Here is a link to a short note of mine in the Mathematical Gazette that was discussed in an earlier post:
https://goo.gl/LnJREO
The note asks how many different shapes formed on a square template using identical, asymmetric edges will tessellate. In simpler language, how many different four-edged puzzle pieces are there that will tessellate if all edges are shaped in the same way? If all edges are asymmetric but identical, there are only fifteen distinct shapes with two "out" edges and two "in" edges and all tessellate.
A more difficult question is how many of the different shapes formed based on the template of a regular hexagon using identical, asymmetric edges will tessellate. A total of 34 tile isohedrally as Heesch types and another 9 tile anisohedrally. The demonstration of this result is in Exploring Tessellations: A Journey through Heesch Types And Beyond.
