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.
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.
Exploring Tessellations is available from CreateSpace and Amazon.