Heesch Numbers of Unmarked Polyforms

Dataset by Craig S. Kaplan (csk@uwaterloo.ca), 2021

This dataset contains text files and images of a selection of non-tiling polyominoes, polyhexes and polyiamonds with non-trivial Heesch numbers. As n increases, the number of n-forms of a given type grows exponentially (up into the low billions in my enumeration), so I do not offer complete data here. Instead, I gradually increase the threshold for inclusion, eliminating for example the many shapes with Heesch number 0 as n increases, while keeping the few with higher Heesch numbers.

This dataset is meant to accompany the paper "Heesch Numbers of Unmarked Polyforms", which is available on arxiv.org. I am also including a local copy. If you like, you can also download a copy of this dataset.

Here's what you need to know about the two types of files included in this dataset.

Text files

The name of the text file tells you the threshold for inclusion. If the name ends in "0up", all non-tiling polyforms of that size are included. If the name ends in, say, "2up", only polyforms with Heesch number at least 2 are included. Note that a shape is included if either H_c (Heesch number with no holes in the outermost corona) or H_h (Heesch number with holes permitted in the outermost corona) is bigger than the threshold.

Within each file, a shape is described over two lines. The first line gives a sequence of x and y coordinates for the units that make up the polyform. When describing polyominoes, the coordinates should be obvious. They're slightly trickier when working with polyhexes and especially with polyiamonds; see the paper for a complete description of how to interpret coordinates in these cases. After that, there is a line of the form Hc = ... Hh = ... giving the Heesch number of the shape when holes are forbidden and when they are permitted in the outermost corona.

PDF files

For each text file there is an accompanying PDF that shows drawings of coronas that demonstrate the reported Heesch numbers. There is one shape per page, and the ordering of the pages corresponds to the ordering of the shapes in the text file, so you can move back and forth between the text description and the drawing.

In practice, most polyforms have H_h = H_c. In that case, a given page will have a single hole-free drawing on it. If instead H_h = H_c + 1, the page will show two drawings: one without holes on top, and one with holes on the bottom.

The data