Heesch Numbers of Unmarked Polyforms

TL;DR

This paper introduces a SAT solver-based method to compute the Heesch numbers of non-tiling polyforms, providing extensive computational data up to high complexity shapes to advance understanding of their layering properties.

Contribution

It presents a novel computational technique using SAT solvers to determine Heesch numbers of polyforms, enabling exhaustive analysis beyond previous experimental limits.

Findings

Heesch numbers up to 19-ominoes, 17-hexes, and 24-iamonds were computed.

Shapes with finite Heesch numbers up to six are common, but higher numbers are rare.

The method facilitates systematic classification of polyforms by their Heesch number.

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.