Not all of the regular 4-polytopes are spherical. Do you mean just those that are? https://en.wikipedia.org/wiki/List_of_regular_polytopes_and_compounds#Spherical_3 ….
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just the platonics, in analogy with the cyclotomic polynomials. (1-z^600) is just a 600-gon because the complex numbers are algebraically complete. so you need a different kind of root, and the unit 3-sphere in quaternionic space has nice SO(4)ish properties to play with
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