2019/03/04 by Michael Mampusti, Mampusti, Michael, Michael F. Whittaker +1
Computer Science · Engineering · Materials Science · Mathematics · #05B45 #Advanced Materials and Mechanics #Cellular Automata and Applications #Combinatorics (math.CO) #Dynamical Systems (math.DS) #FOS: Mathematics #Metric Geometry (math.MG) #Primary: 52C23 #Quasicrystal Structures and Properties #Secondary: 37E25 #math.CO #math.DS #math.MG #msc:05B45 #msc:37E25 #msc:52C23
paper · pdf · doi:10.48550/arxiv.1903.01158
19 pages, final version
openalex publication_date 2019/03/04 · arxiv created 2020/05/22 · arxiv updated 2020/05/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.