An aperiodic hexagonal tile (1003.4279v2)

Abstract: We show that a single prototile can fill space uniformly but not admit a periodic tiling. A two-dimensional, hexagonal prototile with markings that enforce local matching rules is proven to be aperiodic by two independent methods. The space--filling tiling that can be built from copies of the prototile has the structure of a union of honeycombs with lattice constants of $2^n a$, where $a$ sets the scale of the most dense lattice and $n$ takes all positive integer values. There are two local isomorphism classes consistent with the matching rules and there is a nontrivial relation between these tilings and a previous construction by Penrose. Alternative forms of the prototile enforce the local matching rules by shape alone, one using a prototile that is not a connected region and the other using a three--dimensional prototile.

• The paper demonstrates that a single hexagonal tile, combined with specific local matching rules, can tile a plane aperiodically.
• Aperiodicity is rigorously proven through two independent methods: a direct proof showing how rules necessitate larger nonperiodic structures and a substitution proof revealing hierarchical self-similarity.
• This research has implications for theoretical understanding of tilings and practical applications in fields like crystallography, quasicrystals, and computational materials science, including potential 3D extensions.

Overview of "An Aperiodic Hexagonal Tile" by Socolar and Taylor

The paper presented by Joshua Socolar and Joan Taylor introduces an innovative approach to aperiodic tiling using a singular prototile in Euclidean 2D space. They rigorously demonstrate that a single hexagonal tile, paired with specific local matching rules, can cover a plane in a non-overlapping manner while ensuring that the resulting tiling remains nonperiodic. The research employs two independent proofs and examines potential connections to Penrose tilings. Moreover, their paper expands on alternative configurations, including a three-dimensional interpretation, enforcing matching rules through geometric shape alone.

Key Findings and Contributions

The core contribution of this paper revolves around the concept of enforced aperiodicity via a single prototile. Socolar and Taylor offer multiple pathways to prove that with properly defined local matching rules, a space-filling tiling is aperiodic. These rules include continuous edge markings and aligned vertex decorations across adjacent tiles.

Aperiodicity Proofs

They initially provide a direct proof that explores the implications of matching constraints on hexagonal tiles. The theorem established outlines how these rules necessitate the creation of progressively larger lattices, leading to nonperiodic global structures.

The subsequent approach involves substitution rules. The substitution proof focuses on decomposing and reconstructing tiles into larger versions with identical matching constraints, pushing the arrangement towards inherent nonperiodicity. This aligns with the prevalent practice in tiling research where substitution methods reveal hierarchical self-similarity in aperiodic patterns.

Numerical and Structural Insights

Throughout the paper, Socolar and Taylor substantiate their theoretical assertions with compelling numerical foundations, such as examining the relationships between tile orientations within substitution hierarchies. They emphasize the distinction between local isomorphism classes of the tiling: one class producing a myriad of distinct yet equivalent tilings and another constrained by a singular vertex configuration.

Theoretical and Practical Implications

This paper aligns with theoretical endeavors to expand the understanding of tilings beyond periodic structures, contributing to disciplines like crystallography and the paper of quasicrystals. It provides a groundwork for practical applications in computational materials science, where the spatial arrangement in nano-fabricated structures may leverage aperiodicity to enhance material properties or design novel interactive surfaces.

The authors mention potential avenues for further exploration, particularly concerning the topological implications of tiling defects and the dynamics of self-assembly. Through the introduction of a uniquely shaped 3-dimensional tile, Socolar and Taylor also invite future investigations into higher-dimensional applications, possibly affecting fields of spatial computing and architecture.

Conclusion

Socolar and Taylor's work pushes the conceptual boundaries of tiling by precisely illustrating that enforced nonperiodicity can be achieved with a singular prototile. Their paper invokes thoughtful consideration on how mathematical tile arrangements impact both theoretical exploration and practical implementations in material design. Notably, their research continues to inspire the mathematical community to find novel ways of defining and understanding space-filling patterns.