## Monday, September 6, 2010

### M Theory Lesson 353

On the surface across the corner of the cube we can draw the points of the path space up to permutations. There are nine points around the triangle and one central point representing $XYZ$. Let us include an edge whenever a trit is flipped. If an outer edge is weighted by $1/2$ and an inner edge by $1$ we recover a Pythagorean tetractys labeled by both path types and weighted valencies at a node, the latter corresponding to the path count on the cube. This is a more cyclic picture than the use of projective degree that we saw for the associahedron ternary triangle, reflecting the need to replace duality by triality.

1. Note that the Pythagoreans knew about the number $27$, in the sense that $3^3$ appeared mystically in their multiplication table for the tetracytys.

2. In fact, this M Theory tectractys is just the magic hexagon of the Tao. To preserve cyclicity, it would be better to use the mystical symbols rather than the numbers $0,1,2$.

3. So the Koide phase of $6/27 = 2/9$ could be associated to the central (space generating) part, as a fraction of $1$ (rather than $2 \pi$) for the radial (cosmological) time.

4. And here is an earlier post showing how the planar dual of the tetractys is a honeycomb hexagon a la Terence Tao and coauthors.

5. WOW, Kea. I like this!!!

6. Yeah, pretty cool heh? The mathematicians should like this too ...

7. The mass gap? It should be the same thing? A turbulent Fermi surface?
http://arxiv.org/PS_cache/arxiv/pdf/1009/1009.0265v1.pdf

Now here is a clear link to the Platonic solides :)

8. Well, there are several 'mass gaps' that we could talk about. But yes, there is one here.