Snub Disphenoid

The 12-faced convex deltahedra also known as the Siamese dodecahedron, which is also Johnson solid .

It is implemented in the Wolfram Language as PolyhedronData["SnubDisphenoid"].

The coordinates of the polyhedron vertices of a snub disphenoid of unit side length may be found by solving the set of four simultaneous equations

			(1)
			(2)
			(3)
			(4)

for the four unknowns , , , and . The analytic solution requires solving the cubic equation, and the solutions are given by the smallest positive real roots of

			(5)
			(6)
			(7)
			(8)

Numerically,

			(9)
			(10)
			(11)
			(12)

The surface area of the unit snub disphenoid is

(13)

and the volume is given by the positive real root of

(14)

approximately .