Variations on the Three-Sphere: Laves' Labyrinth Lopped

TL;DR

This paper constructs and analyzes a three-dimensional network on the 3-sphere inspired by Laves networks and the gyroid surface, relating it to the 600-cell and srs networks.

Contribution

It introduces a novel Laves network on S^3 with double-twist, connecting it to the 600-cell and srs networks in Euclidean space.

Findings

Constructed a Laves network on S^3 with double-twist.

Related the network to the vertices of the 600-cell.

Described entangled realizations and their relation to srs networks.

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 $\mathbb{R}^3$.