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.

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

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.

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.)

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