Inspired by the new substitution rules that Shigeki Akiyama and Yoshiaki Araki propose (along with a new proof of aperiodicity!), here are some drawings from the last few days. These aren’t quite the same as
Sometime around 1991, Charles Radin commissioned John Conway for a substitution tiling for which the tiles appeared in infinitely many orientations, and soon John obliged with the Pinwheel. The Pinwheel is formed from right triangles,
The trilobite and cross tiles clearly generalize to higher dimensions. Though a simpler method soon appeared”Threelobites” were a first early graphic / mathematical exploration. Here’s a nice pic of a tiling by the trilobite and
Through a local self-assembly rule these tiles are forced to form arbitrarily large patches of Robinson tilings, with some extra markings that are used to manage the process. In the animation, at each green dot,