Holonomy: A Virtual Reality Exploration of Hyperbolic Geometry

TL;DR

The development of Holonomy is discussed, 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.

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.

Figures (12)

• Figure 1: A line $L$ and several parallels to $L$ through a point $P$ in different models of the hyperbolic plane. Left: Beltrami-Klein model in a disk, straight hyperbolic lines remain straight; Center: Poincaré disk model, straight hyperbolic lines become curved; Right: Poincaré half-plane model, straight hyperbolic lines are either vertical or half circles, but angles are kept intact. Reproduced with permission skrodzki2021illustrations.
• Figure 2: The spanning tree of the graph. Nodes represent the hyperbolic squares underneath. Labels indicate the indices of the nodes.
• Figure 3: Top row: A user moving their body through a physical $2\times2$ Euclidean square grid. Bottom row: The movements are mapped to a square tiling in the hyperbolic plane, with five squares meeting at a vertex. Reproduced with permission jochems2023mini-map.
• Figure 4: Screenshots of VR environments, experiences, and games using hyperbolic geometry.
• Figure 5: The user interface shows the current status and the mini-map, helping with navigation.