Deltoidal icositetrahedron

The deltoidal icositetrahedron, also called the strombic icositetrahedron, small lanceal disdodecahedron, or sladid, 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 $$\frac{4-\sqrt2}{2} ≈ 1.2989$$ and the cuboctahedron's edge length is \frac{2\sqrt2-1}{2} ≈ 0.91421.

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