### M Theory Lesson 121

Ma continues with a consideration of $S_4$, the permutations on four letters. Recall that this group describes a 24 vertex permutohedron polytope in three dimensions, which is a hexagonal version of the Stasheff polytope for the pentagon. Ma thinks in terms of a SUSY seesaw model for mass matrices. On reduction of the parameters, his neutrino matrix now becomes a degenerate 1-circulant with $a$ on the diagonal.

We associated $3 \times 3$ 1-circulants with vertices of a hexagon, which comes from the cube. A cube has the symmetry group $S_4$, but in operad land it is more natural to relate the cube to the permutohedron via the Loday-Ronco maps.

We associated $3 \times 3$ 1-circulants with vertices of a hexagon, which comes from the cube. A cube has the symmetry group $S_4$, but in operad land it is more natural to relate the cube to the permutohedron via the Loday-Ronco maps.

## 2 Comments:

Hi Kea, this comment is related to polyhedrons, but from two different perspectives.

I have been reading Paul J Nahin [EE], ‘Chases and Escapes: The Mathematics of Pursuit and Evasion‘. In chapter 3 ‘Cyclic Pursuits’ he discusses the n-Bug Problem.

MathWorld has the Mice Problem for n=3,4,5,6 with dynamic illustrations.

Are these active circulants?

http://mathworld.wolfram.com/

MiceProblem.html

Until reading Nahin, I did not realize that this MathWorld page was related to Pursuit Evasion Games. I should have, since there were links to: Pursuit Curve, Spiral, Tractrix, Whirl.

I like being an old dog trying to learn new tricks. I only wish that my ability in rigor matched what I think is an ability in insight.

Hi Kea, There may be something unusual about the Ma paper?

See Table 3 Perfect Geometric Solids and following paragraph on p6 discussing dualities.

On MathWorld there is a discussion of a Regular Polychoron.

“Of the six regular convex polychora, five are typically regarded as being analogous to the Platonic solids ...” with a strangely similar duality: 120-600 cell duality, 16 cell dual with tesseract, and 24 cell and pentatope self dual.

http://mathworld.wolfram.com/RegularPolychoron.html

Wolfram Demonstrations Project has 120-600 cell download.

http://demonstrations.wolfram.com/600Cell120CellDuality/

Table 4 Character Table of S4, item C3 with n=3, h=8 may have something in common with tesseract: “ composed of 8 cubes with 3 to an edge“

http://mathworld.wolfram.com/Tesseract.html

Although a stretch of the imagination consider:

quadrants with 2 axes

may be expanded to

octrants with three axes.

This may all be coincidence, but it is a curious play on numbers.

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