An aperiodic monotile (2303.10798v3)

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.

• The paper presents the hat polykite, an eight-kite aperiodic monotile that tiles the plane without translational symmetry.
• It extends the concept by introducing a continuum of aperiodic polykites, including the turtle, through systematic variation of side lengths.
• A substitution system and novel geometric coupling argument rigorously prove the aperiodicity, offering key insights for tiling theory and material science.

An Aperiodic Monotile: The Hat Polykite

The paper entitled "An aperiodic monotile" by David Smith et al. addresses a significant problem in the field of tiling theory: finding a single shape, also known as an "einstein," that can tile the plane without forming any periodic patterns. It introduces the "hat" polykite, which is proven to tile the plane aperiodically, thus solving this long-standing open problem.

Core Findings

1. Discovery of the Hat Polykite: The authors present the "hat," an eight-kite aperiodic polykite, capable of tiling the plane without any translational symmetry. This discovery adds to a sparse category of aperiodic monotiles, which have previously only been successful with more complex multi-tile configurations.
2. Continuum of Aperiodic Shapes: Beyond the hat polykite, this paper generalizes the result by demonstrating a continuum of aperiodic polykites formed by varying the lengths of the sides, characterized as $\mathrm{Tile}(a,b)$. Among these is the "turtle," a related $10$-kite tile, confirming the non-trivial breadth of the aperiodicity features of these shapes.
3. Proof Framework:
   • Substitution Systems: The paper introduces a substitution system where the metatiles (or clusters of hat tiles) form supertiles—larger combinatorially equivalent tiles, further emphasizing the inherent aperiodicity.
   • Geometric Coupling Argument: A novel geometric proof technique is utilized, where it is shown that any periodic tiling by the hat would force contradictory periodic conditions in related tilings by tetriamonds and octiamonds, which is impossible.

Implications and Speculations

• Theoretical Implications: The existence of an aperiodic monotile enriches the conceptual understanding of tiling theory, exemplifying the complexity that can emerge from interactions of simple geometric rules. It provides profound insights into the undecidability and complexity constraints of the tiling problem.
• Practical Relevance: The paper’s insights might influence practical areas such as quasicrystals structured in aperiodic tiling formations in material science, contributing to the theoretical underpinnings of their properties.
• Future Directions: This research opens opportunities to explore smaller or simpler aperiodic monotiles, potentially with fewer sides or in domains adhering to stricter constraints such as reflection-free tilings. It posits the existence of reflection-free einsteins as conceivable, prompting further exploration.
• Related Open Questions: The discovery propels further questions on aperiodic tiling, including whether higher-dimensional analogs could yield equally minimal or simpler monotile counterparts, or impact undecidability aspects regarding tile orientations and their periodicity.

The findings in this paper mark an advancement in the paper of aperiodic systems, challenging existing paradigms and extending the frontier beyond established multi-tile sets. This establishes a crucial categorical resolution for mathematical and applied investigations where aperiodicity emerges naturally.