The image below was taken from the Tilings Encyclopedia site (the very last entry) maintained by Dirk Frettlöh and Franz Gähler. The Wheel Tiling was discovered by Hans-Ude Nissen.
https://tilings.math.uni-bielefeld.de/
The Spectre can be can be thought of, as being composed of the two polygons that are used to make up the Wheel Tiling. If you look closely, you’ll also find the Buddha-like figure, the Mystic. This substitution tiling uses both front and backs of the Spectre and leaves gaps now and again (the smaller of the two polygons).
Curiously, I recently found some of my old drawings (dated August 2021) where I had played around with the Wheel Tiling and Spectre polygons (including reflections) but didn’t really do much with them, as you can see below (image requested by Adam Scherlis).
Below are several clusters of Spectres (rotations only) for the propeller (in yellow) and bow-tie (in red) of the Wheel Tiling to display the non-repeating pattern.
There have been several other clever examples of marked Spectres by Yoshiaki Araki and Casey Mann that use certain combinations of hexagons, pentagons, rhombi, squares and equilateral triangles. Here’s one I put together using two polygons but needs four colours.
Andrew Russell on Twitter (@AndrewR88968514) found out that you can overlay regular hexagons, squares and thin rhombi onto Spectres to produce stick-like animals; wild!
The recent Hatfest at The Mathematical Institute in Oxford was a great success. I had a wonderful time meeting up with so many hat enthusiasts, online colleagues, friends and family. A big thank you to Alex, Henna, Nick, Mike and all the backroom staff for making it all happen.