Prismatic compound of antiprisms
# Prismatic compound of antiprisms

Compound of n p/q-gonal antiprisms :

| n=2 5/3-gonal 5/2-gonal ||
| - | - |
| Type | Uniform compound |
| Index | q odd: $UC_{23}$ q even: $UC_{25}$ |
| Polyhedra | n p/q-gonal antiprisms |
| Schläfli symbols (n=2) | ß{2,2p/q} ßr{2,p/q} |
| Coxeter diagrams (n=2) | |
| Faces | 2n {p/q} (unless p/q=2), 2np triangles |
| Edges | 4np |
| Vertices | 2np |
| Symmetry group | nq odd: np-fold antiprismatic $(D_{npd})$ nq even: np-fold prismatic $(D_{nph})$ |
| Subgroup restricting to one constituent | q odd: p-fold antiprismatic $(D_{pd})$ q even: p-fold prismatic $(D_{ph})$ |

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

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

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

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

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)

## Compound of three antiprisms

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

| Three tetrahedra | Three octahedra |
| - | - |
