There are many ways to connect between edges of the tiles that result in complex loops of various lengths. One of them I found interesting is this one, which connects into infinite curves in 3 directions.

We can move the connection points closer to the corners (though not exactly on the corner as then we’d have 4 coincident points and wouldn’t be able to connect the continuous curves up)

I find these interesting because they suggest to me that there might be an analogue to be found to the Ammann bars in Penrose tilings.

A nice article that covers these a bit

Which includes this quote from Grunbaum and Shephard
“it is the system of bars which are fundamental and the only
function of the tiles is to give a practical realization to them.”

The stepped curves do not give us these nice matching rules and alternative generation methods like the Ammann bars, but perhaps there is a way of decorating the hat tile or one of the edge length variants with segments in a way that does?

Re-reading this thread, I realize too late that you were actually asking about Max Allstadt’s second pattern, not Daniel’s. Max said “minimum spanning tree of the centroids, plus multipipe”, which took me a while to get my mind around, since I have been thinking in two dimensions!

So I had a go at it! I did not manage exactly his effect, but connecting all centroid points to their two nearest neighbours and then using multipipe indeed produces much his effect, which is very striking:

Upping the pipe radius until there is more contact might even give something rigid enough for 3-D printing - I could see this being a nice broach, if made in silver!

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.