I would like to read article(s) that provide the “state of the art” on the following open problem:


“Enumerate all convex uniform 5-polytopes.”


This problem is posted on the “Open Problem Garden” (<http://garden.irmacs.sfu.ca/op/convex_uniform_5_polytopes>), but no citation is given. (The same open problem is also listed on Wikipedia’s “List of unsolved problems” (<https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics>), but Wikipedia only cites the same “Open Problem Garden” page).


Does anyone know of a legitimate source stating that this is an open problem—especially, of any source(s) that also provide the current mathematical knowledge on this topic?


[Note: I have already looked at all four articles listed on Wikipedia’s “Uniform 5-Polytope” page (https://en.wikipedia.org/wiki/Uniform\_5-polytope): H.S.M. Coxeter’s “Regular and Semi-Regular Polytopes I, II, and III” and Norman W. Johnson’s dissertation, “The Theory of Uniform Polytopes and Honeycombs.”]



State-of-the-art article on "uniform 5-polytopes?"
There is some information in M.Winter's MSE question, [How many uniform polytopes are there in higher dimensions?](https://math.stackexchange.com/q/3063822/237). There I quote Egon Shulte's "Semiregular and Uniform Convex Polytopes,"
emphasizing [Wythoff's construction](https://en.wikipedia.org/wiki/Wythoff_construction).


So possibly this Coxeter paper will help (I am not familiar with it):



> 
> Coxeter, H. S. M. "Wythoff's construction for uniform polytopes." *Proceedings of the London Mathematical Society* 2, no. 1 (1935): 327-339.
> [DOI](https://doi.org/10.1112/plms/s2-38.1.327).
> 
> 
> 


