Ref: "Non-tiles and Walls - a variant on the Heesch problem" (https://arxiv.org/pdf/1605.09203)
Definitions (adapted from above doc): A non-tile is any polygon that does not tile the Euclidean plane. Given a non-tile P, a wall is a simply connected region formed with infinitely many copies of P with finite width throughout and dividing the plane into 2 semi-infinite and simply connected regions - basically, a wall stretches infinitely across the plane and has no cavities within. If a wall formed with P is such that any line joining the two semi-infinite 'pieces' of the plane has to traverse finite distances through at least n of the P units forming the wall, then, the wall has thickness n - where n is an integer. The thickness number of P is the thickness of the thickest wall that can be formed with P units.
Eg: The Heesch pentagon (formed by attaching an equilateral and a 30-60-90 right triangle to a square) can form a wall as follows and has thickness number 2 (we don’t see how a wall of thickness 3 can be made with this polygon).
Question: Are there convex non-tiles with thickness number greater than 2? If so how to construct them for thickness numbers 3, 4..?
Note 1: Ref above gives some more preliminary results (with figures) on walls and states the present question (and some more).
Note 2: just reiterating, Walls are an attempt to grade non tiles inspired by Heesch and based on a different topology to his ‘coronas’ around a central unit.
