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

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.