Polygonal antiprism

A polygonal antiprism is a polyhedron that consists of two bases in opposite orientations connected by triangles. Polygonal antiprisms are the antiprisms of polygons, and were the original meaning of the word. The dual of an antiprism is a trapezohedron.

Uniform antiprisms exist for any regular polygon {n/d} where n/d > 1.5. When n/d < 2, the triangular faces of the corresponding antiprism cross its rotational symmetry axis, and thus these antiprisms are often called retroprisms or crossed antiprisms.

Antiprisms make up one of the two infinite families of uniform polyhedra, the other being the family of polygonal prisms. These are in fact related, as the {n/d} antiprism can be constructed by alternation of a {2n/d}-gonal prism.

The edges of a uniform antiprism lie on a hyperboloid of one sheet. This is more apparent in star antiprisms where edges cross.

Coordinates
Coordinates for an {n/d} antiprism are given by
 * $$\left(\cos\left(\frac{kd\pi}{n}\right),\ \sin\left(\frac{kd\pi}{n}\right),\ (-1)^kh\right),$$

for k ranging from 0 to 2n – 1. If the antiprism is uniform, then h is specifically given by
 * $$h=\frac12\sqrt{\cos\left(\frac{d\pi}{n}\right)-\cos\left(\frac{2d\pi}{n}\right)}.$$

If furthermore the entire figure has unit edge length, the coordinates need to be scaled down by 2sin(π/n).