Aperiodic Tile adventures

Here’s a nice finding, fresh from the mathematical frontier and surely ripe for a bit of 3D printing - a single tile shape, which can tile the plane but only in a patternless fashion. An aperiodic monotile, AKA an einstein. Look ma, no repeats:

The shape is simply described by cutting a hexagon and stitching together:


I have quickly created this shape using Freecad. Unfortunately I seem not to be able to upload the CAD file or the STL. Here is a picture of it!

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OK I have 3D printed some of these pieces. To fit them together, some have to be turned upside down.

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Well done! Yes indeed, some precise proportion (apparently) need to be flipped. It seems this is fine - which is to say, this result seems accepted as a single-tile solution.

Some nice decoration can be applied, see here for a thread. (Via this readable article, with more examples and links. Notably, perhaps, a lasercut version.)

Do you know what proportion have to be upside down?
I have an idea to turn it into a game.
By the way, does anyone know what FREE graphics package produces clean SVG files for the laser cutter? I tried Inkscape but it is problematic with its infilling. I am having a go at using Processing, but that is a bit problematic too. I may end up editing the SVG using a text editor!

It’s between 1 in 6 and 1 in 7, apparently: it’s 1 in φ^4 which is nicely irrational. About 14.6% in modern terms.

About your SVG trouble, I think you’re not alone. I see there’s a web page which just possibly might be helpful: Cuttle but I’ll also ask on Mastodon… some responses here.

Thanks, will have a look re SVG.
OK I have come up with a game, and written a set of rules. Will bring it on Sunday and we can try it.
I won’t post the details here though!

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In news not entirely unrelated to aperiodic tiling, there’s a nice hypnotic visual animation of a quasicrystal nature here:
Animated Quasicrystals - Jason Davies

Here’s a still image which isn’t the same at all:

There’s a new idea which does the trick with all tiles the same way up by bending the edges of the tile: provisionally called The Spectre. See here (illustrations and designs within)
A chiral aperiodic monotile

Explanation thread here

That is much better. I was bothered that turning the tile upside down effectively creates a different shape.
My 3D printer is temporarily out of action (the small fan cooling the cold end failed, causing the filament to soften prematurely and clog up). I will have a go at printing the Spectre.