Heesch Numbers of Unmarked Polyforms

Abstract

A shape's Heesch number is the number of layers of copies of the shape that can be placed around it without gaps or overlaps. Experimentation and exhaustive searching have turned up examples of shapes with finite Heesch numbers up to six, but nothing higher. The computational problem of classifying simple families of shapes by Heesch number can provide more experimental data to fuel our understanding of this topic. I present a technique for computing Heesch numbers of non-tiling polyforms using a SAT solver, and the results of exhaustive computation of Heesch numbers up to 19-ominoes, 17-hexes, and 24-iamonds.