**Kea Explores Triality**It turns out the interesting solid Kea posted was not the truncated semi regular solid sometimes used in four space lattices. But she inspired me to take a look at this pleasingly vertically symmetric shape. I am not sure what she is doing with this, looking at her past posts I assume it has something to do with M-theory. But I decided to make a model since I had left over materials from making Rowlands. I note that in a sense this can be seen as a seven dimensional structure without a singularity center either. Perhaps it takes 4 rather than 2 of them to do what Rowlands seems to do with his n-order tetrahedra. Sometimes it helps to make a physical model, to make things much easier to see as if we color them- it is difficult for example to keep the clockwise and anti-clockwise directions of labels when one has to view a flattened solid.

Now, to make such structures that do not contain triangles I had to carefully add struts between the square faces and the pentagonal faces. It comes out as a sort of sphere. See the illustration.

I note in Kea's drawing (and I did look up the nature of such numbers and their unique view of space different from say the double factorials which I suspect a topological thing in the high level language of a lampion on my last post) It is not clear to me what the pluses and minuses mean if not the spin of things- yet she did say these labels had to be checked- it a prototype or sketch of the problem.

So, I find first of all it contains 14 points and 9 polygons (6x pentagons, 3x squares) and the edges are indeed 21. Well, I notice also from one point slicing to another we find this sequence of points 1 3 6 3 1 or 3 3 3 for the faces. I also note that Kea labeled or colored them in such a way that were in a square an extra of three colors exists the opposite edge is this color. Interestingly the sense of the double colors of the three squares seem skewed as if a global spin handedness of the shape even if turned on its axis.

Now, from this I realized that we could count the struts in the polyhedra much like we represent shadow polytopes as faces on eutactic star patterns of Coxeter. (I was going to consider the duality of things, faces to points, and things like why there are no 18 face deltahedra, and 14 as a number that comes up say in a point centered representation of a hypercube with 2 superimposed in the center and such ideas but I did not go that direction). In any case I am more prone to wonder about a fifth and truely or naturally four dimensional label ABC* where * is impartially or preferentially implied (after all that seems to be the motive for shifting the ideas of factorials between at least adjacent dimensions and these structures.)

Now, here we get into related speculations from my view in hopes to tie in certain ideas of space and information to my "quasic" view. We have 6 pentagons to which we add five struts each for a total of 30, and 3 squares for a total of 6... to this we add the 21 of the sphereical skeleton (clearly there seems to be an assymmetry of the groups off the idea of more conventional stacking of things like these analogs to pentagonal pyramids) that gives us the interesting number 57. But we can also add thru the natural space seven struts that connects the 14 points, Thus: 64...!

Now, in the three distinct meeting of such a code of 64 as three space ABC in the quasic grid we find a 6 x 6 square for a total of 36 including 6 in the diagonal. These taken away from the 64 leaves those in the periphery of this 6x6 Conway-Mahon-Otto color matching Matrix, of which I interpret these 28 as in a sense to be labled as if two added colors, k and w , black and white, in natural four space. ABC* .

So clearly in my scheme of things the suposed locations of the codons taken as primary to describe what may be happening in the particle physics (which is what Rowlands suggests the other way around) This structure too can be something in its compact labeling that could explain some of the finer effects of gene code reading.

* * *

Note in the models the color coding of the nodes randomly chosen gumdrops. The struts were pieces of spagitti and the 21 edges toothpicks. It was hard to see while the unstable construction how I might have oriented the gumdrops the same way.

Fantastic! This polytope is called a (Stasheff) associahedron, and it has a great deal to with M Theory! Category theorists play with generalisations of this shape, but your simple counting has far more physics in it than most of what they have managed so far ...

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