## Tuesday, October 27, 2015

### A journey through Heesch types and beyond

Months ago I began trying to sort the various tessellations I had developed for mazes into their appropriate types. (See past posts on this blog for background information.) The impetus for this project came from playing with the program TesselManiac! to see how many of the program's templates (or Heesch types) could be used to tessellate standing birds. The project seemed quite limited when I began and I was unsure how I would put together the results, but the more I learned the more I realized how much more there was to learn. Starting with the Heesch classification of 28 types, I quickly learned that there was another classification, Grünbaum and Shepard's 93 isohedral classes. Trying to make sense of the 93 classes took months. An unexpected and short e-mail exchange with another person doing tessellations made me realize that I needed to become comfortable with the 17 symmetry groups. Sorting my letter tessellations, I found some that I could not classify and was introduced to anisohedral tiles and tilings. So far I have not ventured into the topic of aperiodic tessellations and I lack the tools to display them.

Several times I have thought that the project was finished only to have some further insight or find a new way of seeing a topic that required additions or reorganization. I have once again reached a point where the book looks finished but I will not be surprised if further revisions are needed.

In the process of writing the book, which has the title Exploring Tessellations: A Journey through Heesch Types And Beyond, I created many new tessellations. I wanted at least one bird tessellation to illustrate each of the 28 Heesch types, and though a few are weak, I was able to accomplish this. I had no examples for some of the isohedral classes so I needed to develop those. Most of these are abstract, geometric tilings.

The book begins with illustrations and explanations of the 17 wallpaper groups of symmetry and then examines the 28 Heesch types, avoiding as much as possible mathematical jargon. Next is a look at the isohedral classes, followed by themed examples. The book ends with a few mazes to illustrate how tessellations fit maze making. The book is more advanced than the short introductions to tessellations aimed at children but it is simpler than those written for people who want the mathematics of the topic explained. It contains hundreds of illustrations with comments.

Past posts have illustrated many of the bird tessellations done for the book. Below is a shark tessellation that is featured on the book's cover.
The 93 isohedral classes include the 28 Heesch types, 21 classes with unshapeable edges, 24 with both shapeable and unshapeable edges, and 20 that restrict the edges of Heesch types, often resulting in tilings that satisfy more than one Heesch type. The book ignores the 21 groups that have no shapeable edges. They are grids of polygons that often cannot be distinguished if they lack internal markings. In trying to make sense of the rest, I eventually realized that if a class has mirror symmetry of both tile and tiling, a bisection (or in one case a trisection) resulted in a tiling that formed another isohedral class. Below on the top is an example of IH12 with its bisection, IH22, on the bottom.
Exploring different ways to make puzzle pieces, I found that there were only two tiles possible if the template shape was a square and if the edges were identically shaped with central symmetry. However, one of them fit together in several ways.

Playing with different ways to form square tiles, I found that there were only four possible tiles if the sides had identical central rotation. Three of them had symmetry and fit various special isohedral classes. The fourth was asymmetrical. Below is an example.
One final thing I found as I worked on this is only a few people are seriously interested in tessellations. World wide the number is probably only in the hundreds.

Exploring Tessellations is available from CreateSpace and Amazon.