Beaded Super Right Angle Weave Quilt for a Group of Order 18

Here's my latest piece, a quilted wall hanging. The fabric is pieced cotton and silk, and appliqued with bead work. I call it, "Super Right Angle Weave: 18 Patches in 3 Colors and 3 Sizes" because I'm not very creative with titles. Florence suggested I call it "RAW Diamonds," which I like.  So that's it's artsy name. The whole piece measures 13 inches on a side. This quilt was included in Juried Exhibit accompanying the 2014 AMS Special Session on Mathematics and Mathematics Education in Fiber Arts at the Joint Mathematics Meeting in Baltimore, MD. It will be at the Ohio State Mansfield’s Pearl Conard Art Gallery from Monday, November 9 to Tuesday, December 8, 2015 in the Math & Art exhibition, "In the Realm of Forms." This quilt was also featured on the Scientific American website. This piece is SOLD

This piece began as a study in color for what I call Super Right Angle Weave (SRAW), a bead weave based upon the regular tiling by squares. Each beaded patch is 6 square by 6 squares of the tiling.  I weave loops of four beads in each square face and attach these loops across the edges of the tiling with a single bead between the loops.  For this set, I use a coloring with three bead types (two types for the faces and one type for the edges).  I chose three colors in each of  three sizes (albeit the purple beads are slightly different shades of purple across the three sizes), for a total of nine different bead types.  This set of 18 patches answers the following question: What are all of the possibilities if I weave SRAW with three colors, one color in each of three sizes, where the colors are arranged as shown?  The patches are arranged in sets of three, where each row uses the same three bead types, but arranged differently.  Here you can see it as a work in progress before I picked out the fabrics. (You can click on the photos to make them bigger.)

When I found the fabrics that matched the beads, I was delighted.  Each little square of bead work measures about 1 1/8 inches square. Here's a close-up of the beadwork. 

At some point, I realized my little patches formed a nice mathematical set.  When I arranged them in different ways, I found that the columns had things in common, as did all of the diagonals.  They formed sets in the sense that you could pick an attribute, and all three in the set were either the same on that attribute or all different.  It's like that game of Set, but my set has 18 cards instead of 81.  At some point, I realized I had a group of order 18.  Each element in the group corresponds to one patch of beading, up to automorphisms.  This group has two complete copies of 9 elements.  Within each set of 9, you can partition them into cosets vertically, horizontally, and along both diagonals. In other words, my arrangement shows different ways to build cosets with 3 elements in each coset (6 cosets * 3 elements =18). But if you group cosets across the two sets of 9, you only get 2 elements in each coset (9 cosets * 2 elements =18). 

A quick Google search told me there are 5 different groups with 18 elements, but I had no idea which one I found.

Lucky for me, mathematician Tom Davis was kind enough to help me identify which group of 18 elements this is.  After a few highly detailed emails, he concluded that it's the generalized dihedral group for E9.   Here's his argument:

I identified each patch with something like this...
Let me call the three colors Yellow, Green and Blue (Y, B, G), and the position attributes of say, the bottom point: yellow, green, blue (ygb).

Then each pattern can be classified by a 6-character code, like this:

(tuv)
(wxy)

Where t=largest, u = medium, v = small
and
w = bottom, x = left corner, y = other

Fore example,
(YGB)
(bgy)

distinguishes the size and position attributes.

Now, if we consider, say, (YGB) and (ygb) to be the "correct" order of the beads in both categories, then I can treat both the top and bottom parts of the pattern representation as a permutation away from the "correct" patterns. I can combine them into a single permutation, like (YBG)(gyb) or (BG)(gy) where you never mix the capital letters with the lower-case ones since size and position are independent.

Then your group operation is trivial: it's just the multiplication of the permutations, and (using Mathematica) I found that the total group size is, in fact, 18. (In other words, none of the multiplications take you outside of the patches you've made.)

So you CAN consider them to be group elements, but depending on which of them you chose to be the identity, the operation table would be different. I had Mathematica make a group operation table, but it translated the elements into numbers and here it is included as a screen snap.

It's clearly not commutative and it's not the dihedral group, so it's either the direct product of S3 and Z3 or the "generalized dihedral group for E9" whatever in the heck that is :)

I don't think it's S3xZ3. I've also included the screen snap for that (it's the one with the group named "dp" (but it could be: I'm not so good at comparing tables where the elements aren't necessarily listed in the same order).

In fact, it IS the "generalized dihedral group for E9." Here's proof. I had Mathematica draw the Cayley graphs of your group and of S3xZ3 (called "dp") and they're completely different. Included is a screen snap of the Mathematica result:

Now let's give Tom Davis a big round of applause.  Thanks Tom!