### M Theory Lesson 171

As one moves up the $n$-ordinal ladder, perhaps by adding levels to a tree (as in the case of the extension of the associahedra to the permutoassociahedra), the spherical polytopes acquire more and more vertices and faces. That is, they begin to better approximate a sphere. In all dimensions, both the cube and the permutohedron tile $\mathbb{R}^{N}$. The translation lattice for the 3 dimensional permutohedron may be generated by the vectors $(1,1,-3)$, $(1,-3,1)$ and $(1,1,-3)$. The associahedra do not share this property, but recall that they instead tile the real points of interesting moduli spaces.

## 2 Comments:

Adding more faces brings them closer to a sphere. There was a fuss some months ago when some cosmologists suggested the Universe was a dodacahedron. These polygons are useful for approximating a sphere.

Indeed, Louise. Time will no doubt show that you were correct all along.

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