Locked Polyomino Tilings

Enumerates locked polyomino tilings, as described in Section 4 of this paper:

https://arxiv.org/abs/2307.15996

Also see the OEIS sequence for symmetric pentomino tilings:

https://oeis.org/A365862

SVG images of all 4-fold symmetric, locked 5-omino tilings of grids up to size 35 X 35 are in the Images directory. The tiling shown above (S30-P05.svg) is my favorite, one of only ten such tilings of the 30 X 30 grid.

Running the code

The code for enumerating locked polyomino tilings is contained in locked_polyomino_tilings.py. The code is only very lightly documented, containing examples of how to use the core functions at the bottom of the file in 10 numbered code blocks. Not all functionality has corresponding tests that use it; for example, there is an old plotting function that uses matplotlib to visualize tilings, which has now been replaced by a far-superior SVG visualization tool. Such relics are left in the code and mostly still work, but are not essential.

It is instructive to go through an example of how to generate all of the 30 X 30 tilings in the file All5OminoTilings.txt and then use this file to produce all of the images in the Images directory. You should follow along by reading the comments in the Python files. First, run:

python3 locked_polyomino_tilings.py 6

python3 locked_polyomino_tilings.py 7

python3 locked_polyomino_tilings.py 8

These can be run in parallel, and may each take several hours to finish. Important note: If you do not download the .p files, they will be generated the first time the code is run for each polyomino size, which may take several seconds for $n = 3$, several minutes for $n = 4$, and several days for $n = 5$. Outputs will be stored in three separate files in an Outputs directory. Once they are done, run

python3 locked_polyomino_tilings.py 9

and cut/paste from the terminal (or redirect output) into a new file, like 5OminoTilings.txt. Make sure to include a line return at the very end of the file. The file All5OminoTilings.txt was created in this exact same way, except that it also included outputs from runs on the 20 X 20 and 35 X 35 grids (the latter of which took more than a week to run). Let's assume we're working with that file now, which you can download directly from this repository. To create the SVGs and store them in the Images directory, run:

python3 print_to_svg.py All5OminoTilings.txt ColorCode5Ominoes.txt Images

This uses the coloring defined in ColorCode5Ominoes.txt, which you can edit first if you want to change the color scheme.