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Mathematical objects are generally abstract and not very approachable. Illustrations and interactive visualizations help both students and professionals to comprehend mathematical material and to work with it. This approach lends itself…
Euclidean distance geometry is the study of Euclidean geometry based on the concept of distance. This is useful in several applications where the input data consists of an incomplete set of distances, and the output is a set of points in…
Many scientific fields study data with an underlying structure that is a non-Euclidean space. Some examples include social networks in computational social sciences, sensor networks in communications, functional networks in brain imaging,…
The enduring legacy of Euclidean geometry underpins classical machine learning, which, for decades, has been primarily developed for data lying in Euclidean space. Yet, modern machine learning increasingly encounters richly structured data…
In this article, I develop an elementary system of axioms for Euclidean geometry. On one hand, the system is based on the symmetry principles which express our a priori ignorant approach to space: all places are the same to us (the…
We analyse the axioms of Euclidean geometry according to standard object-oriented software development methodology. We find a perfect match: the main undefined concepts of the axioms translate to object classes. The result is a suite of C++…
Multimodal large language models (MLLMs) have made rapid progress in recent years, yet continue to struggle with low-level visual perception (LLVP) -- particularly the ability to accurately describe the geometric details of an image. This…
With the rapid improvement of machine learning (ML) models, cognitive scientists are increasingly asking about their alignment with how humans think. Here, we ask this question for computer vision models and human sensitivity to geometric…
Projective geometry provides the preferred framework for most implementations of Euclidean space in graphics applications. Translations and rotations are both linear transformations in projective geometry, which helps when it comes to…
Mathematical thinking is a fundamental aspect of human cognition. Cognitive scientists have investigated the mechanisms that underlie our ability to thinking geometrically and numerically, to take two prominent examples, and developmental…
Spatial intelligence spans a rich suite of abilities, including visualising and transforming shapes, mentally rotating objects, judging relational positions and containment, and estimating numerosity. However, it still remains a critical…
In the era of foundation models and Large Language Models (LLMs), Euclidean space has been the de facto geometric setting for machine learning architectures. However, recent literature has demonstrated that this choice comes with…
The Euclidean scattering transform was introduced nearly a decade ago to improve the mathematical understanding of the success of convolutional neural networks (ConvNets) in image data analysis and other tasks. Inspired by recent interest…
Understanding geometry relies heavily on vision. In this work, we evaluate whether state-of-the-art vision language models (VLMs) can understand simple geometric concepts. We use a paradigm from cognitive science that isolates visual…
Humans possess a remarkable capacity to recognize and manipulate abstract structure, which is especially apparent in the domain of geometry. Recent research in cognitive science suggests neural networks do not share this capacity,…
In this paper, we propose that 'embodied mathematics' should be studied not only by reduction to the present individual bodily experience but in an historical context as well, as far as the origins of mathematics are concerned. Some early…
Recent literature has shown that symbolic data, such as text and graphs, is often better represented by points on a curved manifold, rather than in Euclidean space. However, geometrical operations on manifolds are generally more complicated…
In this paper, we propose to study a new geometric optimization problem called "geometric prototype" in Euclidean space. Given a set of patterns, where each pattern is represented by a (weighted or unweighted) point set, the geometric…
Many problems in Euclidean geometry, arising in computational design and fabrication, amount to a system of constraints, which is challenging to solve. We suggest a new general approach to the solution, which is to start with analogous…
It is shown that the generalized geometries may be obtained as a deformation of the proper Euclidean geometry. Algorithm of construction of any proposition S of the proper Euclidean geometry E may be described in terms of the Euclidean…