Marjorie Senechal

This summer's Hatfest at Oxford University, UK, was a seamless hybrid of conference and festival. For two joyous days (20 and 21 July), we celebrated the Hat, while papers were presented, arguments percolated and research plans laid.

Just four months before, David Smith, a retired printer and tiling hobbyist in Yorkshire, UK; Joseph Myers, a combinatorics specialist in Cambridge, UK; Craig Kaplan, a computer scientist at the University of Waterloo, Canada; and Chaim Goodman-Strauss, a tiling theory expert at the Museum of Mathematics in New York, USA, had posted a jointly authored 89-page preprint, An Aperiodic Monotile [1], on arxiv.org. The Hat, as the tile was dubbed, made world headlines at the speed of light.

The search for a single shape that tiles the plane, but never periodically, goes back to the early 1970s, when Sir Roger Penrose's tilings by kites and darts showed that 'order' and 'translation' are not synonyms. And if two shapes can tile aperiodically, what about one? Several monotile candidates were proposed over the ensuing years whose features (periodic layers, discontinuities) prompted mutterings: that's not what we were looking for! But now here's the Hat, a simple 13-sided polygon nestling in a periodic tiling of the plane (Fig. 1).

Figure 1. A small portion of an infinite Hat tiling (Fig 1.1 of [1], reproduced with permission).

David Smith (Fig. 2) discovered the Hat while 'messing about' with the PolyForm Puzzle Solver app (https://www.jaapsch.net/puzzles/polysolver.htm).

Figure 2. David Smith at the Hatfest. Photo by Stan Sherer.

He knew that an aperiodic monotile was a tiling holy grail, and as the regions he tiled grew larger and larger with no translations in sight, he thought he might have found one. He contacted Kaplan, who contacted Myers and Goodman-Strauss. Together, the quartet proved that

(a) the Hat is indeed a tile (its copies fill the plane without gaps or overlaps);

(b) all Hat tilings are nonperiodic (they proved this in two very different ways);

(c) the Hat is just one of an infinite family of aperiodic monotiles obtained from it by increasing or decreasing the lengths of its edges (see Fig. 3 and https://cs.uwaterloo.ca/~csk/hat/).

Figure 3. Craig Kaplan's animation of the Hat family (for the link and details, see https://cs.uwaterloo.ca/~csk/hat/).

Moreover, in those four months between the posting and the Hatfest, other scholars showed that Hat tilings are aperiodic crystals (i.e. their diffraction patterns consist of sharp Bragg peaks), that the diffraction patterns have sixfold rotational symmetry (Fig. 4), and that Hat tilings can be generated by cut-and-project methods like those used for Penrose and other aperiodic tilings [2, 3].

Figure 4. Diffraction pattern of a Hat tiling [2, 3].

Still, some muttered: that's not what we were looking for! Hat tilings include – and must include – both the hat and its mirror image, but those are two different tiles! Or are they? Euclid considered them to be the same, and so does the authoritative Tilings and Patterns [4]. Objection overruled! Nevertheless, the four authors quickly tweaked a member of the Hat family into an aperiodic monotile (which they called a spectre) that tiles the plane without its mirror image [5].

The Hatfest was organized by four British scientists: Felix Flicker (physics, Cardiff University), Nick Jones (Oxford Mathematical Institute), Henna Koivusalo (mathematics, University of Bristol) and Mike Whittaker (mathematics, University of Glasgow). Fittingly, it was held in Oxford's Mathematical Institute, whose splendid courtyard is paved with a Penrose tiling (Fig. 5).

Figure 5. Entrance to the Andrew Wiles Building, home of the Mathematical Institute at the University of Oxford. Photograph by Alain Goriely, 2013. (Creative Commons Attribution-Share Alike 3.0.)

The unprecedented excitement over the Hat is due to its remarkable properties, the internet and the generosity of the authors, who freely shared their ideas and software. It was also due to its accessibility from several vantage points. As everyone at the Hatfest could attest, understanding the Hat draws on the full discovery–justification spectrum, from hands-on 'messing about' to case-by-case computation to hefty theory. Everyone could, and did, jump right in.

The first day of the festival was designed for the broadest of audiences. Natalie Frank (Vassar College, Poughkeepsie, NY, USA) brought the audience up to speed on the math. Sir Roger Penrose (University of Oxford) described the development of his two aperiodic tiles and asked whether there might be a completely different type of aperiodic monotile, one that's not situated in a periodic tiling. In my talk, I showed how paradigms baked into tiling theory and crystallography over the centuries had ingrained the periodicity paradigm. Craig Kaplan told the inside story of the discovery of the quartet's collaboration.

There were hands-on workshops, too. David Smith and Jean-Marc Castera (Paris, France) led one on the Hat, and Lorenzo Sadun (University of Texas, Austin, TX, USA) led another on origami. Hatfest participants assembled a Hat tiling (Fig. 6 and Fig. 7) during a reception and discussion with artists. The first day concluded with a public lecture by Chaim Goodman-Strauss and a panel discussion with the day's speakers, chaired by Henna Koivusalo. At one point, I think during Chaim's talk, Felix Flicker told me later: "I looked back over the full lecture theatre (we had around 330 people signed up in the end) and saw people from all walks of life delighted to be hearing about this fascinating discovery. One person was wearing a Hat hat, while another was wearing a Hat dress. People were drinking Hat beer. Outside the theatre, the walls were lined with Hat artwork, including a hand-sewn quilt that had taken thousands of hours to make. People are thrilled about it in a way I haven't seen before for other discoveries."

Figure 6. Participants in the Oxford Hatfest creating a tiling with paper hats. Photograph by Stan Sherer.

Figure 7. The red paper hats in Fig. 6 are mirror images of the white hats.

Talks on the second day – see https://sites.google.com/view/thegrimmnetwork/hatfest for the full list of speakers, affiliations, lecture titles and abstracts – situated the Hat in up-to-the-minute tiling theory and in the growing field of aperiodic order. And Adolfo Grushin (CNRS, Grenoble, France) discussed joint work with Felix Flicker on possible applications in physics and materials science, in which they found that Hat tilings sit "in a previously unoccupied place in crystallography between quasicrystals and crystals." And, by looking at the behavior of electrons in tight binding models based on the Hat, they found properties remarkably similar to those of the 2D-periodic crystal graphene. (Other properties lie closer to those of models of disordered materials and quasicrystals.)

The Hatfest was the second meeting of the Grimm Network (https://sites.google.com/view/thegrimmnetwork/hatfest), founded by the four organizers to honor the memory of the mathematician and physicist Uwe Grimm (Open University, UK), who died unexpectedly in 2021. "Uwe was always really welcoming, bringing together as broad a range of people as possible to create an international community interested in aperiodic tilings," Felix Flicker explained. "We sought to continue his legacy by bringing together mathematicians, physicists, artists, hobbyists, and more. I think Uwe would have been very proud of the event and his legacy."

I am pleased to dedicate this note to Uwe, who was my colleague and friend. I hope that future meetings of the Grimm Network will keep us up to date on this rapidly burgeoning field.


[1] Smith, D., Myers, J. S., Kaplan, C. S. & Goodman-Strauss, C. (2023). arXiv: 2303.10798v1.

[2] Socolar, J. E. S. (2023). arXiv: 2305.01174v1.

[3] Baake, M., Gähler, F. & Sadun, L. (2023). arXiv: 2305.05639v1.

[4] Grünbaum, B. & Shephard, G. (1987). Tilings and Patterns. New York: W. H. Freeman & Co.

[5] Smith, D., Myers, J. S., Kaplan, C. S. & Goodman-Strauss, C. (2023). arXiv: 2305.17743.


Marjorie Senechal is the Louise Wolff Kahn Professor Emerita in Mathematics and History of Science & Technology at Smith College, Northampton, MA, USA.
25 September 2023

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