# Hatfest: celebrating the discovery of an APERIODIC MONOtile

## Hatfest took place at Oxford University on July 20th and 21st 2023

Hatfest was a two-day celebration of The Hat, an aperiodic monotile: see here, here, or here!

The first day contained talks and workshops aimed at the public, while the second contained accessible talks aimed at the broadest possible mathematical audience. There were artworks on display and activities for the duration of the event (click the image above the green text to load the pdf guide to the artworks, produced by Lucy Ward).

## Location

The Mathematical Institute, Oxford University

Address: University of Oxford, Radcliffe Observatory, Andrew Wiles Building, Woodstock Rd, Oxford OX2 6GG

## Schedule

Thursday 20th July 2023

9.00 - 9.50 Prof. Natalie Frank (Vassar College): slides, video

10.00 - 10.50 Prof. Sir Roger Penrose (University of Oxford)

10.50 - 11.20 coffee

11.20 - 12.10 Prof. Marjorie Senechal (Smith College): video

12.10 - 12.40 UKMT and MoMath announcement

12.40 - 14.30 Lunch

14.30 - 15.20 Prof. Craig Kaplan (University of Waterloo): video

15.30 - 16.30 Hands-on workshop by Dave Smith, discoverer of The Hat, and Jean-Marc Castera

15:30 - 16:30 Origami workshop by Lorenzo Sadun

16.30 - 18.00* Record-setting Assembly of Hat tiling

17.00 - 18.00* Drinks reception and discussion with artists

18.00 - 19.30 Panel discussion chaired by Dr Henna Koivusalo. Preceded by an introduction by Prof. Goodman-Strauss

*in parallel, video

Link to Titles and Abstracts: Thursday (also below)

Friday 21st July 2023

9.00 - 9.10 Opening

9.10 - 10.00 Prof. Chaim Goodman-Strauss (National Museum of Mathematics): video

10.00 - 10.30 Dr Jamie Walton (University of Nottingham): video

10.30 - 11.00 Daniel Roca González (KIT): video

11.00 - 11.30 Coffee

11.30 - 12.20 Prof. Jarkko Kari (University of Turku): video

12.20 - 12.50 Dr Adolfo Grushin (Grenoble): video

12.50 - 14.30 Lunch

14.30 - 15.20 Prof. Rachel Greenfeld (Institute for Advanced Study)

15.20 - 15.50 Dr Shrey Sanadhya (Ben Gurion University of the Negev): video

15.50 - 16.10 Coffee

16.10 - 17.00 Prof. Lorenzo Sadun (University of Texas, Austin): video

17.00 - 17.10 Closing

Link to Titles and Abstracts: Friday (also above)

## Organisers

Felix Flicker, Nick Jones, Henna Koivusalo, and Mike Whittaker

## Funding

This event was organised in partnership with the Clay Mathematics Institute and is supported by the Mathematical Institute, University of Oxford, The Engineering and Physical Sciences Research Council, The Heilbronn Institute for Mathematical Research, The Institute of Physics and The London Mathematical Society.

*Header image is due to Samuel Velasco/Quanta Magazine, see here.

## Titles and abstracts: Thursday

9.00 - 9.50 Prof. Natalie Frank (Vassar College): slides

Title: What is an "aperiodic monotile", and why are people so excited about it?

Abstract: The discovery of an aperiodic monotile, announced in March of this year, has been heralded in the New York Times, the Guardian, and multitudes of news sites, blogs, social media, and even made a comic appearance on American late night television. Now we're having a whole conference about it! This talk will to bring you up to speed on what the hubbub is all about. I'll provide historical context and mathematical motivation behind the search for an aperiodic monotile, talk about the many moments of progress prior to the discovery, and discuss the overarching theoretical framework behind the proof of aperiodicity.9.50 - 10.40 Prof. Sir Roger Penrose (University of Oxford)

Title: On non periodic tilings11.00 - 11.50 Prof. Marjorie Senechal (Smith College)

Title: On hiding in plain sight

Abstract: There it is, sitting right among the hexagons! And it's such a simple polygon! Why hadn't "The Hat" , or another member of its family of aperiodic monotiles, ever been found before? In this talk we explore how paradigms baked into tiling theory and crystallography over the centuries pointed in the wrong directions.11.50 - 12.40 Prof. Craig Kaplan (University of Waterloo)

Title: Aperiodic monotiles: new shapes just dropped

Abstract: A set of tiles is aperiodic when they can be used to tile the plane, but never in a way that repeats in a regular pattern of translations. Penrose's "Kite and Dart" tiles, discovered in the 1970s, show that a set of size two can be aperiodic, but since that time nobody had been able to find a set of size one---that is, a single tile---that displays the same behaviour. In 2023, David Smith, Joseph Samuel Myers, Chaim Goodman-Strauss and I showed that a simple shape called the "Hat" is an aperiodic monotile. We later also showed that a related family of shapes called "Spectres" tile aperiodically without using reflected tiles. In this talk I will introduce tilings and aperiodicity, and tell the story of the discovery of these marvellous shapes and of the collaboration that led to the proof of their aperiodicity.

## Titles and abstracts: Friday

9.10 - 10.00 Prof. Chaim Goodman-Strauss (National Museum of Mathematics)

Title: Problems In Tiling, Open and Closed

Abstract: The Hat and Spectre monotiles lay to rest the longstanding question of the existence of an aperiodic monotile. We’ll survey the status of several related decision and existence problems, across a range of settings, such as hyperbolic space, more abstract settings, or tilings by a single monotile.10.00 - 10.30 Dr Jamie Walton (University of Nottingham)

Title: Hexagonal aperiodic tiles

Abstract: Before the discovery of the hat monotile, several small aperiodic prototile sets were found which were based on the simple periodic tessellation of hexagons. These include Penrose’s 1+ε+ε2 tiles, the Taylor–Socolar tile and a tile based on ‘orientational matching rules’. These three fall short of solving the problem of finding a single tile that forces aperiodicity purely via tile geometry. In particular, the latter set is more faithfully considered (from the perspective of shifts of finite type) as a twin pair of tiles with charged edges, the two tiles being identical only up to a charge flip that 2-to-1 factors to aperiodic model sets. I will show the proof of aperiodicity of this tile set, which is notably very simple. The three sets above all factor to the aperiodic arrowed half-hex tilings, whilst the hat, although being related to hexagons, forces aperiodicity in a very different way. So, in discussing these other near misses to the strongest forms of the aperiodic monotile problem, I hope to frame what a remarkable discovery the hat was.10.30 - 11.00 Daniel Roca González (KIT)

Title: Hyperuniformity of hat tilings

Abstract: Hyperuniformity (of Torquato-Stillinger) and number rigidity (of Ghosh-Peres) are two long range order properties of point processes (or tilings), which have attracted a lot of attention by mathematical physicists and probabilists alike due to their manifold connections to material science, chemistry and biology.

While these properties are by now fairly well-understood for one-dimensional tilings, very little was known about them for higher-dimensional tilings until recently. In this talk we establish hyperuniformity for many tilings coming from inflation rules, including hat tilings. For other tilings, such as Penrose tilings and certain tilings associated to the hat, we are also able to prove number rigidity, making them be among the first examples with finite local complexity, hyperuniformity and number rigidity in higher dimensions. Our proof of these facts uses three main ingredients: Spectral characterizations of hyperuniformity and number rigidity, spherical diffraction for the group of motions of the Euclidean plane (also known as powder diffraction) and self-similarity equations for spherical diffraction which generalize self-similarity equations of Baake and Grimm for ordinary diffraction.

Our method applies more generally to substitution tilings of arbitrary dimensions for which there is a sufficiently large eigenvalue gap in the reduced substitution matrix.11.30 - 12.20 Prof. Jarkko Kari (University of Turku)

Title: Low complexity colorings of the two-dimensional grid

Abstract: A two-dimensional configuration is a coloring of the infinite grid Z^2 using a finite number of colors. For a finite subset D of Z^2, the D-patterns of a configuration are the patterns of shape D that appear in the configuration. The number of distinct D-patterns of a configuration is a natural measure of its complexity. We consider low-complexity configurations where the number of distinct D-patterns is at most |D|, the size of the shape. We use algebraic tools to study periodicity of such configurations [1]. In the case D is a rectangle - or in fact any convex shape - we establish that a uniformly recurrent configuration that has low-complexity with respect to shape D must be periodic [2]. This implies an algorithm to determine if a given collection of mn rectangular patterns of size mxn admit a configuration containing only these patterns. (Without the complexity bound the question is the well-known undecidable domino problem.) We also show, for an arbitrary shape D, that a low-complexity configuration must be periodic if it comes from the well-known Ledrappier subshift, or from a wide family of other similar algebraic subshifts [3].

References

[1] J. Kari, M. Szabados. An Algebraic Geometric Approach to Nivat’s Conjecture. Information and Computation 271, pp. 104481 (2020).

[2] J. Kari, E. Moutot. Decidability and Periodicity of Low Complexity Tilings, proceedings of STACS 2020, the 37th International Symposium on Theoretical Aspects of Computer Science, LIPIcs 154, pp.14:1-14:12 (2020).

[3] J. Kari, E. Moutot. Nivat’s conjecture and pattern complexity in algebraic subshifts. Theoretical Computer Science 777, pp. 379–386 (2019).

12.20 - 12.50 Dr Adolfo Grushin (Grenoble)

Title: Quantum physics out of the Hat: graphene shadows, zero-modes and topology

Abstract: In this talk I will discuss the physics of electrons propagating on the Hat. The electron’s energy as a function of its momentum displays striking similarities to that of graphene, but also striking differences. First, the electronic properties differ for the two mirror images of the tiling. Second, certain states have exactly zero energy; their number equals the number of flipped hats when the magnetic flux per tile is half of the flux quantum. Third, the Hat’s electronic spectrum is periodic in magnetic flux, unlike in typical quasicrystals. Additionally, I will show that arbitrary flux values lead to topologically protected boundary modes with universal conductance. To conclude, I will discuss possible experimental realisations of this physics in state-of-the-art experiments.14.30 - 15.20 Prof. Rachel Greenfeld (Institute for Advanced Study)

Title: Tiling by translations

Abstract: Translational tiling is a covering of a space (such as Euclidean space) using translated copies of a building block, called a "translational tile'', without any positive measure overlaps. Here, the tile is not allowed to be rotated or reflected, which imposes a stronger structure on the tilings: translational tilings are conjectured to have properties that tilings by a broader group of motions do not have.

In the talk, we will survey the fascinating study of translational tilings, highlighting its distinctions from more general tilings and its connections with other problems; we will further discuss recent developments, new results and open problems.15.20 - 15.50 Dr Shrey Sanadhya (Ben Gurion University of the Negev)

Title: Periodicity of joint co-tiles in $\mathbb{Z}^d$

Abstract: Given a finite set (a tile) $F \subset \mathbb{Z}^d$, we say that a set $A \subset \mathbb{Z}^d$ is a co-tile of $F$ if the collection of sets $F+a$, for $a \in A$, forms a tiling of $\mathbb{Z}^d$. For finitely many tiles $F_1,...,F_k$, we say that $A$ is a joint co-tile if $A$ is a co-tile of each $F_i$.

In this talk, we will discuss the structure of joint co-tiles in $\mathbb{Z}^d$, particularly their periodicity. We will discuss the connections of this notion to the periodic tiling conjecture (PTC), whose $\mathbb{Z}^2$ case was resolved by Bhattacharya, and a counterexample in high dimension was recently given by Greenfeld-Tao. Our setup extends a theorem of Newman (every co-tile in $\mathbb{Z}$ is periodic) to higher dimensions. It provides characterization for the periodicity of a co-tile for all $d>2$. This is joint work with Tom Meyerovitch and Yaar Solomon [arXiv:2301.11255].16.10 - 16.50 Prof. Lorenzo Sadun (University of Texas, Austin)

Title: Deformations and dynamics of the Hat family of tilings

Abstract: Shape deformations of a tiling are governed by its first Cech cohomology. For the Hat, a simple computation shows that there is a 4-parameter family of tiling spaces, topologically conjugate but not MLD to the original. This includes the 1-parameter family developed by Smith et al, a self-similar CAP tiling, and Socolar's golden Key tiling. The spectrum of all of these tilings is pure point and equals $\mathbb{Z}[\xi, \phi]$, where $\xi = \exp(2 \pi i/6)$ and $\phi$ is the golden mean. All tilings are cut-and-project as well as hierarchical, with the exact same total space and window, only with different projections to physical space. Similar comments apply to the recently discovered Spectre tilings. This is joint work with Michael Baake and Franz Gaehler.