Polygonal antiprism

A polygonal antiprism is a polyhedron that consists of two identical polygonal bases in opposite orientations connected by isosceles triangles. Polygonal antiprisms are the antiprisms of polygons, and were the original meaning of the word. The dual of a polygonal antiprism is an antitegum. The digonal antiprism is equivalent to the tetragonal disphenoid and is a noble polyhedron, while the triangular antiprism is a variant of the octahedron.

They are topologically related to the polygonal gyroprisms, which are half-symmetry variants consisting of two non-opposite and non-parallel bases connected by scalene triangles.

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

${\displaystyle \left(\cos\left(\frac{kd\pi}{n}\right),\ \sin\left(\frac{kd\pi}{n}\right),\ (-1)^kh\right),}$

for k ranging from 0 to 2n – 1. If the antiprism is uniform, then h is specifically given by

${\displaystyle h=\frac12\sqrt{\cos\left(\frac{d\pi}{n}\right)-\cos\left(\frac{2d\pi}{n}\right)}.}$

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