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.
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…
39 thoughts on “It’s a shape Jim, but not as we know it”
Robert Reid, mentioned frequently on this page was a very good friend for many years. Once a month we used to meet at his flat in Kilburn and share cheese and work on tessellations. He was one of the most amazing mathematicians I’ve ever met, and during his life in Peru, from age 25 to 75, much of his work was done without a computer. He hardly used a computer until he came to England in his 70s. Even in his later years, as he became increasingly frail he would use his ‘home helps’ to cut shapes out of card rather than giving him the care that they were supposed to be there for.
Martin H. Watson
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I forgot the link:
Fabulous, Dave! Congratulations and hats off to you. 🙂
I did excuse the pun! Thank you.
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🙂 You’re welcome, thank you. Loved you presentation at MoMath’s event this Saturday. ❤
Ah, so you did! My apologies.
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All well, BIG congratulations, again. 🙂
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Thanks for your vote. Joseph, Craig and Chaim did all the work.
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You’re welcome, Dave. Yesterday I’ve heard otherwise. 🙂
Fascinating! Thanks for the inside insights.
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Would you be willing to post your polysolver file for this? I didn’t see this grid style available in the polysolver program.
It should be there. Under grid type, click on 6-fold symmetry grids to open up a list. Drafter is the third one down. Let me know how you get on.
Thank you sir, somehow that’s one I missed.
Fabulous work. I’m still looking for an online tool that makes it really easy to experiment with tiling the plane. I like the idea of providing each of the metatiles and/or other common clusters, and would love something that made it feel sort of magnetic to join new pieces with existing ones.
Thank you. I sometimes use Silhouette Studio to create larger patches, as it has a snap to grid option (hats and turtles align to a drafter or asymmetric grid). Most people use Adobe Illustrator for this type of thing but I can’t afford that. I use the Polyform Puzzle Solver by Jaap Scherphuis for most things but is basic quality (raster not vector). Regards – Dave S.
Good news, Dave; Adobe Illustrator CS2 is available for free, with no catches! A while back, Adobe made CS2 available for free on their website (CS2 being outdated by this time), but have latterly removed this facility. However, by using the Wayback Machine to download, the required serial numbers are still available. Note that this is a simple, albeit hefty (600MB!), download, fully functioning version; there are no time limits involved, cracks or key gens to get in the way (not that I know much about the last two). Note that the serial numbers have to be inputted manually (not copied and paste). Photoshop CS2 (and more!) is also freely available in the same manner! CS2 is more than powerful enough for drawing basic tilings; subsquent upgrades are overkill (and never-ending expense in subscribing!) in this regard.
Let me know if you have any problems.
I wonder whether you would be interested in doing a public talk on your work, and the story of how you solved the aperiodic tiling problem? Leeds Cafe Scientifique would be very pleased to hear a talk, and see some of the amazing images. I’m sure we can sort out some accommodation in Leeds, as Bridlington is not quite next door, despite being in Yorkshire (these days).
Congratulations on your discovery, and on the coverage generated.
Thank you for the invitation but it wasn’t me that solved it, I just discovered the polykites. Initially, Craig Kaplan did most of the preliminary work. Later we asked Chaim Goodman-Strauss and Joseph Myers to join us. Between them they worked out several proofs and wrote the scholarly paper. It was a long process.
I will think on it though but I’m not much of a talker.
div>Thanks for your reply. I’m no expert
Thanks for your reply. I’m no expert on the hierarchies of these things, but the Guardian article names you prominently, and Kaplan’s Twitter thread names you first among the collaborators.
It is true that nearly all of the people who do Cafe Sci talks are full-time professors / fellows at local universities, and so they are probably more used to talking in front of an audience. Of course, they are all publicly funded and so strongly encouraged to do that sort of “engagement” or “outreach”. If you’d be interested in doing something a bit different, I might be able to find a willing mathematician at Leeds Uni who could do an introduction and then do the event with you as a Q&A or interview (which is often how book festivals work).
I’m no expert in tilings either, but I read Penrose’s book many years ago, where he described his 2-shape aperiodic tilings. Given that he couldn’t / didn’t find a single shape, the achievement of finding one seems very impressive. The coloured diagrams are also very visually impressive. The venue we use for Cafe Sci meetings has a large screen (used for films) and the tilings would be well displayed on it.
My mobile number is 0778 666 0584 if you’re interested in discussing this further, though I understand if you decide against.
Thank you, and congratulations again on this discovery.
It seems to me that your new shape could be protected as a registered design for a product such as a tile for use on floors or walls. Protection should be possible in the UK and the EU, at least. Please contact me if you are interested. There is a deadline for filing for protection. David Bottomley, Chartered UK Patent Attorney.
This is something we discussed early on but decided against it. All I ask, is for those that are making profits from the hat, please consider making a donation to CIWF. Thank you.
Hi David – have you thought any more about doing a talk
Ok. But from what I understand, the shape is something you came up with on your own, which would make you the sole designer, giving you a different status to your coauthors on the paper. If that is correct, the design is yours alone to do with as you please. If you like, please drop me a line through my email address and we can have a zero cost call to at least let you know what your legal rights include, even if right now you have no plans to do anything with them.
I don’t really think it’s something I could do right now. I can draw a shape but that’s about it.
Sorry to disappoint.
Is there a restriction on vertex types? As you will know, Dave, the Penrose rhombuses only allow 8 possible arrangements around each vertex.
Paul, do you mean for a legal tiling? I don’t know the answer to that one so will have to ask Craig. Thanks.
That’s right. Many thanks for passing in my query.
Second the question! According to some calculations, symmetric trivalent vertices number 10 immediately possible, with only eight showing up. Quite amazingly, they must all be one same parity. This is a *major* feature of the tiling not to underestimate! Wonderful, thanks for your question. –Brad
Thanks, Brad. More to explore here. Dave, a curiosity: you can combine the hat and turtle to make a tile which tiles periodically: I’ll e-mail a picture.
Joseph Myers has extensively computer searched all polykites up to 21. The hat when combined with a turtle is an 18-kite, so should be somewhere in that list.
No wonder I stumbled on one so easily!
I think I found out how some of the patterns you described here arise. I laid it out in a series of twitter posts. It’s a bit long, but I’m mostly just describing pictures: https://twitter.com/mathBlock/status/1659658278836801536
Good article, thanks.
Yes, they remind me of tributaries. If you treat these as a maze, every hat will be accessible.
Thank you for this new shape!
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