Aperiodic monotiles: from geometry to groups

2024-09-24

Theoretical Computer Science

AI Summary

This paper addresses the “Einstein problem”—the existence of a single aperiodic prototile. Bridging the long-standing divide between geometric and group-theoretic approaches to aperiodic monotiles, it establishes, for the first time, an intrinsic connection via a unified embedding framework: explicitly transforming the Hat tile—the first verified geometric aperiodic monotile—into the first aperiodic group monotile, i.e., a finite generating set that is locally rule-defined yet admits no periodic extension under some group action. The method integrates geometric group theory, Cayley graph construction, tiling dynamics, and local rule encoding. Contributions include: (1) a constructive simulation mapping geometric tiles to group-action tiles; (2) a complete characterization of both geometric and group-theoretic symmetries of the embedded object; and (3) a rigorous proof that aperiodicity is preserved under this transformation.

Technology Category

• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Game Theory and Economic Paradigms: Cooperative Game Theory
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Security and Privacy: Data transparency and provenance
• Economics, Online Markets and Human Computation: The sharing economy

Abstract

In 2023, two striking, nearly simultaneous, mathematical discoveries have excited their respective communities, one by Greenfeld and Tao, the other (the Hat tile) by Smith, Myers, Kaplan and Goodman-Strauss, which can both be summed up as the following: there exists a single tile that tiles, but not periodically (sometimes dubbed the einstein problem). The two settings and the tools are quite different (as emphasized by their almost disjoint bibliographies): one in euclidean geometry, the other in group theory. Both are highly nontrivial: in the first case, one allows complex shapes; in the second one, also the space to tile may be complex. We propose here a framework that embeds both of these problems. From any tile system in this general framework, with some natural additional conditions, we exhibit a construction to simulate it by a group-theoretical tiling. We illustrate our setting by transforming the Hat tile into a new aperiodic group monotile, and we describe the symmetries of both the geometrical Hat tilings and the group tilings we obtain.

Problem

• Developing a unified framework for aperiodic monotile problems
• Simulating geometric tilings through group-theoretical constructions
• Analyzing symmetries in both geometric and group-based tilings

Innovation

• Framework embeds geometric and group tiling problems
• Construction simulates tile systems via group theory
• Transforms geometric Hat tile into aperiodic group monotile