Unboundedness of the Heesch Number for Hyperbolic Convex Monotiles

TL;DR

This paper resolves the Heesch problem for hyperbolic convex monotiles and introduces the first weakly aperiodic convex monotiles from dual homogeneous tilings.

Contribution

It provides the first resolution of the Heesch problem in hyperbolic geometry and presents novel weakly aperiodic convex monotiles.

Findings

Heesch problem is resolved for hyperbolic convex monotiles.

First example of weakly aperiodic convex monotiles from dual tilings.

Advances understanding of tiling properties in hyperbolic geometry.

Abstract

We provide a resolution of the Heesch problem for homogeneous (also known as semi-regular) tilings, and as a corollary, for tilings by convex monotiles in the hyperbolic plane. We also provide the first known example of weakly aperiodic convex monotiles arising from the dual of homogeneous tilings.