Enumerating Isotopy Classes of Tilings guided by the symmetry of Triply-Periodic Minimal Surfaces

TL;DR

This paper introduces a method to enumerate all distinct tilings of hyperbolic surfaces with symmetries related to triply-periodic minimal surfaces, expanding tiling classification techniques.

Contribution

It generalizes isotopic tiling enumeration by linking mapping class groups with automorphisms, providing explicit descriptions for tilings associated with complex minimal surfaces.

Findings

Explicit enumeration of isotopically distinct tilings

Descriptions of subgroups of mapping class groups

Examples of tilings with primitive, diamond, and gyroid symmetries

Abstract

We present a technique for the enumeration of all isotopically distinct ways of tiling a hyperbolic surface of finite genus, possibly nonorientable and with punctures and boundary. This provides a generalization of the enumeration of Delaney-Dress combinatorial tiling theory on the basis of isotopic tiling theory. To accomplish this, we derive representations of the mapping class group of the orbifold associated to the symmetry group in the group of outer automorphisms of the symmetry group of a tiling. We explicitly give descriptions of certain subgroups of mapping class groups and of tilings as decorations on orbifolds, namely those that are commensurate with the Primitive, Diamond and Gyroid triply-periodic minimal surfaces. We use this explicit description to give an array of examples of isotopically distinct tilings of the hyperbolic plane with symmetries generated by rotations, outlining how the approach yields an unambiguous enumeration.