I have lately been experimenting with shapes composed of Cairo pentagons, with the aid of the Polyform Puzzle Solver by Jaap Scherphuis. One of the most interesting of these, is the fish. Alain Nicholas worked his magic on it, creating a piranha, soon to be shown on his amazing site… http://en.tessellations-nicolas.com/
It can produce aperiodic tilings within a square border of various sizes.
This example displays the frequency and placement of rotations and mirrors.
Smaller motifs can be arranged in different ways. The bottom example had me fooled at first – the two images are actually mirrors of each other.
Here’s another polycairo made up of 12 Cairo pentagons. It is easier to tessellate than the fish, as you only have to deal with rotations, due to its mirror symmetry
… and here’s the artwork that Alain Nicholas created from it – thank you kind Sir!
A chaotic arrangement of the tiles.
I believe this last example can only tile periodically but in an interesting way.
The pattern uses all four rotations and their mirrors.
Thank you to David Bailey for helping out. His site on Escher-like tessellations and Cairo tiling is well worth a visit… http://www.tess-elation.co.uk/new-hom