A New Tiling
With a flip and some twists
Roger Penrose has been floored. And perhaps re-floored. Here he is standing on the floor of the Mitchell Institute of Texas A&M, which features one of the famous aperiodic tilings he discovered in the 1970s.
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Aperiodic means that the set of two rhombus tiles (with notches making them align only in certain ways) can cover the entire plane but only without any globally repeating pattern. Simple tilings, of the kind we covered here, make a finite pattern that can be translated to cover the infinite plane.
Penrose discovered another aperiodic tiling with only two pieces: the kite and dart (again with forced markings). But for over 50 years it was open whether there is a single connected tile that can cover the whole plane but only aperiodically.
Oxford’s new Mathematical Institute—Roger’s home base—has a courtyard paved with another of his tilings. Now there is a new pattern to use for flooring:
The New Pattern and Its Twists
The single tile and its surprising properties were discovered by David Smith, Joseph Samuel Myers, Craig Kaplan, and Chaim Goodman-Strauss.
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| Composite crop from this, this, this, and this. |
The introduction to their recently-posted paper has a nifty sentence that foreshadows why neither the discovery nor the proof of its correctness were easy:
Can one tile embody enough complexity to forcibly disrupt periodic order at all scales?
They have answered this question yes, though with one asterisk: their asymmetric tile needs to call on its mirror twin—obtained by flipping it over—to complete the tilings. Whereas, the tiles in Penrose’s pairs each have mirror symmetry and so work solely by translations. Penrose’s rhombus tiles are also convex, whereas the 13-sided “hat” or “shirt” shape—as with Penrose’s “dart” shape—is not.
After several popular-science articles and blog posts, in turn linked by Gil Kalai, the New York Times covered this in Tuesday’s science section. The story by Siobhan Roberts quoted Marjorie Senechal—who was recently editor of the Mathematical Intelligencer:
“Mathematicians had been searching for such a shape for half a century. It wasn’t even clear that such a thing could exist.”
This applies whether or not one insists on convexity or translation without flips. In fact, the tile is composed of eight hexagonal slices, also referred to as “kites” but thinner than Penrose’s, in a way that forms an asymmetric sub-region of a simple repeating tiling of those regions:
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| source crediting Tilman Piesk |
This does not constitute a proof that the tile can cover the whole plane, let alone that it must do so aperiodically. Penrose is quoted in the NYT article as saying he finds the proofs in the new paper “very complicated.”
One further fact that both augments and aids the task of proving is that all tilings composed of hats can be continuously deformed keeping identical shapes that still tautly cover the plane. One direction of twisting terminates in a chevron pattern and the other in a “comet” composed of a hexagon plus diamond. The chevron and comet can tile repeatedly but here they do not—meanwhile the authors prove that every shape in-between can only tile aperiodically. The animation created by Kaplan is also in the NYT article.
The Proofs
The proofs in the paper need to accomplish two goals:
- Show that the tile (and its flip) can cover the entire plane—no holes or gaps or dead-ends.
- Show that no such covering can be periodic.
Both tasks are hard, but among several proofs of the latter there is a nifty trick. The former works by an argument that rings of the hat tiles have a logical “metastructure” that enables them to scale up while preserving the ability to mesh with the next-inner and next-outer ring. Because the scale expands, this does not constitute a finite period. Also providing slack is the further-proved fact that the hat tile can cover the plane in uncountably many ways, no two isomorphic.
Some visuals for the proofs of both points are summarized in a seven-part telling on twitter by Alex Kontorovich of Rutgers, which is based on a presentation given by the authors at the National Museum of Mathematics, with which Goodman-Strauss is affiliated.
The latter point is given multiple proofs in the paper. One of them, also summarized by Kontorovich from the presentation, has a nifty trick: Suppose the hat gave a periodic tiling. That tiling would admit the continuous twisting action, under which periodicity would be preserved. In particular, the chevron and comet tilings at the endpoints would be periodic after all.
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What the authors call “a new kind of geometric incommensurability argument” then comes into play. It ultimately reaches the conclusion that the simple triangular lattice would have equal-area regions under scalings that differ by a factor of 
The paper also gives a long separate proof of forced aperiodicity by means that are more analogous to proofs with other tile sets, but here it needs computer-assisted enumeration and verification of many cases.
Logic and Complexity Impact?
The subject of tilings has a long history growing out of art and esthetics, but vital connections to biology, crystallography, and logic were discovered more recently. The logical connection ties in the other two and was elevated by Hao Wang’s famous proofs that certain tiling problems—and questions of periodicity—are undecidable.
Just a few months ago, the online magazine Quanta covered recent work by Rachel Greenfield and Terance Tao showing that in certain high dimensions 
The first paper by Greenfield and Tao is titled, “Undecidable translational tilings with only two tiles, or one nonabelian tile,” while the second paper is, “A counterexample to the periodic tiling conjecture” with the above result. This supplements and offsets the geenral result that when all translational tilings must be periodic, whether such a tiling is possible for a given set of tiles becomes decidable.
We wonder what the impact of the new result will be on these connections, and how far the nexus of logic and combinatorics will reach down into computational complexity theory.
Open Problems
In-between the new discovery using a tile and its flip and the impossibility results for translation-only tilings are questions when rotations in the plane but not flips are allowed. Are those questions still wide open? Look at here for lots on this issue in general.
And partly in jest we ask: will Oxford’s Mathematical Institute—or maybe better, MoMath—put in a new floor with the new tiling?
[fixed usage of “kites”, noted Penrose tiles need notches/markings to force aperiodicity]
Re “In fact, the tile is composed of eight of Penrose’s kites”: in fact, Penrose’s kites have 72°–72°–72°-144° angles, but these ones are 60°–90°–90°–120°.
Ah, thanks! I could have twigged on the illogic of one stone being able to tile in ways two stone could not. The status of rotations and Z^2 versus R^2 at the end could also use some more sorting out.
I hope this tweets-thread of mine will help handle the tiling problems, if a bit:
https://twitter.com/koitiluv1842/status/1641767583451070465
This is probably a stupid question, but does this change anything with respect to the four-color theorem ?
There was an article in Quanta on 4CT just two days ago, but I left the idea of mentioning it at the end of the “proofs” section on the cutting-room floor. I did not see any concrete relation. The “new geometric incommensurability argument” may have other applications, however. The claimed human-readable nonconstructive proof of 4CT, which we included toward the end of a post on nonconstructivity, did get withdrawn.
No asterisk is needed. Reflections and glide-reflections are isometries of the Euclidean plane, so it’s accepted for some tiles to be (glide) reflections of others.
Now Penrose’s tiles need decoration (bumps and notches) of some sort, to enforce the matching-rules and prevent periodic tilings. This makes them asymmetrical and concave. The beauty of Smith, Myers, Kaplan and Goodman-Strauss’s tile is that it does not need any such decoration. It is a pure polyform, made, as you say, from 8 kites, or, alternatively, from 4 pentagons of the shape where you get 3 pentagons by trisecting a regular hexagon by 3 apothems.
Thanks! Took me awhile to twig on this—so noted.