Deltoidal icositetrahedron

The deltoidal icositetrahedron, also called the strombic icositetrahedron or small lanceal disdodecahedron, is one of the 13 Catalan solids. It has 24 kites as faces, with 6+12 order-4 and 8 order-3 vertices. It is the dual of the uniform small rhombicuboctahedron.

It can also be obtained as the convex hull of a cube, an octahedron, and a cuboctahedron. If the cube has unit edge length, the octahedron's edge length is $$\frac{4-\sqrt2}{2} \approx 1.29289$$ and the cuboctahedron's edge length is $$\frac{2\sqrt2-1}2 \approx 0.91421.$$

Each face of this polyhedron is a kite with its longer edges $$\frac{4-\sqrt2}{2} \approx 1.29289$$ times the length of its shorter edges. These kites have three angles measuring $$\arccos\left(\frac{2-\sqrt2}{4}\right) \approx 81.57894^\circ$$ and one angle measuring $$\arccos\left(-\frac{2+\sqrt2}{8}\right) \approx 115.26317^\circ$$.

Vertex coordinates
A deltoidal icositetrahedron with dual edge length 1 has vertex coordinates given by all permutations of:
 * $$\left(±\sqrt2,\,0,\,0\right),$$
 * $$\left(±1,\,±1,\,0\right),$$
 * $$\left(±\frac{\sqrt2+4}{7},\,±\frac{\sqrt2+4}{7},\,±\frac{\sqrt2+4}{7}\right).$$