4
$\begingroup$

Is there a neat way to show (or a reference that already proves) that

  • the 4-cube is the only convex 4-polytope in which all facets are regular 3-cubes?
  • the 24-cell is the only convex 4-polytope in which all facets are regular octahedra?
  • the 120-cell is the only convex 4-polytope in which all facets are regular dodecahedra?

Note that I do require that the facets are regular, so e.g. just any cubical polytope does not fit.

I am aware of Gosset's semiregular polytopes (vertex-transitive and all facets are regular polytopes), for which this statement is true, but I do not require vertex-transitivitiy here.

However, if my statement turns out to be wrong in this general form, then I wonder whether it holds when I require that all vertices are on a common sphere.

$\endgroup$

2 Answers 2

5
$\begingroup$

This is Theorem 1 (actually, Satz 1) of Roswitha Blind, Konvexe Polytope mit kongruenten regulären $(n- 1)$-Seiten im $\Bbb{R}^n$ $(n \ge 4)$, Comment. Math. Helvetici 54 (1979) 304--308. The short proof follows from two short lemmas, one of which cites Coxeter's Regular Polytopes and an article by Shephard.

You might also be interested in a related question from last year with helpful links, which references a second 1979 paper of Blind in her program to explore the analogue of Johnson solids for 4-polytopes.

$\endgroup$
0
$\begingroup$

Within 4D we also have the bipyramid on a tetrahedral base, as well as the bipyramid on an icosahedral base, which both use 8 or 40 tetrahedra solely.

In the papers of the Blind couple there are further polytopes, with various regular facets listed too. But your restriction to just a single type of facets and your restriction to 4D comes down to just these 2, except of the six 4D regular polytopes - at least if you'd further restrict to convexity.

--- rk

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.