Compound of n p/q-gonal antiprismsn=2 5/3-gonal5/2-gonalTypeUniform compoundIndexq odd: UC23 q even: UC25Polyhedran p/q-gonal antiprismsSchläfli symbols(n=2)ß{2,2p/q}ßr{2,p/q}Coxeter diagrams(n=2)Faces2n {p/q} (unless p/q=2), 2np trianglesEdges4npVertices2npSymmetry groupnq odd: np-fold antiprismatic (Dnpd) nq even: np-fold prismatic (Dnph)Subgroup restricting to one constituentq odd: p-fold antiprismatic (Dpd) q even: p-fold prismatic (Dph) In geometry, a prismatic compound of antiprism is a category of uniform polyhedron compound. Each member of this infinite family of uniform polyhedron compounds is a symmetric arrangement of antiprisms sharing a common axis of rotational symmetry.
Infinite family This infinite family can be enumerated as follows: For each positive integer n ≥ 1 and for each rational number p/q > 3/2 (expressed with p and q coprime), there occurs the compound of n p/q-gonal antiprisms, with symmetry group: Dnpd if nq is odd Dnph if nq is even
When p/q = 2, or equivalently p = 2, q = 1, the component is the tetrahedron (or dyadic antiprism). In this case, if n = 2 then the compound is the stella octangula, with higher symmetry (Oh).
Compounds of two antiprisms Compounds of two n-antiprisms share their vertices with a 2n-prism, and exist as two alternated set of vertices.
Cartesian coordinates for the vertices of an antiprism with n-gonal bases and isosceles triangles are
with k ranging from 0 to 2n−1; if the triangles are equilateral,
+ Compounds of 2 antiprisms2 digonalantiprisms(tetrahedra)2 triangularantiprisms(octahedra)2 squareantiprisms2 hexagonalantiprisms2 pentagrammiccrossedantiprism
Compound of two trapezohedra (duals) The duals of the prismatic compound of antiprisms are compounds of trapezohedra: Two cubes(trigonal trapezohedra)
Compound of three antiprisms For compounds of three digonal antiprisms, they are rotated 60 degrees, while three triangular antiprisms are rotated 40 degrees. Three tetrahedraThree octahedra
References .