Deltoidal hexecontahedron

The deltoidal hexecontahedron, also called the strombic hexecontahedron, small lanceal ditriacontahedron, or sladit, 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.