation studies conducted on widely used point cloud datasets demonstrate the effectiveness of our method. On the real-world ScanObjectNN(-PB) datasets, the average classification accuracy outperforms the SOTA method by 2.08%(0.66%), respectively, indicating that the proposed HPR has great generalization capability and strong robustness against perturbed data. Our code is available at: https://github.com/Jonathan-UCAS/HPR.

Self-Assembly and Transport Phenomena of Colloids: Confinement and Geometrical Effects

AbstractColloidal dispersions exhibit rich equilibrium and nonequilibrium thermodynamic properties, self-assemble into diverse structures at different length scales, and display transport behavior under bulk conditions. In confinement or under geometrical restrictions, new phenomena emerge that have no counterpart when the colloids are embedded in an open, noncurved space. In this review, we focus on the effects of confinement and geometry on the self-assembly and transport of colloids and fluidized granular systems, which serve as model systems. Our goal is to summarize experiments, theoretical approximations and molecular simulations that provide physical insight on the role played by the geometry at the mesoscopic scale. We highlight particular challenges, and show preliminary results based on the covariant Smoluchowski equation, that present promising avenues to study colloidal dynamics in a non-Euclidean geometry.

Editorial Foreword

Non-Euclidean Geometry in Modern Physics and Mathematics is a series of biennial conferences initiated by the Bogolyubov Institute for Theoretical Physics of the National Academy of Sciences of Ukraine in 1997, dedicated to the heritage of J´anos Bolyai, Carl Friedrich Gauss, and Nikolai Ivanovich Lobachevski (BGL, alphabetically), and held at various sites connected with the founders of the new geometry; for history see: https://indico.cern.ch/event/586799/page/8964-former-blg-conferences.

Geodesic conformal gradient device based on a torus.

In recent years, optical fields on non-Euclidean geometry have become a hot topic. The geodesic conformal transformation theory, linking curved surfaces with planar gradient refractive indices, holds unique advantages in controlling curved optical fields. However, this theory has not yet addressed surfaces with non-trivial topology with a certain genus. In this work, we design a gradient planar device based on the geodesic conformal transformation theory for toroidal surfaces, which can achieve Gaussian beam focusing. Unlike traditional angle-preserving geodesic theories, the non-zero genus results in the one-to-two discontinuous boundaries in the device, and we utilize inversion transformations to rectify this drawback.

Bilateral Symmetric non-Euclidean multi-frequency invisibility.

Light propagation in non-Euclidean geometry has become a hot topic in recent years, while transformation optics theory demonstrates unique advantages in this respect. A notable application of transformation optics in non-Euclidean space is non-Euclidean invisibility cloak which avoids the challenges of negative refraction and anisotropic materials. In this work, we propose another configuration for non-Euclidean invisibility, capable of achieving invisible across a wide spectrum. Using coordinate transformation, we convert this non-Euclidean invisibility into planar gradient medium and validate its effects through full wave simulations. We also discover that the corresponding gradient medium can further relax the material parameters. Our findings suggest diverse strategies for non-Euclidean invisibility and planar gradient media, potentially advancing optical invisibility and transformation optics in non-Euclidean spaces.

Dirac walks on regular trees

The study of matter fields on an ensemble of random geometries is a difficult problem still in need of new methods and ideas. We will follow a point of view inspired by probability theory techniques that relies on an expansion of the two point function as a sum over random walks. An analogous expansion for Fermions on non-Euclidean geometries is still lacking. Casiday et al (2022 Linear Multilinear Algebr. 72 325–65) proposed a classical ‘Dirac walk’ diffusing on vertices and edges of an oriented graph with a square root of the graph Laplacian. In contrast to the simple random walk, each step of the walk is given a sign depending on the orientation of the edge it goes through. In a toy model, we propose here to study the Green functions, spectrum and the spectral dimension of such ‘Dirac walks’ on the Bethe lattice, a d-regular tree. The recursive structure of the graph makes the problem exactly solvable. Notably, we find that the spectrum develops a gap and that the spectral dimension of the Dirac walk matches that of the simple random walk ( ds=1 for d = 2 and ds=3 for d⩾3 ).

Non-Euclidean geometries in and for the mathematics teacher education: A literature review

Science challenges the relevance of school geometry in its treatment of space. Therefore, the research discusses the incorporation of non-Euclidean geometries in the initial and continuing education of mathematics teachers. Hence, this literature review aims to synthesize and describe, in a broad and current range, the recent state of research in educational mathematics on these geometries in teacher education. To do this, we configured a method in three moments: the search, evaluation, selection, and organization of the sources found and reading, analyzing, and writing results. The results highlight a regional and temporal interest that prioritizes teaching practice. Furthermore, a deficit and prescriptive perspective of teachers' knowledge and a minority perspective of the teacher as a subject of knowledge are recognized. We conclude with the need for research on various phenomena, including Euclidean generalizations.

«Здесь геометрия становится уже религией»: метафорика неэвклидовой геометрии в теоретических построениях Д.С. Мережковского

The article examines Dmitry Merezhkovsky’s tendency – previously unnoticed, but variously shown – to present his theoretical reasoning through images and concepts referring to non-Euclidean geometry, as well as through the concepts of multidimensional spaces, mystically interpreted. It is shown that such an approach is a stable figurative-theoretical method of Dmitry Merezhkovsky, to which he resorts in a large number of works – from the earliest to the lates. Geometric and spatial metaphorics dating back to Fyodor Dostoevsky and Vladimir Solovyov, and actively engaged by his contemporaries, is an essential feature of his worldview. It is emphasized that, in contrast to the authors who used the concepts of the “fourth dimension” and “non-Euclidean space” to illustrate occult, mediumistic or astral concepts, Merezhkovsky applies them to Christian mystics, describing the phenomena of “miracle”, “kingdom of God”, “eternal life” and so on. The evolution of the corresponding views of D. Merezhkovsky from 1907 to 1939 is traced, the sources that influenced their formation are established, among which are Fyodor Dostoevsky, Nikolai Bugaev, Anton Kartashev, Ernest Renan, Christian apocrypha. It is noted that in a number of cases the corresponding reasoning and images of Merezhkovsky were reflected in the works of Andrei Bely.

Conceptualization and Acceptance of Non-Euclidean Geometries

Geometry is made up of theorems and propositions that are based on postulates induced by experience. Until the 18th century, the fundamental postulates of Greek geometry were five. However, the fifth postulate became the most studied and discussed. The objective is to demonstrate that the great debates were fruitful and that the impossibility of proving the fifth postulate led to the creation of other consistent and possible geometries. The systemic and analytical method that we use allows us to see the real applications of non-Euclidean geometries, atomic nuclei, speed of sea waves, catenary, air routes, notable elements and others. Finally, relativistic theories about the configuration of space and time lead to the conclusion, the great challenge of determining whether real physical space is a Euclidean space.

A Lesson with Francis Bacon Forced Me to See Outside the Software Box

Abstract Computers became more standard in the early 1980s, and science and technology were influencing artists like the author. He began using a supercomputer and FORTRAN 77 software to define a non-Euclidean geometry model using conformal mapping to create art. A series of twisted and curved space images was manifested, reflective of Albert Einstein’s theories of general and special relativity. It was the author’s encounter with Edward Teller, “father of the H-bomb,” that led the author to artist Francis Bacon. It was a private lesson with the artist that forced the author out of the box and into his own visual world.

Statistical inference for wavelet curve estimators of symmetric positive definite matrices

In this paper we treat statistical inference for a wavelet estimator of curves of symmetric positive definite (SPD) using the log-Euclidean distance. This estimator preserves positive-definiteness and enjoys permutation-equivariance, which is particularly relevant for covariance matrices. Our second-generation wavelet estimator is based on average-interpolation (AI) and allows the same powerful properties, including fast algorithms, known from nonparametric curve estimation with wavelets in standard Euclidean set-ups. The core of our work is the proposition of confidence sets for our AI wavelet estimator in a non-Euclidean geometry. We derive asymptotic normality of this estimator, including explicit expressions of its asymptotic variance. This opens the door for constructing asymptotic confidence regions which we compare with our proposed bootstrap scheme for inference. Detailed numerical simulations confirm the appropriateness of our suggested inference schemes.

Art etching of graphene.

The growth of graphene on a metal substrate using chemical vapor deposition (CVD), assisted by hydrocarbons such as CH4, C3H8, C2H6, etc. leads to the formation of carbon clusters, amorphous carbon, or any other structure. These carbon species are considered as unwanted impurities; thus a conventional etching step is used simultaneously with CVD graphene growth to remove them using an etching agent. Meanwhile, art etching is a specific method of producing controlled non-Euclidean and Euclidean geometries by employing intricate and precise etching parameters or integrated growth/etching modes. Agents such as H2, O2, CH4, Ar, and others are applied as art etching agents to support the art etching technology. This technique can generate nanopores and customize the properties of graphene, facilitating specific applications such as nanodevices, nanosensors, nanofilters, etc. This comprehensive review investigates how precursor gases concurrently induce graphene growth and art etching during a chemical vapor deposition process, resulting in beautifully etched patterns. Furthermore, it discusses the techniques leading to the creation of these patterns. Finally, the challenges, uses, and perspectives of these non-Euclidean and Euclidean-shaped art etched graphene geometries are discussed.

Adaptive Log-Euclidean Metrics for SPD Matrix Learning.

Symmetric Positive Definite (SPD) matrices have received wide attention in machine learning due to their intrinsic capacity to encode underlying structural correlation in data. Many successful Riemannian metrics have been proposed to reflect the non-Euclidean geometry of SPD manifolds. However, most existing metric tensors are fixed, which might lead to sub-optimal performance for SPD matrix learning, especially for deep SPD neural networks. To remedy this limitation, we leverage the commonly encountered pullback techniques and propose Adaptive Log-Euclidean Metrics (ALEMs), which extend the widely used Log-Euclidean Metric (LEM). Compared with the previous Riemannian metrics, our metrics contain learnable parameters, which can better adapt to the complex dynamics of Riemannian neural networks with minor extra computations. We also present a complete theoretical analysis to support our ALEMs, including algebraic and Riemannian properties. The experimental and theoretical results demonstrate the merit of the proposed metrics in improving the performance of SPD neural networks. The efficacy of our metrics is further showcased on a set of recently developed Riemannian building blocks, including Riemannian batch normalization, Riemannian Residual blocks, and Riemannian classifiers.

Exploring elliptic geometry in neural networks : The concept of elliptic neural networks (ENN)

The exploration of non-Euclidean geometries in the context of neural network architectures presents a novel avenue for enhancing the processing of complex data structures. This paper introduces the concept of Elliptic Neural Networks (ENN), a framework that integrates the intrinsic properties of elliptic geometry into the fabric of neural computation. Characterized by constant positive curvature and closed geodesic paths, elliptic geometry provides a unique perspective for representing data, especially advantageous for configurations exhibiting cyclic and dense hierarchical interconnections. We begin by delineating the fundamental aspects of elliptic geometry, focusing on its impact on conventional notions of distance and global structural interpretation within data sets. This foundational understanding paves the way for our proposed mathematical schema for ENNs. Within this framework, we articulate the methodologies for embedding data in elliptic spaces and adapt neural network operations by encompassing neuron activation, signal propagation, and learning paradigms to conform to the topological idiosyncrasies of elliptic geometry. In this paper addressing the implementation of ENNs, we discuss the computational challenges and the development of novel algorithms requisite for navigating the elliptic geometric landscape. The prospective applications of ENNs are extensive and diverse, ranging from the enhancement of cyclic pattern recognition in time-series analysis to more sophisticated representations in graph-based data and natural language processing tasks. This theoretical exposition aims to set the groundwork for subsequent empirical research to validate the proposed model and assess its practical performance relative to conventional neural network architectures. By harnessing the distinctive characteristics of elliptic geometry, ENNs mark a significant stride towards enriching the arsenal of machine learning methodologies, potentially leading to more versatile and efficacious computational tools in data analysis and artificial intelligence.

Sliced Inverse Regression in Metric Spaces

In this article, we propose a general nonlinear sufficient dimension reduction (SDR) framework when both the predictor and response lie in some general metric spaces. We construct reproducing kernel Hilbert spaces whose kernels are fully determined by the distance functions of the metric spaces, then leverage the inherent structures of these spaces to define a nonlinear SDR framework. We adapt the classical sliced inverse regression of \citet{Li:1991} within this framework for the metric space data. We build the estimator based on the corresponding linear operators, and show it recovers the regression information unbiasedly. We derive the estimator at both the operator level and under a coordinate system, and also establish its convergence rate. We illustrate the proposed method with both synthetic and real datasets exhibiting non-Euclidean geometry.

Toward the Relational Formulation of Biological Thermodynamics.

Classical thermodynamics employs the state of thermodynamic equilibrium, characterized by maximal disorder of the constituent particles, as the reference frame from which the Second Law is formulated and the definition of entropy is derived. Non-equilibrium thermodynamics analyzes the fluxes of matter and energy that are generated in the course of the general tendency to achieve equilibrium. The systems described by classical and non-equilibrium thermodynamics may be heuristically useful within certain limits, but epistemologically, they have fundamental problems in the application to autopoietic living systems. We discuss here the paradigm defined as a relational biological thermodynamics. The standard to which this refers relates to the biological function operating within the context of particular environment and not to the abstract state of thermodynamic equilibrium. This is defined as the stable non-equilibrium state, following Ervin Bauer. Similar to physics, where abandoning the absolute space-time resulted in the application of non-Euclidean geometry, relational biological thermodynamics leads to revealing the basic iterative structures that are formed as a consequence of the search for an optimal coordinate system by living organisms to maintain stable non-equilibrium. Through this search, the developing system achieves the condition of maximization of its power via synergistic effects.