[Paper Review] An aperiodic monotile

David J. Smith, Joseph Samuel Myers+2 more|arXiv (Cornell University)|Mar 20, 2023

Quasicrystal Structures and Properties39 references21 citations

TL;DR

The paper proves that the hat polykite is an aperiodic monotile (einstein) for topological disks, providing two proofs and a continuum of aperiodic shapes Tile(a, b).

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.

Motivation & Objective

• Motivate the search for a single aperiodic monotile (einstein) among disk-shaped tiles.
• Show that the hat polykite admits plane tilings but none that are periodic.
• Demonstrate two independent proofs of aperiodicity (one constructive via metatiles, one Berger-style).
• Explore a continuum of related shapes Tile(a, b) that tile the plane with combinatorially equivalent tilings.

Proposed method

• Construct tilings of the hat by identifying clusters called metatiles arising from local hat configurations.
• Form a substitution system on metatiles that yields supertiles with the same combinatorial structure as the metatiles.
• Prove aperiodicity by showing no tiling by hats can be strongly periodic and by analyzing hierarchical tilings.
• Provide a second, combinatorial/phased proof using a Bergif-style argument on metatiles and hierarchical structure.
• Use computer-assisted case enumeration to support the substitution framework and ensure all tilings follow the same structure.

Experimental results

Research questions

• RQ1 Can a single topological-disk tile tile the plane only non-periodically (is an einstein)?
• RQ2 Do the hat and its continuum Tile(a, b) enforce a hierarchical, non-periodic tiling structure?
• RQ3 Can metatiles derived from hat clusters support a substitution system yielding aperiodic tilings?
• RQ4 Do both constructive and Berger-style arguments jointly establish aperiodicity for the hat and related shapes?

Key findings

• The hat polykite is an aperiodic monotile: it tiles the plane but admits no translationally periodic tilings.
• Every tiling by hats contains a unique hierarchy of supertiles that enforces non-periodicity.
• Metatiles derived from hat clusters admit a substitution system and tile the plane.
• There exists a continuum Tile(a, b) of shapes exhibiting combinatorially equivalent tilings to the hat, all aperiodic except for some degenerate cases.
• A secondary, computer-assisted proof confirms that the hat must form hierarchical—and hence aperiodic—tilings.