Overview
Research addressed the Heesch problem within the hyperbolic plane, focusing on two distinct categories: homogeneous (also referred to as semi-regular) tilings and convex monotiles. The investigation resulted in resolving the Heesch problem for both types of tilings. Specifically, the study covered edge-to-edge restrictions for homogeneous tilings and no edge-to-edge restriction for convex monotiles. Furthermore, the work involved the construction of an infinite family of weakly aperiodic convex monotiles, characterized by interior angles that are rational multiples of $\pi$.
Research Context
The Heesch problem is a concept in tiling theory. The research explicitly addresses this problem within the geometric context of the hyperbolic plane. The study considered homogeneous tilings, which are defined as edge-to-edge tilings constructed from regular polygons where every vertex exhibits the same cyclic type. For convex monotiles, the problem was investigated without imposing an edge-to-edge restriction.
Findings
- The Heesch problem in the hyperbolic plane was resolved for homogeneous tilings.
- The Heesch problem in the hyperbolic plane was resolved for convex monotiles, specifically in a context without any edge-to-edge restriction.
- An infinite family of weakly aperiodic convex monotiles was constructed. The interior angles of these monotiles are rational multiples of $\pi$.