Snub square antiprism

The snub square antiprism is one of the 92 Johnson solids (J85). It consists of 8+16 triangles and 2 squares.

It can be constructed from a square antiprism by expanding the two halves outward and inserting a set of 16 triangles in between the halves.

Coordinates
Coordinates for a snub square antiprism with unit edge length are given by where A is the second-to-greatest root of
 * (±1/2, ±1/2, C/2),
 * (±√2A/2, 0, B/2),
 * (0, ±√2A/2, B/2),
 * (±A/2, ±A/2, −B/2),
 * (0, ±√2/2, −C/2),
 * (±√2/2, 0, −C/2),
 * $$x^6-2x^5-13x^4+8x^3+32x^2-8x-4,$$

and where B and C are given by
 * $$B=\sqrt{1-\left(1-\frac{\sqrt2}{2}\right)}A,$$
 * $$C=\sqrt{2+2\sqrt2 A-2A^2}+B.$$

From these coordinates, its volume can be calculated as ξ ≈ 3.60122, where ξ is the greatest real root of
 * $$531441 x^{12}- 85726026 x^8- 48347280 x^6 + 11588832 x^4 + 4759488 x^2-892448.$$

Related polyhedra
The snub square antiprism can be considered to be the square case in the family of snub antiprisms. The snub triangular antiprism is the regular icosahedron, and the snub disphenoid or snub digonal antiprism is another Johnson solid. No other members of this family can be made convex and regular-faced (the snub pentagonal antiprism can be made with regular faces, but is concave).