# Deltoidal icositetrahedron

Deltoidal icositetrahedron
Rank3
TypeUniform dual
Notation
Coxeter diagramm4o3m ()
Conway notationoC
Elements
Faces24 kites
Edges24+24
Vertices6+8+12
Vertex figure8 triangles, 6+12 squares
Measures (edge length 1)
Dihedral angle${\displaystyle \arccos \left(-{\frac {7+4{\sqrt {2}}}{17}}\right)\approx 138.11796^{\circ }}$
Central density1
Number of external pieces24
Level of complexity4
Related polytopes
DualSmall rhombicuboctahedron
ConjugateGreat deltoidal icositetrahedron
Abstract & topological properties
Flag count192
Euler characteristic2
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryB3, order 48
ConvexYes
NatureTame

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 ${\displaystyle {\frac {4-{\sqrt {2}}}{2}}\approx 1.29289}$ and the cuboctahedron's edge length is ${\displaystyle {\frac {2{\sqrt {2}}-1}{2}}\approx 0.91421.}$

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

## Vertex coordinates

A deltoidal icositetrahedron with dual edge length 1 has vertex coordinates given by all permutations of:

• ${\displaystyle \left(\pm {\sqrt {2}},\,0,\,0\right),}$
• ${\displaystyle \left(\pm 1,\,\pm 1,\,0\right),}$
• ${\displaystyle \left(\pm {\frac {{\sqrt {2}}+4}{7}},\,\pm {\frac {{\sqrt {2}}+4}{7}},\,\pm {\frac {{\sqrt {2}}+4}{7}}\right).}$