Bridges 2020 paper: Design of a Sampler of Isohedral Tilings of the Pied-de-poule Tile

Abstract: We describe the design of a sampler of all isohedral tilings of the classic pied-de-poule tile. The sampler is a Hawaiian shirt, also known as Aloha shirt. This allows us to explore the wider applicability of the pied-de-poule motif, yet respecting the design principles of classic Hawaiian shirts. We formulate the precise inclusion and exclusion criteria of the set of tilings to be enumerated. By checking the various combinations of symmetries, we find that there are eight distinct tilings, to be included in the sampler. We also show representations of the three orbifolds of the corresponding symmetry groups, which are added to the shirt in the form of phantasy sea creatures, based on a torus, a Klein bottle and a twisted pillow.