Classifying Tessellations
The definitive work on all possible isohedral tilngs was done by Graunbaum and Shephard who proved that there are 93 isohedral tile types, now known as the IH types. For a nice tutorial on the IH types see Tom McLean’s IH Tutorial
Alain Nicolas in his book Parcelles de’infini identifies the 35 IH tile types that are the best suited for doing Escher-like tessellations. The set includes the 28 Heesch types and the 7 IH types with a single interior mirror. TesselManiac! implements these 35 plus 1 more additional type which has two mirrors.
Heinrich Heesch (1906-1995) was a German mathematician who in the 1930’s proved that there are 28 ways to tile the the plane with asymmetric tiles in a isohedral manner.
TesselManiac! creates thirty-six types of isohedral (IH) tilings. I use Heesch’s notation for classification of the twenty-eight, asymmetric isohedral tiles. I (with the help of Dr. Schattschneider) extend the notation to include the eight additional IH tiles with internal mirror symmetry to get the thirty-six.
Historical Software Note: When I started programming my first tessellation program, TesselMania!, in 1993 I was using Escher’s tile classification system. Dr. Schattschneider suggested I switch to Heesch’s system which turned out to be a terrific suggestion. At the time I thought I was the first to do an Escher-like tiling program using Heesch’s notation. Recently I learned that William Chow in 1977 wrote a computer program for the “the automatic generation of interlocking shapes” which is based on Heesch’s system. He was an Asst. Professor at the University of Illinois at the time. He programmed it in ForTran on a Tektronix 4010 graphics terminal, no small feat!