Thursday, July 5, 2018

The ongoing journey

Exploring Tessellations: A Journey through Heesch Types And Beyond was published in October, 2015 and then for the next year was repeatedly updated as mistakes were corrected, new material added, and sections reorganized. The last of those updates was at the end of 2016. Since that last update I have designed an activity book and another maze book as well as updating two earlier maze books. The work on these books provided enough new material to justify going back and reviewing the content and organization of Exploring Tessellations.

This revision of June 2018 adds many new tilings as examples and deletes some old tilings that were substandard. It also reorganizes the section on Heesch types. The original organization of that section was based on efforts to find standing birds that tessellated. That whimsical organization was replaced with a more standard organization that stresses Heesch families. Heesch and his co-author showed that their list of 28 types consisted of nine main types from which the other 19 types could be derived by shrinking edges to zero. For example, type TTTT results when one of the TT pairs in type TTTTTT shrinks away.

These chess pieces were added as an example of isohedral class IH69. The tile has symmetry over a diagonal. It simultaneously fits Heesch types CCCC, G1G1G2G2, and CCGG. In this example two edges are straight, but that is not a requirement of class IH69.

This next image has been in the book from the start as an illustration of isohedral class IH91. It also has reflective symmetry. It is based on an isosceles triangle and one edge must be straight for it to have that symmetry. It fits both Heesch type CCC and CGG.
Finally, below is a new example of Heesch type CC3C3 (IH39), which is the hardest of the Heesch type to form into Escher-like tessellations. It could be a leaf on a twig or a rosebud.
The book has hundreds of other illustrations and is one of the few sources that explain both the Heesch classification and the isohedral classification of Gr├╝nbaum and Shepard.

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