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.
posted by Kea | 9:44 AM
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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