The Path to Aperiodic Monotiles

2025/09/02 by Kaplan, Craig S.
#05B45 #52C20 (Primary) 05B50 (Secondary) #Combinatorics (math.CO) #F.2.2 #FOS: Mathematics #G.2.1 #History and Overview (math.HO)

paper · doi:10.48550/arxiv.2509.12216

Abstract

This article, written for undergraduate mathematics students, provides an accessible introduction to a few key problems in tiling theory: Heesch's problem, the isohedral number problem, and the existence of an aperiodic monotile. I contributed to the solution of the last of these problems in 2023, but many related questions remain open and worthy of study. My goal is to get more students excited about studying tiling theory.

Citations

• Tiling with Three Polygons is Undecidable
• A chiral aperiodic monotile
• An aperiodic monotile
• A counterexample to the periodic tiling conjecture (announcement)
• A Quasilinear-Time Algorithm for Tiling the Plane Isohedrally with a Polyomino

Related

• Direct Construction of Aperiodic Tilings with the Hat Monotile
• An aperiodic monotile for the tiler
• A self-ruling monotile for aperiodic tiling
• An aperiodic monotile that forces nonperiodicity through dendrites
• Ordine privo di periodicità: il fascino matematico delle tassellazioni