Combinatorial coordinates for the aperiodic Spectre tiling| www.chiark.greenend.org.uk
Aperiodic Tilings V: the Refinable Frontier| www.chiark.greenend.org.uk
Wolfram Language function: Generate the hat tiling using combinatorial hexagons. Complete documentation and usage examples. Download an example notebook or open in the cloud.| resources.wolframcloud.com
Wolfram Language function: Generate the hexagonal tiling pattern of the hat and its supertiles. Complete documentation and usage examples. Download an example notebook or open in the cloud.| resources.wolframcloud.com
The recently discovered "hat" aperiodic monotile mixes unreflected and reflected tiles in every tiling it admits, leaving open the question of whether a single shape can tile aperiodically using translations and rotations alone. We show that a close relative of the hat -- the equilateral member of the continuum to which it belongs -- is a weakly chiral aperiodic monotile: it admits only non-periodic tilings if we forbid reflections by fiat. Furthermore, by modifying this polygon's edges we ob...| arXiv.org
The recently discovered Hat tiling admits a 4-dimensional family of shape deformations, including the 1-parameter family already known to yield alternate monotiles. The continuous hulls resulting from these tilings are all topologically conjugate dynamical systems, and hence have the same dynamics and topology. We construct and analyze a self-similar element of this family called the CAP tiling, and we use it to derive properties of the entire family. The CAP tiling has pure-point dynamical s...| arXiv.org
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 th...| arXiv.org
Two algorithms for randomly generating aperiodic tilings| www.chiark.greenend.org.uk
Simon Tatham's Portable Puzzle Collection| www.chiark.greenend.org.uk
Beyond “Beyond the wall”| www.chiark.greenend.org.uk