faces of type {p}, edge figures {r}, and vertex figures {q,r}. The existence of a regular 4-polytope {p,q,r} is constrained by the existence of the regular polyhedra {p,q} which becomes cells, and {q,r} which becomes the vertex figure. Existence as a finite 4-polytope is dependent upon an inequality: The 16 regular 4-polytopes, with the property that all cells, faces, edges, and vertices are congruent: 6 regular convex 4-polytopes: 5-cell {3,3,3}, 8-cell {4,3,3}, 16-cell {3,3,4}, 24-cell {3,4,3}, 120-cell {5,3,3}, and 600-cell {3,3,5}. 10 regular star 4-polytopes: icosahedral 120-cell {3,5,5/2}, small stellated 120-cell {5/2,5,3}, great 120-cell {5,5/2,5}, grand 120-cell {5,3,5/2}, great stellated 120-cell {5/2,3,5}, grand stellated 120-cell {5/2,5,5/2}, great grand 120-cell {5,5/2,3}, great icosahedral 120-cell {3,5/2,5}, grand 600-cell {3,3,5/2}, and great grand stellated 120-cell {5/2,3,3}. Convex uniform 4-polytopes Symmetry of uniform 4-polytopes in four dimensions There are 5 fundamental mirror symmetry point group families in 4-dimensions: A4 = , B4 = , D4 = , F4 = , H4 = . There are also 3 prismatic groups A3A1 = , B3A1 = , H3A1 = , and duoprismatic groups: I2(p)×I2(q) = . Each group defined by a Goursat tetrahedron fundamental domain bounded by mirror planes. Each reflective uniform 4-polytope can be constructed in one or more reflective point group in 4 dimensions by a Wythoff construction, represented by rings around permutations of nodes in a Coxeter diagram. Mirror hyperplanes can be grouped, as seen by colored nodes, separated by even-branches. Symmetry groups of the form [a,b,a], have an extended symmetry, [[a,b,a]], doubling the symmetry order. This includes [3,3,3], [3,4,3], and [p,2,p]. Uniform polytopes in these group with symmetric rings contain this extended symmetry. If all mirrors of a given color are unringed (inactive) in a given uniform polytope, it will have a lower symmetry construction by removing all of the inactive mirrors. If all the nodes of a given color are ringed (active), an alternation operation can generate a new 4-polytope with chiral symmetry, shown as "empty" circled nodes", but the geometry is not generally adjustable to create uniform solutions. Enumeration There are 64 convex uniform 4-polytopes, including the 6 regular convex 4-polytopes, and excluding the infinite sets of the duoprisms and the antiprismatic prisms. 5 are polyhedral prisms based on the Platonic solids (1 overlap with regular since a cubic hyperprism is a tesseract) 13 are polyhedral prisms based on the Archimedean solids 9 are in the self-dual regular A4 [3,3,3]