Holonomy: A Virtual Reality Exploration of Hyperbolic Geometry

TL;DR

HOLONOMY is a virtual reality system enabling users to physically walk through an infinite hyperbolic space, demonstrating novel rendering and navigation techniques for non-Euclidean environments.

Contribution

This work introduces HOLONOMY, a VR platform that allows natural exploration of hyperbolic geometry, overcoming technical challenges in rendering and navigation in non-Euclidean spaces.

Findings

Successful implementation of hyperbolic environment rendering

User ability to navigate hyperbolic space physically

Preliminary evidence of learning outcomes in hyperbolic geometry

Abstract

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 the controller, eliminating the need for teleportation or other artificial VR locomotion methods. This paper discusses the development of HOLONOMY, highlighting the technical challenges faced and overcome during its creation, including rendering complex hyperbolic environments, populating the space with objects, and implementing algorithms for finding shortest paths in the underlying non-Euclidean geometry. Furthermore, we present a proof-of-concept implementation in the form of a VR navigation game and some preliminary learning outcomes from this implementation.