## Friday, December 9, 2011

### Theory Update 132

Letting the quadrant signs determine the qubit component of the $216 = 2^3 3^3$ dimensions, we observe a parity cube, defined by the eight tetractys centres. The hexagon cores of the eight tetractys faces on the octahedron form the eight hexagonal faces of a $24$ vertex permutohedron. Both cubes and permutohedra tile three dimensional space. The six squares of the permutohedron may be labeled by the planes they lie in, as in $XY^{\pm}$. Each vertex of the permutohedron carries $3$ quark paths from the (signed) tetractys.

1. That octahedron is an interesting construct. If you have a six-dimensional space of possible masses, for (d,u,s,c,b,t), and mark out the (000,001,010,100) tetrahedron for each of the triples (d,u,s), (u,s,c), (s,c,b), (c,b,t), you get a chain of tetrahedra joined by faces and edges with the same general shape as a "tower tetracube" (only the components are tetrahedra, not cubes).

What I'm observing is that the octahedron can be decomposed into eight tetrahedra (joined at the coordinate axes), which can be divided into two sets of four, each of which is twists like an S-tetromino bent into the third dimension, and that one of these "bent tetrominos" has the same connectivity as the six-dimensional chain of Koide tetrahedra that I mentioned above, but collapsed into three dimensions.

While I'm mentioning six dimensions, another thing I've figured out is the six-dimensional counterpart of Foot's cone. It should be a five-dimensional "cone" with a T^4 cross-section, in which each S^1 factor is associated with one of the four tetractyses listed above. Instead of a single six-dimensional vector of quark masses, you have to think of a four-dimensional "ray" which is the product of the four three-component mass vectors listed above.

It's a bit hard to describe this stuff purely verbally, but hopefully some sense gets through...

2. Cool, Mitchell! Yes, I was hoping this would explain the $(b,c,s)$ type triplets. Slowly we are getting there ...

3. and let's not forget Vafa's (hyperbolic) braids-in-tetrahedra program ...

4. See page 50, where they draw a (BPS chiral) particle braid inside one tetrahedron. Note that the parity cube is a categorified instance of the Mac Lane pentagon associahedron used by Vafa et al.

5. Mitchell, note that opposite tetrahedra only differ by a global sign flip. The 'tetronimo' is built from two such pairs, and the gluing from the 3D tiling would give one of those nice knot tetrahedrons.