The first is the joint work of Prof. Shigeki Akiyama and Yoshiaki Araki.
https://arxiv.org/abs/2307.12322
And here’s another similar but simpler proof by someone mysteriously known as mathblock8128.
mathblock8128.wordpress.com/2023/08/15/the-turtle-prototile-is-not-periodic-a-simple-proof-2/
The image below was taken from the Tilings Encyclopedia site (the very last entry) maintained by Dirk Frettlöh and Franz Gähler. The Wheel Tiling was discovered by Hans-Ude Nissen.
https://tilings.math.uni-bielefeld.de/
The Spectre can be can be thought of, as being composed of the two polygons that are used to make up the Wheel Tiling. If you look closely, you’ll also find the Buddha-like figure, the Mystic. This substitution tiling uses both front and backs of the Spectre and leaves gaps now and again (the smaller of the two polygons).
Curiously, I recently found some of my old drawings (dated August 2021) where I had played around with the Wheel Tiling and Spectre polygons (including reflections) but didn’t really do much with them, as you can see below (image requested by Adam Scherlis).
Below are several clusters of Spectres (rotations only) for the propeller (in yellow) and bow-tie (in red) of the Wheel Tiling to display the non-repeating pattern.
There have been several other clever examples of marked Spectres by Yoshiaki Araki and Casey Mann that use certain combinations of hexagons, pentagons, rhombi, squares and equilateral triangles. Here’s one I put together using two polygons but needs four colours.
Andrew Russell on Twitter (@AndrewR88968514) found out that you can overlay regular hexagons, squares and thin rhombi onto Spectres to produce stick-like animals; wild!
The recent Hatfest at The Mathematical Institute in Oxford was a great success. I had a wonderful time meeting up with so many hat enthusiasts, online colleagues, friends and family. A big thank you to Alex, Henna, Nick, Mike and all the backroom staff for making it all happen.
So, for those that thought the hat was two shapes, then I have a surprise for you. The Spectre doesn’t need reflections to tile non-periodically, making it a true or chiral einstein (actually two, as you also have a mirrored version). But remarkably nobody noticed it!
For everyone else, we need to deform the sides in a consistent way. A basic form of this is alternating notches (protrusions/intrusions). This Spectre has 55 sides.
But let’s backtrack a little first. It all started with the discovery of the hat then soon after, the turtle. Then we fast forward onto Joseph’s discovery of the evolutionary timeline (JET) from chevron to comet. Slap bang in the middle of it was the Spectre but seemingly boring, so got overlooked (it was periodic after all).
Inspired by Yoshiaki Araki’s fun aperiodic tile maker app https://www.t3puzzle.com/a/, it occurred to me that this polygon had more freedom than the others. So what would happen if I were to restrict it to unreflected tiles only? A pile of machine cut shapes from card followed.
After some study, Craig and I noticed that the angle placement of the middle tile (the “virtual” reflected tile) is offset by 30° relative to those that surround it, which in turn produces a small arc. I named it the Spectre (a spirit or presence perhaps?).
One of Craig’s early observations using Joseph’s hex cluster arrangements.
This example started out as a brute force computational drawing by Craig. He noticed certain groupings of Spectres that were consistent throughout, containing eight and nine Spectres respectively which he highlighted. I added the white line marking to show a wide periodic band from left to right. The green line marking was added later by Craig which heads off at a right angle.
Moving on, Joseph found out that you could simulate the way in which the Spectre tessellates by mixing hats with turtles (hat dominant) and turtles with hats (turtle dominant). It was all coming together nicely.
Then another important breakthrough. Craig produced a brilliant substitution tiling system unearthed a Buddha (a Mystic in the paper), which was a combination of two tiles that produced a symmetrical outline. And also, where every other iteration produced a reflected version. I remember being so thrilled when I saw it.
There are actually two Buddhas/Mystics in a cluster and forming a gentle arc.
A small patch of Spectres incorporating ‘chains’. Those in blue are the ‘oddball’ Spectres that take the place of reflected tiles. Long chains or worms that appear in the hat and turtle tilings do not appear in a Spectre tiling. They have more of a curved nature.
Colours indicate the ‘oddball’ Spectres. Those in red are one of the six orientations to give an idea of the complexity.
Mystics only and with a little image filtering, to help display the organised chaos.
By dilating the Mystics enough times, those ones nearest to one another, converge and create larger areas I call butterflies.
Here’s one way to draw the Spectre. Remember all sides are of unit length 1 except the longer side which is of unit length 2.
The only thing now was to alter the edges of the Spectre, so that unreflected tiles and reflected tiles could not be used in the same tessellation. That was the easy bit, as there are lots of ways to accomplish this.
Below a fantastic example by Yoshiaki Araki of how to manipulate the sides of a straight-edged Spectre to produce bizarre mythical animals.
There was a frantic push at the end for Craig, Joseph and Chaim to get it out there (I just cracked the whip), but we did it! I cannot thank them enough for their efforts.
The journey is finally complete!
More here… https://cs.uwaterloo.ca/~csk/spectre/
‘Sphere’ is an applet I found on Jaap’s Puzzle Page some years ago… https://www.jaapsch.net/puzzles/sphere.htm
I used it to create fictitious petanque boules (although one or two I recognise) with mostly ‘over the top’ designs. Striations help players tell their boules apart. I played the game for about 15 years.
I used a couple of filters in Paintshop Pro 7 to vary the colour and to create some depth.
One of a few polykites that I found around the time of the hat. It reminds me of a woodlouse or perhaps some sort of dinosaur.
Woodlice can form attractive radial patterns with 2, 3 and 4-fold symmetries. Each use all six rotations and their mirrors.
Two types of 2-fold symmetry
Two types of 3-fold symmetry
4-fold symmetry (left) and next to it, a star shaped hole at the centre.
Small and large hexagonal holes.
A decorated radial to finish.
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 non-periodically. 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…
Drafters are 30-60-90 degree triangles and can produce some neat patterns. The examples that follow are all grid based. From left to right, duck, meerkat and electric eel.
Three ducks can be assembled into boomerang arrangements. These can be placed adjacent to one another within a concertina strip. Ducks in light grey and gold on the outer edges are reflected tiles.
The concertina strips can be shuffled around to create subtle variations.
Meercats produce a ‘ring’ around an offset central six-pointed star resulting in a regular hexagon – very nice. Colours denote left and right versions.
A paved structure without the colours.
The electric eel tessellation is comprised of jagged orbs of both non-reflected and reflected tiles intermixed with six-pointed star-like motifs (only one completed).
Within the same pattern, there are also large regular hexagon clusters that overlap. The one in dark grey is constructed differently to the other two on show (which are 180 degree rotations of each other).
I used the Polyform Puzzle Solver by Jaap Scherphuis for all of these. An application I have been using for many a year, thanks Jaap.
I’ve spent many hours playing with this text prompt to image AI generator – it’s amazing… https://www.artbreeder.com/
Here’s just a few of the images I have created (in no particular order). Click on one to enlarge, then use arrow keys.
I may change the gallery from time to time.
I was going through some stuff around Christmas time and found some really old dot matrix prints. I think I had a Atari 520 STFM and a Lexmark printer at the time. The first is a Heighway dragon curve discovered by John Heighway in 1966.
I’m not familiar with the other two but they look like they are based on six sided stars. They are interesting to study, as they are both made up of a single line from start to finish. Very decorative.
I put together these two polydrafters using the Polyform Puzzle Solver by Jaap Scherphuis… https://www.jaapsch.net/puzzles/polysolver.htm
The red squirrel can produce an attractive periodic tessellation. It is made up of a large motif containing all six rotations and their mirrors.
Six grey squirrels that rotate around a common point, produce a polyhex (a regular heptahex). These motifs, whether mirrored or not, can abut one another in endless arrangements. The greyed out areas denote reflections.
Also, polyhex motifs can be aligned differently around a central polyhex, which will alter the overall appearance.
Here’s another pattern that can only be put together one way (two, if you include its mirrored counterpart).
hedraweb - David Smith 