hatvalidate

This repository contains Python code to accompany the article "An aperiodic monotile". The goal of this code is to validate the case-based computations in Section 4 of the paper, which establish that hat polykites are forced to cluster correctly into the metatiles that are used as one of the proofs of aperiodicity. Here are some quick instructions.

1. It's helpful to have a chart showing all possible neighbours of a central hat, together with the affine transformation matrices associated with those neighbours (assuming the central hat uses the identity matrix). The viz_neighbours script generates such a chart:

   $ python viz_neighbours.py > neighbours.ps
   $ pstopdf neighbours.ps
   

   The provided file neighbours.pdf is the result of executing the commands above. There are 54 potential neighbours. Many of these cannot actually occur in a tiling by hats (or even in a 1-patch around a central hat), but it doesn't hurt to have all of them. Appendix B of the paper includes diagrams that show how to eliminate many of the 54 potential neighbours.

2. The starting point for the case-based analysis is to generate all "surroundable 2-patches". These are the subset of the 2380 possible 2-patches that can themslves be surrounded to yield a 3-patch. There are 188 such patches. They can be generated using the "surround" script:

   $ python surround.py > 2patches.txt
   

   The computation proceeds by brute force, and deliberately avoids optimization for the sake of clarity. It completes in about 32 minutes on a 2019 laptop (2.4 GHz 8-Core Intel Core i9). The provided file 2patches.txt is the result of executing the command above. The file is a sequence of 2-patches. Each one first gives the number N of tiles in the patch on its own line, then N lines, each containing a corona number and a transformation matrix for a tile in the patch (the top two rows of a 2D affine transformation; see geom.py for details).

3. The actual case-based analysis is performed by matching.py:

   $ python matching.py < 2patches.txt
   

   This script will read in patches, and for each one verify that it satisfies all the conditions of Section 4. It assigns labels to all tiles in the 0- and 1-coronas using the rules of Figure 4.2, and then verifies the within-cluster (Figure 4.3) and between-cluster (Figure 4.4) matching conditions. If everything checks out (which it does!), the result is a bit of an anticlimax: the program will simply print "188/188 patches passed checks" and stop. Of course, That's a cause for celebration!

For more information, including a link to the paper, see my web page about this project.