An aperiodic monotile that forces nonperiodicity through dendrites
| Autor: | Michael Mampusti, Michael F. Whittaker |
|---|---|
| Rok vydání: | 2020 |
| Předmět: |
Hexagonal crystal system
Plane (geometry) General Mathematics 010102 general mathematics Metric Geometry (math.MG) Dynamical Systems (math.DS) Type (model theory) Isometry (Riemannian geometry) 01 natural sciences Combinatorics Mathematics - Metric Geometry Aperiodic graph FOS: Mathematics Mathematics - Combinatorics Primary: 52C23 Secondary: 37E25 05B45 Combinatorics (math.CO) Tree (set theory) Mathematics - Dynamical Systems 0101 mathematics QA Translational symmetry Prototile Mathematics |
| Zdroj: | Bulletin of the London Mathematical Society. 52:942-959 |
| ISSN: | 1469-2120 0024-6093 |
| DOI: | 10.1112/blms.12375 |
| Popis: | 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 local growth 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 local growth rule initiates a new method to produce tilings of the plane. 19 pages, final version |
| Databáze: | OpenAIRE |
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