Prismatic compound of antiprisms

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Compound of n p/q-gonal antiprisms
(n=2, p=5, q=3) (n=2, p=5, q=2)
Type	Uniform compound
Index	q odd: UC23 q even: UC25
Polyhedra	n p/q-gonal antiprisms
Faces	2n {p/q} (unless p/q=2), 2np triangles
Edges	4np
Vertices	2np
Symmetry group	nq odd: np-fold antiprismatic (Dnpd) nq even: np-fold prismatic (Dnph)
Subgroup restricting to one constituent	q odd: p-fold antiprismatic (Dpd) q even: p-fold prismatic (Dph)

Each member of this infinite family of uniform polyhedron compounds is a symmetric arrangement of antiprisms sharing a common axis of rotational symmetry.

This infinite family can be enumerated as follows:

• For each positive integer n>1 and for each rational number p/q>3/2 and p/q≠2, 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
• For each positive integer n>1, there occurs the compound of n tetrahedra (as antiprisms, corresponding to p/q=2 in the previous case), with symmetry group:
  • D_2nd if n is odd
  • D_2nh if n is even

In the latter case, the compound with n=2 has greater symmetry (O_h): it is the stella octangula.

[edit] References

• John Skilling, Uniform Compounds of Uniform Polyhedra, Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 79, pp. 447-457, 1976.