Exploring Tessellations: A Journey through Heesch Types And Beyond

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From the Introduction:

Many years ago I combined two interest, computer programming and typeface design, to developed a maze construction kit. Computer programs generated mazes that were displayed with special typefaces that I designed. It did not take long to realize that using tessellation patterns was one way to display mazes in a visually interesting way.

In 2015 while working on Holiday Mazes, I discovered the program Tesselmaniac, a successor and improved version of the once-popular program TesselMania. I used it in designing a few pages of Holiday Mazes but also found interesting tessellating patterns that did not fit that book. Playing with the program, I began for the first time to seriously study the various Heesch types. (German mathematician Heinrich Heesch [1906-1995] showed that all single-tile tessellations that meet certain criteria can be sorted into 28 types and those types are named after him.) As I played with the program, I decided to see how many of Heesch types I could use to create a shape that was recognizable as a bird standing on the back of another bird. Eventually I found designs for 17 of them and that exercise was the genesis of this book.

Many people who are interested in tessellations arrive at the topic via symmetry because tessellations are a subtopic of the study of symmetry. Quilters, for example, find that symmetry is the key to designing visually interesting quilt patterns. Patterned squares can be arranged in many ways but not all give pleasing results. In my case, the process was reversed: I started with tessellations and they led me to trying to understand symmetry. There is no reason to repeat my mistake so we begin this tessellation adventure by looking at symmetry.

Next we will explore one of the most popular ways of categorizing tessellations, by Heesch type. Each of the 28 Heesch types will be illustrated with at least one bird pattern. Almost all will also have examples of other patterns, most of which I used in maze books. Standing-bird tiles will illustrate 17 of the types. Because these shapes are similar, they illustrate how these Heesch types differ. The other 11 types are also illustrated with at least one birdlike shape. For each type there is an explanation of how edges are related with translation, rotation, and glide moves. We will pay attention to the valence, which is the number of lines meeting at each intersection or vertex, and to how many tiles form what is called a translation unit, a group of tiles that can fill the plane without rotation or flipping.

An alternative classification of tessellations was developed by Grünbaum and Shepard who found that there were 93 classes of isohedral tilings. These classes include the Heesch types and also some Heesch types with restrictions on how edges are formed. In sorting the various patterns I have used in the past, I was surprised by how many of them fit into these restricted classes. Some of the 93 classes force edges to be straight and allow flips that violate the Heesch types. Next, discussions of various themed tessellations, including tessellating arrows, crosses, bugs, planes, puzzle pieces, and people both review and develop the ideas presented in the earlier parts of the book. In one of these themes and near the end in an examination of letter or alphabet tessellations, there are short discussions of non-isohedral or anisohedral tilings.

The most famous name in the field of tessellations is M. C. Escher (1989-1972), a Dutch artist who created scores of beautiful prints and drawings based on tessellating animals and humans. An Escher-like tessellation is one that resembles a real-world object rather than being an abstract, geometrical shape. I found most of the Escher-like tessellations illustrated in these pages by playing with programs such as TesselMania or Tesselmaniac, by trial and error in Fontographer, or by thinking through designs on paper. I have tried to acknowledge when my tile has relied on a method used by others and have largely omitted designs that are fairly accurate reproductions of designs of others. I have undoubtedly omitted credit that is due on some of these designs. I do not make the same claim for abstract, geometrical tiles. Many of them are widely known and have been independently discovered many times. In many cases I have copied what I considered well-known patterns.

Disclaimer: The study of tessellations is a branch of mathematics. Mathematicians speak about the subject in ways that require more mathematical knowledge than I have. This book attempts of unravel a few of the insights of mathematicians and to illustrate them with examples. I am not an expert on the subject of tessellations and this account almost certainly contains significant errors. I found and corrected many errors in the process of preparing this book, some of which are described in the text, and I doubt that I found and corrected all of them. Please excuse whatever typos remain; I corrected many but know that some remain.

Robert Schenk
October 2015

The journey is not ended. In addition to several small revisions since October 2015, there were major revisions in January and May of 2016 that broke the book into parts, adding many pages to the material that is in the "Explorations" chapters. I also corrected many mistakes but undoubtedly missed many and also introduced new ones.

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