Tiling of Hyperbolic Surface by a Single Tile

TL;DR

This paper develops an algorithm to find all possible single tile tilings of hyperbolic surfaces with negative Euler characteristic, providing new insights into their structure for small genus surfaces.

Contribution

It introduces a novel algorithm for identifying all single tile tilings of hyperbolic surfaces, expanding understanding of their finite tiling configurations.

Findings

Algorithm successfully finds all single tile tilings for small genus surfaces

Identifies finite set of tilings for surfaces with negative Euler characteristic

Provides a comprehensive classification of single tile tilings in hyperbolic geometry

Abstract

Tilings of a surface of negative Euler characteristic by n-gons with n\ge 7 is a finite problem. One extreme of the finite problem is single tile tilings. We develop the algorithm for finding all the single tile tilings and present the results for surfaces of small genus.