Deltoidal hexecontahedron

The deltoidal hexecontahedron, also called the strombic hexecontahedron or small lanceal ditriacontahedron, is one of the 13 Catalan solids. It has 60 kites as faces, with 12 order-5, 20 order-3, and 30 order-4 vertices. It is the dual of the uniform small rhombicosidodecahedron.

It can also be obtained as the convex hull of a dodecahedron, an icosahedron, and an icosidodecahedron. If the dodecahedron has unit edge length, the icosahedron's edge length is $$\frac{7+\sqrt5}{6} ≈ 1.53934$$ and the icosidodecahedron's edge length is $$\frac{4-\sqrt5}2 ≈ 0.88197$$.

Each face of this polyhedron is a kite with its longer edges $$\frac{7+\sqrt5}{6} ≈ 1.53934$$ times the length of its shorter edges. These kites have one angle measuring $$\arccos\left(-\frac{5+2\sqrt5}{20}\right) ≈ 118.26868°$$, the opposite angle measuring $$\arccos\left(\frac{9\sqrt5-5}{40}\right) ≈ 67.78301°$$, and the other two angles measuring $$\arccos\left(\frac{5-2\sqrt5}{10}\right) ≈ 86.97416°$$.

Related polytopes
The deltoidal hexecontahedron is topologically equivalent to the rhombic hexecontahedron which has golden rhombi for faces instead of kites.