Hypernom: Mapping VR Headset Orientation to S^3

TL;DR

Hypernom is a VR game that maps headset orientation to a 4D sphere (S^3) using quaternions, allowing players to explore 4D polytopes through head movements and interact with their cells.

Contribution

It introduces a novel method of mapping VR headset orientation to 4D space using quaternion projections, enabling immersive exploration of 4D geometries.

Findings

Players can navigate 4D polytopes by head movements.

The game demonstrates practical application of quaternion-based orientation mapping.

Players successfully eat all cells by exploring all orientations.

Abstract

Hypernom is a virtual reality game. The cells of a regular 4D polytope are radially projected to S^3, the sphere in 4D space, then stereographically projected to 3D space where they are viewed in the headset. The orientation of the headset is given by an element of the group SO(3), which is also a space that is double covered by S^3. In fact, the headset outputs a point of this double cover: a unit quaternion. The positions of the cells are multiplied by this quaternion before projection to 3D space, which moves the player through S^3. When the player is sufficiently close to a cell, they eat it. The aim of the game is to eat all of the cells of the polytope, which, roughly speaking, is achieved by moving one's head through all possible orientations, twice.