group (5-cell) family. 9 are in the self-dual regular F4 [3,4,3] group (24-cell) family. (Excluding snub 24-cell) 15 are in the regular B4 [3,3,4] group (tesseract/16-cell) family (3 overlap with 24-cell family) 15 are in the regular H4 [3,3,5] group (120-cell/600-cell) family. 1 special snub form in the [3,4,3] group (24-cell) family. 1 special non-Wythoffian 4-polytope, the grand antiprism. TOTAL: 68 − 4 = 64 These 64 uniform 4-polytopes are indexed below by George Olshevsky. Repeated symmetry forms are indexed in brackets. In addition to the 64 above, there are 2 infinite prismatic sets that generate all of the remaining convex forms: Set of uniform antiprismatic prisms - sr{p,2}×{ } - Polyhedral prisms of two antiprisms. Set of uniform duoprisms - {p}×{q} - A Cartesian product of two polygons. The A4 family The 5-cell has diploid pentachoric [3,3,3] symmetry, of order 120, isomorphic to the permutations of five elements, because all pairs of vertices are related in the same way. Facets (cells) are given, grouped in their Coxeter diagram locations by removing specified nodes. The three uniform 4-polytopes forms marked with an asterisk, *, have the higher extended pentachoric symmetry, of order 240, because the element corresponding to any element of the underlying 5-cell can be exchanged with one of those corresponding to an element of its dual. There is one small index subgroup [3,3,3]+, order 60, or its doubling +, order 120, defining an omnisnub 5-cell which is listed for completeness, but is not uniform. The B4 family This family has diploid hexadecachoric symmetry, [4,3,3], of order 24×16=384: 4!=24 permutations of the four axes, 24=16 for reflection in each axis. There are 3 small index subgroups, with the first two generate uniform 4-polytopes which are also repeated in other families, [1+,4,3,3], [4,(3,3)+], and [4,3,3]+, all order 192. Tesseract truncations 16-cell truncations (*) Just as rectifying the tetrahedron produces the octahedron, rectifying the 16-cell produces the 24-cell, the regular member of the following family. The snub 24-cell is repeat to this family for completeness. It is an alternation of the cantitruncated 16-cell or truncated 24-cell, with the half symmetry group [(3,3)+,4]. The truncated octahedral cells become icosahedra. The cubes becomes tetrahedra, and 96 new tetrahedra are created in the gaps from the removed vertices. The F4 family This family has diploid icositetrachoric symmetry, [3,4,3], of order 24×48=1152: the 48 symmetries of the octahedron for each of the 24 cells. There