Moduli of Parallelogram Tilings and Curve Systems

TL;DR

This paper characterizes the topology of the moduli space of periodic parallelogram tilings, linking combinatorial data to geometric structures and constructing canonical tilings for each data set.

Contribution

It determines the topology of the moduli space and provides a method to construct canonical tilings from combinatorial data.

Findings

The moduli space is homotopy equivalent to a circle.

All tilings with the same combinatorial data form an open subset.

Canonical tilings can be explicitly constructed for any combinatorial data.

Abstract

We determine the topology of the moduli space of periodic tilings of the plane by parallelograms. To each such tiling, we associate combinatorial data via the zone curves of the tiling. We show that all tilings with the same combinatorial data form an open subset in a suitable Euclidean space that is homotopy equivalent to a circle. Moreover, for any choice of combinatorial data, we construct a canonical tiling with these data.