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