are uniform polyhedra. The faces of a uniform 4-polytope must be regular. A 4-polytope is scaliform if it is vertex-transitive, and has all equal length edges. This allows cells which are not uniform, such as the regular-faced convex Johnson solids. A regular 4-polytope which is also convex is said to be a convex regular 4-polytope. A 4-polytope is prismatic if it is the Cartesian product of two or more lower-dimensional polytopes. A prismatic 4-polytope is uniform if its factors are uniform. The hypercube is prismatic (product of two squares, or of a cube and line segment), but is considered separately because it has symmetries other than those inherited from its factors. A tiling or honeycomb of 3-space is the division of three-dimensional Euclidean space into a repetitive grid of polyhedral cells. Such tilings or tessellations are infinite and do not bound a "4D" volume, and are examples of infinite 4-polytopes. A uniform tiling of 3-space is one whose vertices are congruent and related by a space group and whose cells are uniform polyhedra. Classes The following lists the various categories of 4-polytopes classified according to the criteria above: Uniform 4-polytope (vertex-transitive): Convex uniform 4-polytopes (64, plus two infinite families) 47 non-prismatic convex uniform 4-polytope including: 6 Convex regular 4-polytope Prismatic uniform 4-polytopes: {} × {p,q} : 18 polyhedral hyperprisms (including cubic hyperprism, the regular hypercube) Prisms built on antiprisms (infinite family) {p} × {q} : duoprisms (infinite family) Non-convex uniform 4-polytopes (10 + unknown) 10 (regular) Schläfli-Hess polytopes 57 hyperprisms built on nonconvex uniform polyhedra Unknown total number of nonconvex uniform 4-polytopes: Norman Johnson and other collaborators have identified 2189 known cases (convex and star, excluding the infinite families), all constructed by vertex figures by Stella4D software. Other convex 4-polytopes: Polyhedral pyramid Polyhedral bipyramid Polyhedral prism Infinite uniform 4-polytopes of Euclidean 3-space (uniform tessellations of convex uniform cells) 28 convex uniform honeycombs: uniform convex polyhedral tessellations, including: 1 regular tessellation, cubic honeycomb: {4,3,4} Infinite uniform 4-polytopes of hyperbolic 3-space (uniform tessellations of convex uniform cells) 76 Wythoffian convex uniform honeycombs in hyperbolic space, including: 4 regular tessellation of compact hyperbolic 3-space: {3,5,3}, {4,3,5}, {5,3,4}, {5,3,5} Dual uniform 4-polytope (cell-transitive): 41 unique dual convex uniform 4-polytopes 17 unique dual convex uniform polyhedral prisms infinite family of dual convex uniform duoprisms (irregular tetrahedral cells) 27 unique convex dual uniform honeycombs, including: Rhombic dodecahedral honeycomb Disphenoid tetrahedral honeycomb Others: Weaire–Phelan structure periodic space-filling