Variations on the Three-Sphere: Laves' Labyrinth Lopped

Abstract

Inspired by the structure of $srs$ Laves networks in ${\mathbb{R}}^3$ that underpin the celebrated gyroid surface, we construct a Laves network of identical three-coordinated vertices on $S^3$ with double-twist. This network is a subset of the vertices and edges of the 600-cell, and can be viewed as a bipartite graph of disjoint 24-cell vertices inscribed in the 600-cell. We describe mutually entangled realizations of this network on $S^3$, and describe their relation to the well-known $srs$ Laves network structure in ${\mathbb{R}}^3$.