Tiling of Hyperbolic Surface by Multiple Tiles

TL;DR

This paper develops an algorithm to find all possible tilings of hyperbolic surfaces with a fixed number of tiles, focusing on small genus surfaces with two tiles, and explores edge length variations.

Contribution

It introduces a new algorithm for enumerating hyperbolic surface tilings with multiple tiles, specifically addressing surfaces of small genus and analyzing edge length diversity.

Findings

Algorithm successfully enumerates tilings for small genus surfaces with two tiles.

Identifies the finite nature of tilings with n-gons for n ≥ 7 on hyperbolic surfaces.

Discusses the variation in edge lengths within multiple tile tilings.

Abstract

Tilings of a surface of negative Euler characteristic by n-gons with n\ge 7 is a finite problem. We develop the algorithm for finding all the tilings for fixed number of tiles and present the calculation for tilings of surfaces of small genus by two tiles. We also discuss the number of distinct edge lengths in multiple tile tilings.