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An aperiodic monotile that forces nonperiodicity through dendrites
Author(s) -
Mampusti Michael,
Whittaker Michael F.
Publication year - 2020
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms.12375
Subject(s) - aperiodic graph , mathematics , substitution tiling , plane (geometry) , penrose tiling , dendrite (mathematics) , tree (set theory) , isometry (riemannian geometry) , combinatorics , symmetry (geometry) , quasicrystal , geometry , pure mathematics
We introduce a new type of aperiodic hexagonal monotile; a prototile that admits infinitely many tilings of the plane, but any such tiling lacks any translational symmetry. Adding a copy of our monotile to a patch of tiles must satisfy two rules that apply only to adjacent tiles. The first is inspired by the Socolar–Taylor monotile, but can be realised by shape alone. The second is a dendrite rule; a direct isometry of our monotile can be added to any patch of tiles provided that a tree on the monotile connects continuously with a tree on one of its neighbouring tiles. This condition forces tilings to grow along dendrites, which ultimately results in nonperiodic tilings. Our dendrite rule initiates a new method to produce tilings of the plane.