Making aperiodic monotiles from layered card stock

[pcallahan]

How often does a result in pure mathematics arrive just in time to practice a new crafting skill? Only once in my lifetime so far.

I have a Cricut, which cuts thin material very well, but I don't have a 3D printer, and I have been getting around this by coming up with ways to make thicker pieces out of layered card stock. I finally have a reliable method, though it is still rather labor-intensive for making a lot of tiles. It takes at least a few minutes per piece, mostly in the gluing stage. Cutting also takes time, but I can do something else while it happens. Since I'm using 8 layers of card stock per piece, even the process of peeling layers off the Cricut mat cannot be ignored.

Still, the opportunity was too good to pass up. Here is a patch of a tiling based on 14 of the recently discovered aperiodic monotile.

20230325_190447.jpg (209.72 KiB) Viewed 589 times

These pieces are pretty small. The long edge of the kite is 7mm.

You may be wondering why they all have +-shaped holes. That's an artifact of the construction process. I assemble the layers on a Lego technics axle, and dip the whole thing in thinned-down white glue (Elmers) with the layers slightly apart. That gets the glue between all the layers. Next I use a spinner, also built of Lego technics pieces to spin off the excess glue.

20230325_124129.jpg (110.44 KiB) Viewed 589 times

The gear ratios are such that the piece spins 45 times each time I turn the handle, which provides sufficient centrifugal force. Once the glue is mostly off, I can squeeze the layers together and let it dry. Fortunately, this is non-toxic even if it's a little messy. I spin it lowered into the cup or I would be splattering glue all over the place.

Finally, here are the layers after cutting.

20230325_125209.jpg (209.94 KiB) Viewed 589 times

I removed the sheet they're cut out of, which is about 0.25mm thick. Another step is getting all those little +-signs off the Cricut mat. I have been saving them as confetti, maybe to include in a clear resin coaster.

Note: The holes can be covered with two more layers of card stock top and bottom. Or if I want to get fancy, I could fill them with air-dry clay and paint them over with acrylic paint. It would make sense to color top and bottom differently so flipped tiles would be visible.

[pcallahan]

EVA foam (the kind sold for kids' crafts) isn't my favorite material, but this shape is simple enough to cut on the Cricut with the knife blade when the long kite edge is 10mm. It's a slow process, making 4 passes through this soft but thick material. However, I'm not the one doing the work, so I just have to keep an eye on it instead of fiddling around with glue. (I far prefer the results of layered card stock and could probably make some very nice game tiles and pieces with it. When the glue is dry, the pieces click when they hit the table top. Anyway, back to the foam...)

I originally wanted to make pieces that fit tab in slot, but I realized that the symmetries don't really work with this. Without any way to connect the pieces, they slide around. A layer of rubber cement works well as a temporary bond, and I assembled these tiles on a piece of cardboard thinly coated with rubber cement. They hold in place well. I got a little frustrated placing these. They are either less intuitive than Penrose tiles or I haven't built up enough intuition. Anyway, you can see I still have a lot left in the bag. Maybe I'll finish it later.

20230326_190518.jpg (142.59 KiB) Viewed 532 times

Update: All right, I finished placing all the tiles I made. I copied the layout from one of the articles, but I wonder in general if you just spiral around placing a new tile where it fits, what's the longest you may have to backtrack if you placed one badly.

This reminds me of a lichen, no doubt because of the color I picked:

20230326_211430.jpg (225.68 KiB) Viewed 517 times

I picked that color out of all the foam I have because I am saving colors I like better and wasn't sure how this would turn out.

[dvgrn]

I copied the layout from one of the articles, but I wonder in general if you just spiral around placing a new tile where it fits, what's the longest you may have to backtrack if you placed one badly.

I've been wondering about that -- and also what additional tiles are forced by the placement of a given einstein tile. Some hints from the original paper would seem to say that there must be multiple different ways to extend any given patch of einsteins. Is the number of distinct infinite einstein tilings uncountably infinite but locally indistinguishable, same as for infinite Penrose tilings?

I'm thinking that I can describe at least two distinct infinite but locally indistinguishable einstein tilings -- one with three-way symmetry around a particular point, one without any such symmetry. This all came up in a Discord ConwayLife Lounge #other-math discussion the other day. But papers and articles so far don't seem to say anything very clear about this.

For example, the original paper implies that the inflation rule for H7 and H8 clusters produces "the same tiling by hats" as the four-supertile inflation rule. But it seems as if repeatedly inflating a triangular supertile, or inflating the H7 and H8 clusters, would produce a tiling with no rotational symmetry anywhere (e.g., a single einstein right in the center of inflation -- hopeless). But repeatedly inflating a fylfot made of three F tiles would ... I would think ... produce an infinite tiling with 120-degree rotational symmetry. In which case, "the same tiling by hats" seems like it might not be well-defined.