Uniform 6-polytope

In six-dimensional geometry, a uniform 6-polytope is a six-dimensional uniform polytope. A uniform polypeton is vertex-transitive, and all facets are uniform 5-polytopes. The complete set of convex uniform 6-polytopes has not been determined, but most 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-Dynkin diagrams. Each combination of at least one ring on every connected group of nodes in the diagram produces a uniform 6-polytope. The simplest uniform polypeta are regular polytopes: the 6-simplex {3,3,3,3,3}, the 6-cube (hexeract) {4,3,3,3,3}, and the 6-orthoplex (hexacross) {3,3,3,3,4}. 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 polytera) in his publication On the Regular and Semi-Regular Figures in Space of n Dimensions. Convex uniform polytopes: 1940: The search was expanded systematically by H.S.M. Coxeter in his publication Regular and Semi-Regular Polytopes. Nonregular uniform star polytopes: (similar to the nonconvex uniform polyhedra) Ongoing: Jonathan Bowers and other researchers search for other non-convex uniform 6-polytopes, with a current count of 41348 known uniform 6-polytopes outside infinite families (convex and non-convex), excluding the prisms of the uniform 5-polytopes. The list is not proven complete. Uniform 6-polytopes with reflective symmetry can be generated by these four Coxeter groups, represented by permutations of rings of the Coxeter-Dynkin diagrams. There are four fundamental reflective symmetry groups which generate 153 unique uniform 6-polytopes. Uniform prism There are 6 categorical uniform prisms based on the uniform 5-polytopes.