How many uniform polytopes are there in higher dimensions?

Question

I am not really interested in the exact numbers, but more in the richness of the class of uniform (convex) polytopes in higher dimensions. Wikipedia contains the followin statement:

In five and higher dimensions, there are $3$ regular polytopes, the hypercube, simplex and cross-polytope. [...] Most uniform higher-dimensional polytopes are obtained by modifying the regular polytopes, or by taking the Cartesian product of polytopes of lower dimensions.

In six, seven and eight dimensions, the exceptional simple Lie groups, $E_6$, $E_7$ and $E_8$ come into play.[...]

I am interested in a quantification of the emphasized sentence above. In what sense are these most uniform polytopes? Does the world of uniform polytopes become "boring" in, say, more than $30$ dimensions, because there only remain simple variations on regular polytopes and cartesian products? Or are there exceptional polytopes expected in higher dimensions too?

I would also be satisfied with something like "this seems to be unknown", preferably with some reference.

Answer 1

This chapter by Egon Schulte,

Egon Schulte, "Symmetry of polytopes and polyhedra." In Handbook of Discrete and Computational Geometry. J. E. Goodman and J. O'Rourke, editors CRC Press, 2017.

has a section on "Semiregular and Uniform Convex Polytopes," including these paragraphs:

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I see that @Dr.RichardKlitzing also mentions Wythoff's construction.

Answer 2

To answer onto your question about that "most":

It is that uniform polytopes have a vertex transitivity wrt. some Coxeter reflection group. Or some mere rotational subgroup thereof. Excluding the latter, i.e. snubs and eg. that 4D grand antiprism, you would be left with the set of Wythoffian polytopes, which are coded via symbols (ringed or unringed) to the nodes of Coxeter-Dynkin diagrams (themselves encoding right these mentioned groups). In fact all those Wythoffian polytopes can be reconstructed from that symbolical coding via Wythoffs kaleidoscopical construction device. - Therefore the number of Wythoffian polytopes is a mere combinatoric quest of selecting subsets of nodes on base of the already known group symbols.

Wrt. to the general number I'd think that it is still open, at least for the higher dimensions. Eg. Coxeters famous article on the uniform polyhedra once was a mere conjecture on completeness, which thereafter was proved only by means of a computer aided research.

--- rk