Saturday, April 8, 2023

Aperiodic Monotile


Some folks recently came up with a single shape tile (a monotile, or einstein tile) that can tile a surface without the possibility of generating a repeating pattern. There's a simple explanation here:

I first saw mention of it on Hack-a-day (I think, but I can't seem to find it there now), followed the included links and immediately drew a copy in Fusion360 so I could print a bunch of the tiles and play with.  A couple days later the NY Times ran an article about it and then Stephen Colbert was joking about it on the Late Show (another one I can't find a link for - don't get old!).

The basic shape, called a "hat" (I think it looks more like a V-neck tee shirt) by the inventor is derived from 3 hexagons, which is how I created the shape in Fusion360.

Here's one of the images that I used to create the Fusion360 model:

Notice that the dark blue tile is a mirror image of the other, lighter tiles that are all the same shape.

I drew a single tile in Fusion360, used one of the diagrams to lay out a bunch of the tiles, then pulled back the perimeter by 0.3 mm. Since each tile was a copy of the original, modifying the original pulled back all the perimeters by 0.3 mm. Then I printed them.

The dark blue tiles in the image above are mirror image of the other tiles that are all the same shape, so I printed a set of those by using the mirror function in Prusa Slicer.

The tile layout I used in Prusa Slicer- these are the mirror image tiles (dark blue ones from the drawing).

I printed the tiles on UMMD in red, white, and blue (mirror) PETG so I could play with them a little. UMMD has a 1 mm nozzle and printed in 0.5 mm layers. The base of each tile is just 1 mm thick and the walls are 1 mm thick, so it takes very little filament to print. They print quickly, so you can produce hundreds of them in a couple hours.

A single tile- the walls are 1 mm thick and 5 mm tall, the base is 1 mm thick (2 print layers). These print fast and use little filament- about 1.3 g per tile.

Here are some of the tiles laid out on Arrakis. Note the hexagonal holes in the pattern- that's because I did not use any mirror image tiles in the layout. If you don't use any mirror image tiles, I think you'll always have holes of some sort in the tiling. Using the mirror image tiles allows patterns that have no holes or gaps.

The paper that describes these tiles and the mathematical proof of aperiodicity is here.

My Fusion360 file is here in STEP format and here in f3d format

Update: 4/23/23

I made a storage box for the tiles, shaped like... what else?

The box is an oversized tile with a 15 degree twist, with a matching lid.

Have fun!