Prismatic compound of antiprisms

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:
  • D_npd if nq is odd
  • D_nph if nq is even

Where p/q=2, the component is the tetrahedron (or dyadic antiprism). In this case, if n=2 then the compound is the stella octangula, with higher symmetry (O_h).

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

• ${\displaystyle \left(\cos {\frac {k\pi }{n}},\sin {\frac {k\pi }{n}},(-1)^{k}h\right)}$
• ${\displaystyle \left(\cos {\frac {k\pi }{n}},\sin {\frac {k\pi }{n}},(-1)^{k+1}h\right)}$

with k ranging from 0 to 2n−1; if the triangles are equilateral,

${\displaystyle 2h^{2}=\cos {\frac {\pi }{n}}-\cos {\frac {2\pi }{n}}.}$

Compounds of 2 antiprisms

2 digonal antiprisms (tetrahedra)	2 triangular antiprisms (octahedra)	2 square antiprisms	2 hexagonal antiprisms	2 pentagrammic crossed antiprism

Compound of two trapezohedra (duals)

The duals of the prismatic compound of antiprisms are compounds of trapezohedra:

Two cubes (trigonal trapezohedra)

For compounds of three digonal antiprisms, they are rotated 60 degrees, while three triangular antiprisms are rotated 40 degrees.

Three tetrahedra	Three octahedra

• Skilling, John (1976), "Uniform Compounds of Uniform Polyhedra", Mathematical Proceedings of the Cambridge Philosophical Society, 79 (3): 447–457, doi:10.1017/S0305004100052440, MR 0397554.