Hat types from equilateral Cairo pentagons
I created these shapes in the grid below, using the Polyform Puzzle Solver by Jaap Scherphuis. I then reconstructed them in VectorEngineer Pro-Tools using angles of 69.295, 90 and 114.295 that were suggested to me by Jaap. The DXF files were converted using Bob Nungester's small but effective DXF to TXT utility, for use in Arnaud Chéritat's dedicated tiling applet (links at the foot). The Som brero: This tile can simulate the chair tiling if reflections are allowed: Crumpled hat: A group of four tiles can produce an attractive periodic tiling: A six-tile motif combined with its reflection, which is also rotated, produces a striped pattern: Beret: This tile when used with its reflection, produces strips that can be glued together in different ways. However, there's only one configuration that the beret can tile if reflections are not used. Novelty hat: This polygon can produce misshapen mesh-like structures, if allowing reflections. Links: Bob Nunges...