# How many uniform polytopes are there in higher dimensions?

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.

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

• Indeed, several polytopes can be given wrt. various Coxeter reflection groups. E.g. the octahedron can be given wrt. $A_3$ or wrt. $C_3$, or the 24-cell could be given wrt. $C_4$, wrt. $D_4$, and wrt. $F_4$. Thus, several of the Stott transformations (i.e. expansion or contraction) would respect some of those various symmetries, but not necessarily all. But uniformity is being defined as a transitive action of some symmetry group. And in this context this refers to Coxeter reflection groups or their rotational subgroups. It's just a matter of appropriate selection. --- rk Jun 28, 2019 at 10:26