It’s a shape Jim, but not as we know it

Thanks to the super human effort of Craig Kaplan, Joseph Myers and Chaim Goodman-Strauss, the hat and turtle polykites have finally arrived on the scene… https://arxiv.org/abs/2303.10798

This article is intended to be a temporary ‘scrapbook’ of how the polykites were conceived. It will contain many raw images and is likely to look messy until I get round to producing something more presentable.

The hat and turtle can only tile the infinite plane aperiodically. They do not permit periodic tilings. They should be treated with respect!

The shaded areas below indicate the similarities between the two polykites. The hat is composed of 8 kites and the turtle, 10.

The hat (left) and the turtle (right)

Jaap’s polyform software came in very handy (I have been using it for years)… https://www.jaapsch.net/puzzles/polysolver.htm

As you can see in one solution, there’s not much to go on. There’s a few clumps so worth investigating further.

Smaller grid but basically same end product.

I left Jaap’s solver program to run a bit longer, time to place them manually.

There’s bound to be a periodic pattern in there somewhere, right? Or, maybe a non tiler perhaps with a large Heesch number?

It was around this time when I asked for Craig Kaplan’s help, as I knew of his Sat Solver program that deals with Heesch numbers. Next it was time to get some shapes cut on the Silhouette card cutter.

I then tried a few radial patterns with three-fold symmetry. The example far right is a section taken out from the one adjacent to it, as I noticed signs of periodicity but the tessellation could not continue.

In the meantime, Craig’s Sat Solver was crunching away at the numbers (it would have usually spat out any periodic tilings by this time). He threw some of the computational drawings my way (I remember it to be an exciting time). We both started to observe some interesting formations and at the same time, looking out for any repetitions.

Below is another large patch of hat tiles generated and drawn by Craig Kaplan’s software. I added the rectangular white markers to indicate the “odd” side of the small and large triangular arrangements. As you can see, they are never opposite one another.

At a glance the patches below may look the same but the directional flow on the outer rings are different.

The poor quality print out on the right was from a section from another of Craig’s drawings that does not quite display three-fold symmetry. I had an idea to first modify it (below left) and then extend the pattern outwards over to one side. Then copy, paste, rotate and move this new batch of tiles to join to the other two sides but it wasn’t an exact match. I don’t think it is possible for any radial symmetries to tile the plane but will check that with Joseph.

Chains produce a mesh that have triangle and parallelogram like features. I noticed a deep periodic band from left to right, so I played around with 180 degree rotations of it to see if they would fit together but the orientations of chains top and bottom stayed the same.

A computerised Craig Kaplan special on the left. Adjacent to that is a clever substitution rule by Craig which didn’t make it to the paper. I joined up the small triangles (that were there to fill the gaps) which produced “2” like constellations.

It was around this time that I tried other polykites. This one below really stood out in Jaap’s solver program, but putting it together was quite difficult at first.

So, I got some cut and sent a picture of it to Craig with the message “… look familiar?!”

More of Craig’s sensational computational drawings followed. I re-coloured the one on the right to show off the clusters. Notice how they never overlap but sometimes leave gaps. These can be filled with lone tiles or combinations of tiles

Below is an artwork that I made up for “A Hat for Einstien” MoMath talk just recently. The clusters of one reflected tile surrounded by six unreflected tiles is a single piece, really to speed up the work. I have played with both the hat and turtle for a long while now but I have still yet to master them. Even though I know what sorts of configurations I expect to see, I still get stuck and have to back track quite a bit.

More to follow soon…

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