Domino tilings beyond 2D

TL;DR

This paper reviews the foundational results of domino tilings in two dimensions and surveys recent advances and open problems in higher-dimensional domino tilings, focusing on the connectivity of tiling spaces.

Contribution

It provides a comprehensive overview of the transition from 2D to higher-dimensional domino tilings and highlights key open problems in the field.

Findings

Summary of 2D domino tilings and their properties

Recent progress in understanding higher-dimensional tilings

Identification of open problems in tiling connectivity

Abstract

There is a rich history of domino tilings in two dimensions. Through a variety of techniques we can answer questions such as: how many tilings are there of a given region or what does the space of all tilings look like? These questions and their answers become significantly more difficult in dimension three and above. Despite this curse of dimensionality, there have been exciting recent advances in the theory. Here we briefly review foundational results of two-dimensional domino tilings and highlight where challenges arise when generalizing to higher dimensions. We then survey results in higher dimensional domino tilings, focusing on the question of connectivity of the space of tilings, and the open problems that still remain.