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
( cos k π n , sin k π n , ( − 1 ) k h ) {\displaystyle \left(\cos {\frac {k\pi }{n}},\sin {\frac {k\pi }{n}},(-1)^{k}h\right)}
( cos k π n , sin k π n , ( − 1 ) k + 1 h ) {\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,
2 h 2 = cos π n − cos 2 π n . {\displaystyle 2h^{2}=\cos {\frac {\pi }{n}}-\cos {\frac {2\pi }{n}}.}
Compound of two trapezohedra (duals) The duals of the prismatic compound of antiprisms are compounds of trapezohedra:
Compound of three antiprisms For compounds of three digonal antiprisms, they are rotated 60 degrees, while three triangular antiprisms are rotated 40 degrees.
References Skilling, John (1976), "Uniform Compounds of Uniform Polyhedra", Mathematical Proceedings of the Cambridge Philosophical Society, 79 (3): 447–457, Bibcode:1976MPCPS..79..447S, doi:10.1017/S0305004100052440, MR 0397554.
