Deltoidal icositetrahedron

Deltoidal icositetrahedron
	(3D model); (OFF file)
Rank	3
Type	Uniform dual
Space	Spherical
Notation
Bowers style acronym	Sladid
Coxeter diagram	m4o3m
Elements
Faces	24 kites
Edges	24+24
Vertices	6+8+12
Vertex figure	8 triangles, 6+12 squares
Measures (edge length 1)
Dihedral angle
Central density	1
Related polytopes
Army	Sladid
Regiment	Sladid
Dual	Small rhombicuboctahedron
Conjugate	Great deltoidal icositetrahedron
Abstract properties
Euler characteristic	2
Topological properties
Surface	Sphere
Orientable	Yes
Genus	0
Properties
Symmetry	B3, order 48
Convex	Yes
Nature	Tame

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 $\arccos\left(-\frac{2+\sqrt2}{8}\right) \approx 115.26317^\circ$ .