## Thursday, May 12, 2016

### Z Z Z Z and IH9

I have struggled to understand isohedral class IH9. I could see that it fit Heesch type TG1G1TG2G2 but was not sure it always fit TG1G2TG2G1. Playing with it in the process of yet another update to Exploring Tessellations: A Journey through Heesch Types and Beyond, I realized that IH9 is the class for tiles that simultaneously fit TG1G1TG2G2 and TG1G2TG2G1. It can fit both because flipping a IH9 tile over a horizontal line give the same result as flipping it over a vertical line. Both TG1G1TG2G2 and TG1G2TG2G1 have pg symmetry, but IH9 has pgg symmetry, so that the tiling can be rotated 180 degrees and it will reproduce itself.

An earlier post reported that a way to construct IH9 tiles in Tesselmaniac! is to use the CGCG template and ignore one of the C lines. An alternative is to use the CCGG template, also ignoring one of the C lines. Below are two ways of tessellating the letter “Z” with screen shots from Tesselmaniac! showing how each can be created in CGCG and CCGG.

In this first picture showing CGCG tiles, the green line is eliminated to combine two tiles into one IH9 tile.
In the CCGG case, the yellow line is eliminated to combine two tiles into one IH9 tile. In all the cases shown here, the midpoints of the C lines are the centers of two-fold rotation.
The second Z fits in a different way. Below the yellow line is the line to eliminate to get the IH9 tile.
In this fourth picture, the yellow line that is eliminated connects across the stem of the letter.
To see why IH9 can be both TG1G1TG2G2 and TG1G2TG2G1, consider how the glide lines in each are oriented. Below is a simple tile that tessellates as TG1G1G2G2. The purple lines are lines of glide reflection. They are perpendicular to the edges that are the TT edges.
In contrast, the lines of glide reflection are parallel to the two TT edges in this TG1G2TG2G1 tiling.
The shape of a class IH9 tile allows both sets of glide-reflection lines. Flipping the tile horizontally gives the same result as flipping it vertically.

The May revision of Exploring Tessellations: A Journey through Heesch Types and Beyond was substantial. I corrected dozens of mistakes, most small but a few larger (such as misclassifications of tessellation patterns) and added about a dozen pages. I also reorganized the book, moving sections and dividing the Explorations chapter into two chapters. I doubt if this will be the last revision; I am sure that many mistakes remain.