Periodic Steiner networks minimizing length

TL;DR

This paper identifies the shortest triply periodic graph in 3D space that spans a fundamental domain with fixed volume, revealing the srs network as the optimal structure related to the gyroid surface.

Contribution

It determines the length-minimizing triply periodic graph in Euclidean 3-space, specifically the srs network, among all graphs with the same fundamental domain volume.

Findings

The srs network minimizes length among triply periodic graphs.

The srs network spans the body centered cubic lattice.

It is related to the gyroid triply periodic surface.

Abstract

We study a problem of geometric graph theory: We determine the triply periodic graph in Euclidean 3-space which minimizes length among all graphs spanning a fundamental domain of 3-space with the same volume. The minimizer is the so-called srs network with quotient the complete graph on four vertices $K_4$. The network spans the body centred cubic lattice and is related to the gyroid triply periodic surface.