Another curiosity of the Cairo tiling is that it can be seen in an instance of polyhedra, specifically of the space-filling Bisymmetric Hendecahedron. Such a polyhedron is a relatively recent discovery, asreported in a 1996 paper by Guy Inchbald. For further details see:

http://www.steelpillow.com/polyhedra/five_sf/five.htm

Although the Cairo tiling as an entity was not directly noticed in Inchbald’s paper, an architect, Jak Drinnan, happened to noticed this, and being familiar with the Cairo tiling made the association.

Although I am interested in polyhedra, this is very much secondary to my interest in tessellation. Indeed, I lack the mathematics here for a true mathematical treatment. Would any reader more familiar with this have anything to say?

For more on Jak Drinnan and the bisymmetric hendecahedron per se, see: