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?
I originally aksed this question on codidact.
