Square antiprism

The square antiprism, or squap, is a prismatic uniform polyhedron. It consists of 8 triangles and 2 squares. Each vertex joins one square and three triangles. As the name suggests, it is an antiprism based on a square.

Vertex coordinates
A square antiprism of edge length 1 has vertex coordinates given by:
 * $$\left(±\frac12,\,±\frac12,\,\frac{\sqrt[4]{8}}{4}\right),$$
 * $$\left(0,\,±\frac{\sqrt2}{2},\,-\frac{\sqrt[4]{8}}{4}\right),$$
 * $$\left(±\frac{\sqrt2}{2},\,0,\,-\frac{\sqrt[4]{8}}{4}\right).$$

Representations
A square antiprism has the following Coxeter diagrams:


 * s2s8o (alternated octagonal prism)
 * s2s4s (alternated ditetragonal prism)
 * xo4ox&#x (bases considered separately)

General variant
The square antiprism has a general isogonal variant of the form xo4ox&#y that maintains its full symmetry. This variant uses isosceles triangles as sides.

If the base edges are of length b and the lacing edges are of length l, its height is given by $$\sqrt{l^2-b^2\frac{2-\sqrt2}{2}}$$ and its circumradius by $$\sqrt{\frac{2b^2+4l^2+b^2\sqrt2}{8}}$$.

The bases of the square antiprism are rotated from each other by an angle of 45°. If this angle is changed the result is more properly called a square gyroprism.

A notable case occurs as the alternation of the uniform octagonal prism. This specific case has base edges of length $$\sqrt{2+\sqrt2}$$ and side edges of length $$\sqrt2$$.

In vertex figures
A square antiprism with base edges of length 1 and side edges of length $$\sqrt2$$ occurs as the vertex figure of the small prismatotetracontoctachoron. One using side edges of length $$\sqrt{2-\sqrt2}$$ occurs as vertex figures of the great distetracontoctachoron.

Related polyhedra
A square pyramid can be attached to a base of the square antiprism to form the gyroelongated square pyramid. If a second square pyramid is attached to the other base, the result is the gyroelongated square bipyramid.

Two non-prismatic uniform polyhedron compounds are composed of square antiprisms:


 * Great snub cube (3)
 * Great disnub cube (6)

There are also an infinite amount of prismatic uniform compounds that are the antiprisms of compounds of squares.

Thirteen square antiprisms can be joined in a ring at their triangular faces if the faces are very slightly distorted. This is because $$2\pi / (2 \theta - \pi) \approx 13.0094$$ where $$\theta = \arccos \left( \frac{\sqrt{3} - \sqrt{6}}{3} \right)$$ is the dihedral angle between the square and the triangle for the uniform square antiprism.