Concept
In geometry, a uniform 5-polytope is a five-dimensional uniform polytope. By definition, a uniform 5-polytope is vertex-transitive and constructed from uniform 4-polytope facets. The complete set of convex uniform 5-polytopes has not been determined, but many can be made as Wythoff constructions from a small set of symmetry groups. These construction operations are represented by the permutations of rings of the Coxeter diagrams. Regular polytopes: (convex faces) 1852: Ludwig Schläfli proved in his manuscript Theorie der vielfachen Kontinuität that there are exactly 3 regular polytopes in 5 or more dimensions. Convex semiregular polytopes: (Various definitions before Coxeter's uniform category) 1900: Thorold Gosset enumerated the list of nonprismatic semiregular convex polytopes with regular facets (convex regular 4-polytopes) in his publication On the Regular and Semi-Regular Figures in Space of n Dimensions. Convex uniform polytopes: 1940-1988: The search was expanded systematically by H.S.M. Coxeter in his publication Regular and Semi-Regular Polytopes I, II, and III. 1966: Norman W. Johnson completed his Ph.D. Dissertation under Coxeter, The Theory of Uniform Polytopes and Honeycombs, University of Toronto Non-convex uniform polytopes: 1966: Johnson describes two non-convex uniform antiprisms in 5-space in his dissertation. 2000-2023: Jonathan Bowers and other researchers search for other non-convex uniform 5-polytopes, with a current count of 1297 known uniform 5-polytopes outside infinite families (convex and non-convex), excluding the prisms of the uniform 4-polytopes. The list is not proven complete. List of regular polytopes#Five Dimensions Regular 5-polytopes can be represented by the Schläfli symbol {p,q,r,s}, with s {p,q,r} 4-polytope facets around each face. There are exactly three such regular polytopes, all convex: {3,3,3,3} - 5-simplex {4,3,3,3} - 5-cube {3,3,3,4} - 5-orthoplex There are no nonconvex regular polytopes in 5 dimensions or above.
