Non-Tiles and Walls - A Variant on the Heesch Problem

TL;DR

This paper explores a new variant of the Heesch problem by examining the topology of walls—simply connected regions dividing the plane into two parts—and presents initial findings and conjectures.

Contribution

It introduces the concept of walls as a new topological approach to the Heesch problem and provides preliminary results and conjectures.

Findings

Initial results on wall topologies

Conjectures about wall properties

New perspective on tiling problems

Abstract

The Heesch problem 'grades' polygons that fail to tile the plane in terms of the number of layers (or corollas) of copies of it that can be formed around a central unit. We study the different topology of ' walls', which we define to be simply connected regions that divide the plane exactly into two simply connected regions. We present preliminary results and conjectures.