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.
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 Hc (Heesch number with no holes in the outermost corona) or Hh (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.
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 Hh = Hc. In that case, a given page will have a single hole-free drawing on it. If instead Hh = Hc + 1, the page will show two drawings: one without holes on top, and one with holes on the bottom.
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n | non-tilers | threshold | number included | text file | PDF file |
7 | 3 | 0 | 3 | 07omino_0up.txt | 07omino_0up.pdf |
8 | 20 | 0 | 20 | 08omino_0up.txt | 08omino_0up.pdf |
9 | 198 | 0 | 198 | 09omino_0up.txt | 09omino_0up.pdf |
10 | 1390 | 0 | 1390 | 10omino_0up.txt | 10omino_0up.pdf |
11 | 9474 | 2 | 110 | 11omino_2up.txt | 11omino_2up.pdf |
12 | 35488 | 2 | 41 | 12omino_2up.txt | 12omino_2up.pdf |
13 | 178448 | 2 | 220 | 13omino_2up.txt | 13omino_2up.pdf |
14 | 696371 | 2 | 202 | 14omino_2up.txt | 14omino_2up.pdf |
15 | 2721544 | 2 | 164 | 15omino_2up.txt | 15omino_2up.pdf |
16 | 10683110 | 2 | 305 | 16omino_2up.txt | 16omino_2up.pdf |
17 | 41334494 | 2 | 528 | 17omino_2up.txt | 17omino_2up.pdf |
18 | 155723774 | 2 | 324 | 18omino_2up.txt | 18omino_2up.pdf |
19 | 596182769 | 2 | 767 | 19omino_2up.txt | 19omino_2up.pdf |
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n | non-tilers | threshold | number included | text file | PDF file |
6 | 4 | 0 | 4 | 06hex_0up.txt | 06hex_0up.pdf |
7 | 37 | 0 | 37 | 07hex_0up.txt | 07hex_0up.pdf |
8 | 381 | 0 | 381 | 08hex_0up.txt | 08hex_0up.pdf |
9 | 2717 | 2 | 192 | 09hex_2up.txt | 09hex_2up.pdf |
10 | 18760 | 2 | 662 | 10hex_2up.txt | 10hex_2up.pdf |
11 | 116439 | 3 | 79 | 11hex_3up.txt | 11hex_3up.pdf |
12 | 565943 | 3 | 34 | 12hex_3up.txt | 12hex_3up.pdf |
13 | 3033697 | 3 | 52 | 13hex_3up.txt | 13hex_3up.pdf |
14 | 14835067 | 3 | 46 | 14hex_3up.txt | 14hex_3up.pdf |
15 | 72633658 | 3 | 324 | 15hex_3up.txt | 15hex_3up.pdf |
16 | 356923880 | 3 | 158 | 16hex_3up.txt | 16hex_3up.pdf |
17 | 1746833634 | 3 | 570 | 17hex_3up.txt | 17hex_3up.pdf |
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n | non-tilers | threshold | number included | text file | PDF file |
7 | 1 | 0 | 1 | 07iamond_0up.txt | 07iamond_0up.pdf |
8 | 0 | 0 | 0 | ||
9 | 20 | 0 | 20 | 09iamond_0up.txt | 09iamond_0up.pdf |
10 | 103 | 0 | 103 | 10iamond_0up.txt | 10iamond_0up.pdf |
11 | 594 | 0 | 594 | 11iamond_0up.txt | 11iamond_0up.pdf |
12 | 1192 | 0 | 1192 | 12iamond_0up.txt | 12iamond_0up.pdf |
13 | 6290 | 2 | 53 | 13iamond_2up.txt | 13iamond_2up.pdf |
14 | 18099 | 2 | 25 | 14iamond_2up.txt | 14iamond_2up.pdf |
15 | 54808 | 2 | 95 | 15iamond_2up.txt | 15iamond_2up.pdf |
16 | 159048 | 2 | 36 | 16iamond_2up.txt | 16iamond_2up.pdf |
17 | 502366 | 2 | 175 | 17iamond_2up.txt | 17iamond_2up.pdf |
18 | 1374593 | 2 | 81 | 18iamond_2up.txt | 18iamond_2up.pdf |
19 | 4076218 | 2 | 159 | 19iamond_2up.txt | 19iamond_2up.pdf |
20 | 11378831 | 2 | 142 | 20iamond_2up.txt | 20iamond_2up.pdf |
21 | 32674779 | 2 | 178 | 21iamond_2up.txt | 21iamond_2up.pdf |
22 | 93006494 | 2 | 149 | 22iamond_2up.txt | 22iamond_2up.pdf |
23 | 264720498 | 2 | 346 | 23iamond_2up.txt | 23iamond_2up.pdf |
24 | 748062099 | 2 | 524 | 24iamond_2up.txt | 24iamond_2up.pdf |