Hypernom: Mapping VR Headset Orientation to S3

Vi Hart, Andrea Hawksley, Henry Segerman and Marc ten Bosch
Proceedings of Bridges 2015: Mathematics, Music, Art, Architecture, Culture (2015)
Pages 387–390

Abstract

Hypernom is a virtual reality game. The cells of a regular 4D polytope are radially projected to S3, 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 S3. 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 S3. 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.

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