# 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.

## 2 Answers

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.

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

• "It is that uniform polytopes have a vertex transitivity wrt. some Coxeter reflection group. Or some mere rotational subgroup thereof". Why is it clear, that their symmetry group cannot be a more general subgroup of a Coxeter reflection group? Jun 27, 2019 at 8:30
• Sorry, but I do not see how this answers the question in my comment. I asked about the cited statement, which is about subgroups of reflection groups and not about representations of a polytope by different reflection groups. Jun 28, 2019 at 10:23
• 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
• Or even taken the other way round: if you'd had a uniform polytope, which would belong to a deeper subgroup of some Coxeter group, then the mere symmetry group of this polytope would provide an according new (smaller) Coxeter group. - But those are already known... Jun 28, 2019 at 10:30
• "... then the mere symmetry group of this polytope would provide an according new (smaller) Coxeter group." I do not understand. Not all symmetry groups of vertex transitive polytopes are Coxeter groups (I am pretty sure about this statement, but please correct me if I am wrong). Jun 28, 2019 at 11:23