One very simple tile decoration that joins into interesting structures on the tiling is this
which results in
Another thing that looks interesting to explore further is these tree structures from which metatile each hat comes from
It’s slow, but here’s what I do:
There are a number of plugins that will create a “minimum spanning tree”, Parakeet has one, but it’s slower than the SuperDelaunay component, which you can find on Food4Rhino. That component has various options for processing a bunch of points, including Delaunay meshes, outlines, and also a minimum spanning tree.
I took all the centroid points of all the tiles, and ran them through minimum spanning tree, which outputs a bunch of line segments.
Then I ran those segments through the Multipipe component.
Then I intersected the gigantic SubD that the multipipe component with the XY plane, and got the single curve that you see. This part is sloooooowww. like 20 minutes on a laptop.
The multipipe+intersect method is my workaround for the fact that you can’t “smooth out” a tree of lines by joining and rebuilding as a degree 3 curve.
wow, amazing.
Now we have fractal omelette theory 
Know that you are responsible for this!
More like sunny-side up!
Eggcellent!
eggcentric
Trying to understand the approach used for the animation, as shown in the NY times article here, by Craig Kaplan, which appears to show an entire class of tilings
Yes indeed (this is the animation without the NYT paywall Aperiodic monotile animation - YouTube)
I was thinking it would be good to make a definition including this edge length variation
Here’s a definition showing the variation for a single tile. Next step - combine this with the substitution system to make the whole tiling.
monotile_family.gh (5.8 KB)
Brilliant, thank you. A simple definition. I will try to understand the derivation of the initial points.
What is amazing to me is how simple the shape is, and the elegance of the two proofs presented in the papers. Add that to how many centuries! people have been staring at this problem, starting with Islamic and Oriental designs.
And then we have the new work of yours, and others, of ‘dividing a square into similar rectangles’.
Just when you think everything has been found, and looked at, here are 2 new major advances of geometrical thought.
ah. from your diagram of 11 days ago, i see the derivation, 4/3 of a hexagon
It’s 3 weeks i keep seeing this subject in the news here and there, then found this chat subject, and this morning this dropped…
I think the Quanta one is probably the best of the articles I’ve seen on this
Time passes… yesterday, Simon Tatham published a very nice interactive blog page on this, which is worth a look: Two algorithms for randomly generating aperiodic tilings.
Bob
Reading the paper, and seeing the truncated triangle, I am reminded of Benitoite crystals from the San Diego area.
Just dropped in to say thanks to Daniel Piker for his description of the H7/H8 substitution method for creating a hat tiling. Helped me greatly when I needed a large tiling for an app I wrote that explores various cellular automata rules on a hat tiling. (More details here if anybody wants to play with it.)
A new development in this research -
An aperiodic tiling without flips!
https://cs.uwaterloo.ca/~csk/spectre/
For example, this means you could produce glazed tiles of only a single type (unlike the previous work, where you could use a single cutter shape, but would have to glaze some of the tiles on the other side).
This also means that you now have more freedom to alter the boundary shape
Here’s a little definition that maps a given curve to the edges of this monotile and a patch of the tiling
chiralmonotile.gh (17.5 KB)
