2023/03/20 by David Smith, David J. Smith, Joseph Samuel Myers +2 · 6 voices · 37 citations
Engineering · Mathematics · #Advanced Materials and Mechanics #Aperiodic graph #Combinatorics #Computer science #Mathematics
paper · pdf · open access · doi:10.5070/c64163843
published in Combinatorial Theory 4(1)
openalex publication_date 2024/07/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
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.Mathematics Subject Classifications: 05B45, 52C20, 05B50Keywords: Tilings, aperiodic order, polyforms