FoAMMar 6
July 2024. A terrace in Istria. Tiles half-laid, some already fixed, a template that doesn't match the floor, an unanswered voice call from Brussels. 480 limestone pieces, CNC-cut from a shape proven mathematically a few months before. One constraint: no tile can be flipped.

Three months, a long hot summer to find out if the pattern held.

https://anarchive.fo.am/silver/spectres/

#aperiodic #tiling #spectre #anarchive #aperiodicmonotile #mathematics #appliedmathematics #reimaginingtechnology #patterns
Show threadMark DingemanseJan 7

Is it really almost three years since the 'hat' aperiodic monotile was discovered? Found out there was an Einstein Mad Hats competition with stunningly creative uses of the 'hat', e.g. this tetris concept by William Fry and a tiling of ghibli-esque figures by Mia Fan-Chiang
https://momath.org/hatcontest/#gcwinners

#aperiodicmonotile

Christian Lawson-PerfectDec 3

You know that cool hoodie I have, with the Spectre #AperiodicMonotile on it?

Well, now you can get your own, thanks to the people at @mathsgear

https://mathsgear.co.uk/products/aperiodic-spectre-tile-unisex-hoodie

I think there's still plenty of time to get one delivered in time for Christmas 😉 🎅

Aperiodic Spectre Tile Unisex Hoodie

Show threadR. Sunder-RajOct 23

Here is a grouping of similarly made turtle tiles.

#mathart #mathart #aperiodicMonotile #monotile #tiling

R. Sunder-RajOct 14

Some of my patterns are based on tilings with can be thought of as having overlapping parts, this overlap often consists of something that looks like a border.

An example is this sequence in which the “borders” are unclear until they resolve into borders of a snub square tiling, or its Cairo-type tiling dual.

https://mathstodon.xyz/@HypercubicPeg/109043217999286054

————

Often when I do something like this, I can find an infinite class of “tiles” where the pattern along the border can be incremented in some predictable way.

A little while back, I had a go at trying to interpret the Hat tile in a similar way using edge-touching dodecagons. Here is one of the versions that I liked.

#mathart #mathsart #aperiodicMonotile #monotile #tiling #tilingTuesday

Christian Lawson-PerfectMay 7, 2025
This looks like it could work!
#AperiodicMonotile
John M. GambleFeb 2, 2025

Back in probably 2023, someone posted their monotile tile for 3D printers, for the purpose of replacing their hexagon tiles in, I think, the game Fjord.

I don't have a 3D printer, and I'm not close to anyone who does, but it doesn't matter, because The Game Crafter has them! They are listed as 53 and 43 mm across, I presume by measuring their longest and second-longest cross-sections.

#monotile #AperiodicMonotile #BoardGames

https://www.thegamecrafter.com/parts?query=infunity%20tiles%20hat

Pieter MostertDec 22, 2024

New preprint on the Spectre aperiodic monotile by Baake, Gähler, Mazáč and Sadun: https://arxiv.org/abs/2411.15503

#AperiodicMonotile #tiling

On the long-range order of the Spectre tilings

The Spectre is an aperiodic monotile for the Euclidean plane that is truly chiral in the sense that it tiles the plane without any need for a reflected tile. The topological and dynamical properties of the Spectre tilings are very similar to those of the Hat tilings. Specifically, the Spectre sits within a complex $2$-dimensional family of tilings, most of which involve two shapes rather than one. All tilings in the family give topologically conjugate dynamics, up to an overall rescaling and rotation. They all have pure point dynamical spectrum with continuous eigenfunctions and may be obtained from a $4:2$ dimensional cut-and-project scheme with regular windows of Rauzy fractal type. The diffraction measure of any Spectre tiling is pure point as well. For fixed scale and orientation, varying the shapes is MLD equivalent to merely varying the projection direction. These properties all follow from the first Čech cohomology being as small as it possibly could be, leaving no room for shape changes that alter the dynamics.

arXiv.org
Pieter MostertDec 17, 2024

Last week I described a substitution for the achiral aperiodic monotile that generates worms - sequences of tiles that can be rearranged to tile the same region in two ways: https://mathstodon.xyz/@pieter/113630380887099482

Earlier today I gave the substitution for another kind of worm found in the achiral aperiodic monotile. To summarise, if we ignore the white tiles, the combined worm substitution takes four tiles (two even and two odd) to patches made of the same four tiles, and takes another two tiles (one even and one odd) to patches made of the same two tiles. These two groups of tiles generate the wriggly and straight line worms respectively.

For the chiral aperiodic monotile (a.k.a. the Spectre), there are also two types of worms, one wrigglier than the other, and the tiles involved in generating them can also be split into two groups. The wrigglier worms are made of purple, blue and green tiles, which are even, even and odd respectively (in this context, we call a tile _odd_ if it occurs with the less frequent orientation modulo \(60^\circ\); since we've chosen to use the hats-in-turtles representation, odd tiles are hats and even tiles are turtles). The less wriggly worms are made of red, orange and yellow tiles, which are even, odd and odd respectively. Unlike the achiral monotile, however, the substitution switches these two groups of tiles, as shown in the last two figures.

(1/2)

#TilingTuesday
#AperiodicMonotile

Pieter Mostert (@[email protected])

Attached: 3 images Some tilings of the plane by the turtle aperiodic monotile contain infinite connected sequences of tiles that can be rearranged to tile the same path in two different ways. Call these paths 'worms', after Conway worms in Penrose tilings (see https://www.ams.org/bookstore/pspdf/car-36-prev.pdf for example). Given a turtle tiling with a worm, you'd expect that its inflation or deflation would also contain a worm, but how does one worm generate another? There are a few ways of describing the process; the one I describe here involves negative tiles, but if you prefer you can group them to get a system where all tiles are conventional. Tiles are called even or odd according to whether their handedness is more frequent or not. We'll consider tilings where the even tiles are left-facing turtles. The substitution depends on using two colours for both even and odd tiles, as shown in the first figure (well, tiles can also be white - these are tiles that don't form part of the worm in the deflated tiling - and their deflation is the same as the deflations of the coloured tiles, except all tiles are white.) The second and third figures show tilings of a hexagon surrounded by turtles, with two tilings of the worms. While this isn't a tiling of the plane, it has the nice property that the six turtles surrounding the hexagon fit inside their deflation, which means every deflation fits inside the next. If you're wondering if you can do a similar thing for Spectre tilings, I'm pretty sure you can, but I haven't worked out the details yet. This is a bit more interesting in that there are two types of worms, one wrigglier than the other, and deflation sends one type to the other. #TilingTuesday #AperiodicMonotile

Mathstodon
Show threadPieter MostertDec 17, 2024

Actually, turtle tilings also have two types of worms - the type shown above, and straight line worms. While straight line worms are visually less interesting, it's worth presenting the substitution since this will bring out the parallels with the Spectre worm substitution later on, and there's a nice connection to Fibonacci numbers.

The substitution is shown in the first figure, while the second shows the result of iterating it three times, starting from a single red turtle. There are some obvious truncated straight line worms in the second image, which are not highlighted. The four largest ones (two horizontal and two diagonal) are produced by two iterations of the substitution applied to the four white tiles in the first figure. This allows you to construct a hierarchical structure of truncated straight line worms, as shown in the following preprint by @jsmith, which builds on work by Erhard Künzel and Yoshiaki Araki: https://arxiv.org/pdf/2403.01911. Naturally, a similar hierarchical structure exists for the wriggly worms.

If you encode the sequence of red and orange tiles using the letters \(X\) and \(Y\) respectively, moving in an upwards direction relative to the turtles, the substitution corresponds to the free group homomorphism
\[(X, Y) \mapsto (XYX^2, X^{-1})\] or equivalently, for \(Z = YX\), to
\[(X, Z) \mapsto (XZX, ZX).\]
Since the latter homomorphism is two iterations of the Fibonacci substitution
\[(X, Z) \mapsto (ZX, X),\]
the number of red and orange tiles in each iterate are given by Fibonacci numbers, in this case 21 and 8.

#TilingTuesday
#AperiodicMonotile