An aperiodic monotile

TL;DR

Addressing whether a single planar shape can enforce aperiodicity, the paper proves the hat polykite is an aperiodic monotile, concretely realized as $Tile(1,\sqrt{3})$, and shows a continuum of combinatorially equivalent aperiodic tiles $Tile(a,b)$. It introduces two independent proofs: a geometric coupling argument that rules out any strongly periodic tilings by relating hat tilings to coupled polyiamond tilings, and a Berger-style, computer-assisted substitution via four metatiles $H,T,P,F$ that yields a hierarchical tiling with an inflation factor $\phi^2$ and mutually consistent edge rules. The results demonstrate uncountably many tilings with identical local matching structure and provide a new mechanism—metatile substitution—for deriving aperiodicity from pure geometry. They also discuss a broader continuum of related shapes, including a turtle, and outline open questions about Heesch and isohedral numbers as well as undecidability in tiling theory.

Abstract

A longstanding open problem asks for an aperiodic monotile, also known as an "einstein": a shape that admits tilings of the plane, but never periodic tilings. We answer this problem for topological disk tiles by exhibiting a continuum of combinatorially equivalent aperiodic polygons. We first show that a representative example, the "hat" polykite, can form clusters called "metatiles", for which substitution rules can be defined. Because the metatiles admit tilings of the plane, so too does the hat. We then prove that generic members of our continuum of polygons are aperiodic, through a new kind of geometric incommensurability argument. Separately, we give a combinatorial, computer-assisted proof that the hat must form hierarchical -- and hence aperiodic -- tilings.

Figures (83)

• Figure 1.1: The grey "hat" polykite tile is an aperiodic monotile, also known as an "einstein". Copies of this tile may be assembled into tilings of the plane (the tile "admits" tilings), but none of those tilings can have translational symmetry. In fact, the hat admits uncountably many tilings. In Sections \ref{['sec:substitution']}, \ref{['sec:clusters']}, and \ref{['sec:subst']} we describe how these tilings all arise from substitution rules, and thus all have the same local structure.
• Figure 1.2: The high-level structure of the first proof of aperiodicity in this paper
• Figure 1.3: The high-level structure of the second proof of aperiodicity in this paper
• Figure 2.1: A computer-generated $10$-patch of 391 hats (left), arranged in ten concentric rings around a central shaded hat. The tiles can be coloured (right), showing that the reflected hats (dark blue) are sparsely distributed and each is surrounded by a congruent "shell" of three unreflected hats (light blue). A thickened outline shows the boundary of the maximal cluster of tiles that appears congruently around every reflected tile.
• Figure 2.2: Long chains of similarly oriented tiles pass through reflected tiles in six directions (left). We can merge each reflected tile with one of its neighbours in its chain (centre), yielding a structure that can be placed into one-to-one correspondence with a patch of regular hexagons (right).

Theorems & Definitions (18)

• Theorem 1.1
• Theorem 2.1
• Theorem 3.1
• Theorem 4.1
• Theorem 5.1
• Theorem 6.1
• Lemma A.1
• proof
• Lemma A.2
• proof