Comments on What is special about the 11-cell and 57-cell?
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What is special about the 11-cell and 57-cell?
Reading about the 11-cell and 57-cell I find two facts implied often:
- They are particularly notable among the abstract regular 4-polytopes.
- They are related to each other.
I'll establish why I think they are notable:
- Both polytopes are notable enough to have their own articles on Wikipedia.
- Freeman Dyson apparently remarked of the 11-cell "Plato would have been delighted to know about it"
- Both Branko Grünbaum and Donald Coxeter indpendently discovered the 11-cell (people don't generally independently discover and publish on boring objects)
- Richard Klitzing, calls the 11-cell "special" and implies the 57-cell is special as well on his page
However I'm not sure what exactly is notable about them. There are a lot of abstract regular 4-polytopes. The Atlas of small abstract regular polytopes has 2912 non-degenerate abstract regular 4-polytopes.
These polytopes aren't particularly small either. The 57-cell is even too large for the atlas. There are so many ARPs in their size class it would seem like they would have to be very special, to warrant the attention they receive.
The 11-cell has some historical notability. It is the shape that prompted the invention of abstract polytopes in the first place. And the 57-cell also predates the invention of abstract polytopes in their modern form.
But is that it? Are they just historically notable, or is there something mathematically notable about them?
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The abstract polytopes which admit realizations are special, and the very concept of an abstract polytope exists to generalize and categorize the polytopes first understood through their realizations.
The classical proof by Euclid that there are only five regular convex polyhedra is considered so beautiful, so divine, that Kepler incorporated it into his proposed laws of planetary motion. The orbits of the 5 planets are inscribed/circumscribed by consecutive tetrahedron, cube, octahedron, dodecahedron, icosahedron.
(Of course that theory had to eventually be discarded after it couldn't be made to fit the data, and more than 5 planets were discovered.)
So discovering a new geometric regular polytope is, for a certain medieval mindset, like discovering a new planet or a new god in heaven. Even if you just like classifying shapes or counting unique combinatorial objects, it's notable.
Regular tessellations of the plane are a mild generalization of regular convex polyhedra, in the sense that polyhedra are what you get if you arrange polygons in a vertex with angles adding up to strictly less than 360º, while a tessellation is polygons arranged adding up to exactly 360º. Also polyhedra can be viewed as tessellations of the sphere, which leads to tessellations of the other symmetric spaces such as the projective plane and hyperbolic space. From that point of view regular tessellations of the plane might even be viewed as more natural than regular polyhedra. But tessellations of noncompact space, viewed as polytopes, have infinitely many cells. As polytopes therefore they are called apeirotopes.
Anyway, once you allow for those mild generalizations, let's recap the well-known results for rank 3 polytopes.
We have the Platonic solids: tetrahedron, cube, octahedron, dodecahedron, icosahedron. For the tilings of the sphere, you have these same five cases, as well as two infinite families of n-gonal hosohedra and n-gonal dihedra; and you have the regular tessellations of the plane, the square tiling, the triangle tiling, and the hexagonal tiling. You also have the apeirogonal dihedronal and apeirogonal hosohedronal tiling. There are infinitely many tilings of the hyperbolic plane. And of the tilings of the sphere, four of them descend to tilings of the projective plane, the hemicube, hemi-octohedron, hemi-dodecahedron, and the hemi-icosahedron. The tetrahedron does not because it is not centrally symmetric.
In addition to these cases, you can drop the convexity and find stellated regular polyhedra, and star tilings, but let's skip over that.
Now when we move to rank 4, we find a pretty similar picture. Six regular convex polychora with polyhedra faces. Of those, four also descend to projective space. There's only one tiling of flat space, the cubic honeycomb. Now there are only finitely many tilings of hyberbolic space, which is interesting.
And what Grünbaum and Coxeter discovered, is that you can also make regular polychora out of the projective polyhedra, the 11-cell out of hemi-icosahedra, and the 57-cell out of hemi-dodecahedra. They are not projective polytopes, but they are locally projective. What's special about them among all abstract rank 4 regular polytopes is that they admit geometric realizations, and are built out of fundamental regular polytopes with just some mild generalizations from the classical case.
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