Prismatic compound of antiprisms explained

bgcolor=#e7dcc3 colspan=2	Compound of n p/q-gonal antiprisms
align=center colspan=2	n=2
Type	Uniform compound
Index	q odd: UC23 q even: UC25
Polyhedra	n p/q-gonal antiprisms
Schläfli symbols (n=2)	ß ßr
Coxeter diagrams (n=2)
Faces	2n (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)

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:
  • 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 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

\left( \cos (k\pi)/(n) , \sin (k\pi)/(n) , ( -1 ) ^{k} h \right) \left( \cos (k\pi)/(n) , \sin (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 (\pi)/(n)} - \cos (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

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