these in his book Regular Polytopes. McMullen added six in his paper New Regular Compounds of 4-Polytopes. Self-duals: Dual pairs: Uniform compounds and duals with convex 4-polytopes: The superscript (var) in the tables above indicates that the labeled compounds are distinct from the other compounds with the same number of constituents. Compounds with regular star 4-polytopes Self-dual star compounds: Dual pairs of compound stars: Uniform compound stars and duals: Compounds with duals Dual positions: Group theory In terms of group theory, if G is the symmetry group of a polyhedral compound, and the group acts transitively on the polyhedra (so that each polyhedron can be sent to any of the others, as in uniform compounds), then if H is the stabilizer of a single chosen polyhedron, the polyhedra can be identified with the orbit space G/H – the coset gH corresponds to which polyhedron g sends the chosen polyhedron to. Compounds of tilings There are eighteen two-parameter families of regular compound tessellations of the Euclidean plane. In the hyperbolic plane, five one-parameter families and seventeen isolated cases are known, but the completeness of this listing has not been enumerated. The Euclidean and hyperbolic compound families 2 {p,p} (4 ≤ p ≤ ∞, p an integer) are analogous to the spherical stella octangula, 2 {3,3}. A known family of regular Euclidean compound honeycombs in any number of dimensions is an infinite family of compounds of hypercubic honeycombs, all sharing vertices and faces with another hypercubic honeycomb. This compound can have any number of hypercubic honeycombs. There are also dual-regular tiling compounds. A simple example is the E2 compound of a hexagonal tiling and its dual triangular tiling, which shares its edges with the deltoidal trihexagonal tiling. The Euclidean compounds of two hypercubic honeycombs are both regular and dual-regular. Footnotes External links MathWorld: Polyhedron Compound Compound polyhedra – from Virtual Reality Polyhedra Uniform Compounds of Uniform Polyhedra Skilling's 75 Uniform Compounds of Uniform Polyhedra Skilling's Uniform Compounds of Uniform Polyhedra Polyhedral Compounds http://users.skynet.be/polyhedra.fleurent/Compounds_2/Compounds_2.htm Compound of Small Stellated Dodecahedron and Great Dodecahedron {5/2,5}+{5,5/2} References . . . . . . Regular Polytopes, (3rd edition, 1973), Dover edition, p. 87 Five regular compounds .