Translation covers of some triply periodic platonic surfaces

TL;DR

This paper investigates translation covers of triply periodic Platonic surfaces, analyzing their symmetry groups and asymptotic counts of saddle connections and cylinders, contributing to the mathematical understanding of these complex structures.

Contribution

It provides new descriptions of the symmetry groups and quadratic asymptotics for counting geometric features on triply periodic surfaces with multiple algebraic and geometric representations.

Findings

Computed affine symmetry groups of the surfaces.

Derived quadratic asymptotics for saddle connections.

Counted cylinders weighted by area.

Abstract

We study translation covers of several triply periodic polyhedral surfaces that are intrinsically Platonic. We describe their affine symmetry groups and compute the quadratic asymptotics for counting saddle connections and cylinders, including the count of cylinders weighted by area. The mathematical study of triply periodic surfaces was initiated by Novikov, motivated by the study of electron transport. The surfaces we consider are of particular interest as they admit several different explicit geometric and algebraic descriptions, as described, for example, in the second author's thesis.