Disdyakis triacontahedron

Disdyakis triacontahedron
	(3D model); (OFF file)
Rank	3
Type	Uniform dual
Space	Spherical
Notation
Bowers style acronym	Siddykit
Coxeter diagram	m5m3m ()
Conway notation	mD
Elements
Faces	120 scalene triangles
Edges	60+60+60
Vertices	12+20+30
Vertex figure	12 decagons, 20 hexagons, 30 squares
Measures (edge length 1)
Dihedral angle
Central density	1
Number of external pieces	120
Level of complexity	6
Related polytopes
Army	Siddykit
Regiment	Siddykit
Dual	Great rhombicosidodecahedron
Conjugate	Great disdyakis triacontahedron
Abstract & topological properties
Flag count	720
Euler characteristic	2
Surface	Sphere
Orientable	Yes
Genus	0
Properties
Symmetry	H3, order 120
Convex	Yes
Nature	Tame

The disdyakis triacontahedron, also called the small disdyakis triacontahedron, is one of the 13 Catalan solids. It has 120 scalene triangles as faces, with 12 order-10, 20 order-6, and 30 order-4 vertices. It is the dual of the uniform great 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 $3\frac{3+\sqrt5}{10} ≈ 1.57082$ and the icosidodecahedron's edge length is $3\frac{4+\sqrt5}{22} ≈ 0.85037$ .

Each face of this polyhedron is a scalene triangle. If the shortest edges have unit edge length, the medium edges have length $3\frac{3+\sqrt5}{10} ≈ 1.57082$ and the longest edges have length $\frac{7+\sqrt5}5 ≈ 1.84721$ . These triangles have angles measuring $\arccos\left(\frac{5-2\sqrt5}{30}\right) ≈ 88.99180°$ , $\arccos\left(\frac{15-2\sqrt5}{20}\right) ≈ 58.23792°$ , and $\arccos\left(\frac{9+5\sqrt5}{24}\right) ≈ 32.77028°$ .