Related papers: Non-euclidean virtual reality I: explorations of $…
We describe our initial explorations in simulating non-euclidean geometries in virtual reality. Our simulation of the product of two-dimensional hyperbolic space with one-dimensional euclidean space is available at http://h2xe.hypernom.com.
This article presents virtual reality software designed to explore the Sol geometry. The simulation is available on 3-dimensional.space/sol.html
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…
We describe a method of rendering real-time scenes in Nil geometry, and use this to give an expository account of some interesting geometric phenomena. You can play around with the simulation at www.3-dimensional.space/nil.html.
HOLONOMY is a virtual environment based on the mathematical concept of hyperbolic geometry. Unlike other environments, HOLONOMY allows users to seamlessly explore an infinite hyperbolic space by physically walking. They use their body as…
In the era of foundation models and Large Language Models (LLMs), Euclidean space is the de facto geometric setting of our machine learning architectures. However, recent literature has demonstrated that this choice comes with fundamental…
In this paper, we study hypersurfaces of Euclidean spaces with arbitrary dimension. First, we obtain some results on $\mbox{H}$-hypersurfaces. Then, we give the complete classification of $\mbox{H}$-hypersurfaces with 3 distinct curvatures.…
Hyperbolic geometry has recently found applications in social networks, machine learning and computational biology. With the increasing popularity, questions about the best representations of hyperbolic spaces arise, as each representation…
This paper introduces a method of calculating and rendering shapes in a non-Euclidean 2D space. In order to achieve this, we developed a physics and graphics engine that uses hyperbolic trigonometry to calculate and subsequently render the…
This paper deals with various topics in analysis on hyperbolic spaces. It surveys some recent progress in non-Euclidean Fourier Analysis and proves some new results for the geodesic Radon transform on hyperbolic spaces.
Data representation in non-Euclidean spaces has proven effective for capturing hierarchical and complex relationships in real-world datasets. Hyperbolic spaces, in particular, provide efficient embeddings for hierarchical structures. This…
Higher-dimensional spaces are ubiquitous in applications of mathematics. Yet, as we live in a three-dimensional space, visualizing, say, a four-dimensional space is challenging. We introduce a novel method of interactive visualization of…
Deep representation learning is a ubiquitous part of modern computer vision. While Euclidean space has been the de facto standard manifold for learning visual representations, hyperbolic space has recently gained rapid traction for learning…
Orbifolds are a modern mathematical concept that arises in the research of hyperbolic geometry with applications in computer graphics and visualization. In this paper, we make use of rooms with mirrors as the visual metaphor for orbifolds.…
We describe local similarities and global differences between minimal surfaces in Euclidean 3-space and constant mean curvature 1 surfaces in hyperbolic 3-space. We also describe how to solve global period problems for constant mean…
How can we convince students, who have mainly learned to follow given mathematical rules, that mathematics can also be fascinating, creative, and beautiful? In this paper I discuss different ways of introducing non-Euclidean geometry to…
In machine learning, data is usually represented in a (flat) Euclidean space where distances between points are along straight lines. Researchers have recently considered more exotic (non-Euclidean) Riemannian manifolds such as hyperbolic…
Non-isotropic geometries are of interest to low-dimensional topologists, physicists and cosmologists. However, they are challenging to comprehend and visualize. We present novel methods of computing real-time native geodesic rendering of…
In this paper we review some author's results about Weingarten surfaces in Euclidean space $\r^3$ and hyperbolic space $\h^3$. We stress here in the search of examples of linear Weingarten surfaces that satisfy a certain geometric property.…
This paper investigates the notion of learning user and item representations in non-Euclidean space. Specifically, we study the connection between metric learning in hyperbolic space and collaborative filtering by exploring Mobius…