# Deltoidal icositetrahedron

Deltoidal icositetrahedron

Animated
Type Catalan
Conway notation oC or deC
Coxeter diagram
Face polygon
kite
Faces 24
Edges 48
Vertices 26 = 6 + 8 + 12
Face configuration V3.4.4.4
Symmetry group Oh, BC3, [4,3], *432
Rotation group O, [4,3]+, (432)
Dihedral angle 138°07′05″
arccos(−7 + 42/17)
Dual polyhedron rhombicuboctahedron
Properties convex, face-transitive

Net

In geometry, a deltoidal icositetrahedron (also a trapezoidal icositetrahedron, tetragonal icosikaitetrahedron,[1], tetragonal trisoctahedron[2] and strombic icositetrahedron) is a Catalan solid. Its dual polyhedron is the rhombicuboctahedron.

## Dimensions

The 24 faces are kites. The short and long edges of each kite are in the ratio 1:(2 − 1/2) ≈ 1:1.292893...

If its smallest edges have length a, its surface area and volume are

{\displaystyle {\begin{aligned}A&=6{\sqrt {29-2{\sqrt {2}}}}\,a^{2}\\V&={\sqrt {122+71{\sqrt {2}}}}\,a^{3}\end{aligned}}}

## Occurrences in nature and culture

The deltoidal icositetrahedron is a crystal habit often formed by the mineral analcime and occasionally garnet. The shape is often called a trapezohedron in mineral contexts, although in solid geometry that name has another meaning.

### Orthogonal projections

The deltoidal icositetrahedron has three symmetry positions, all centered on vertices:

Projectivesymmetry Image Dualimage [2] [4] [6]

## Related polyhedra

The great triakis octahedron is a stellation of the deltoidal icositetrahedron.

### Dyakis dodecahedron

The deltoidal icositetrahedron is topologically equivalent to a cube whose faces are divided in quadrants. It can also be projected onto a regular octahedron, with kite faces, or more general quadrilaterals with pyritohedral symmetry. In Conway polyhedron notation, they represent an ortho operation to a cube or octahedron.

In crystallography a rotational variation is called a dyakis dodecahedron[3][4] or diploid.[5]

Octahedral, Oh, order 24 Pyritohedral, Th, order 12

## Related polyhedra and tilings

Spherical deltoidal icositetrahedron

The deltoidal icositetrahedron is one of a family of duals to the uniform polyhedra related to the cube and regular octahedron.

When projected onto a sphere (see right), it can be seen that the edges make up the edges of an octahedron and cube arranged in their dual positions.

This polyhedron is topologically related as a part of sequence of deltoidal polyhedra with face figure (V3.4.n.4), and continues as tilings of the hyperbolic plane. These face-transitive figures have (*n32) reflectional symmetry.

*n42 symmetry mutation of dual expanded tilings: V3.4.n.4
Symmetry
*n32
[n,3]
Spherical Euclid. Compact hyperb. Paraco.
*232
[2,3]
*332
[3,3]
*432
[4,3]
*532
[5,3]
*632
[6,3]
*732
[7,3]
*832
[8,3]...
*∞32
[∞,3]
Figure
Config.

V3.4.2.4

V3.4.3.4

V3.4.4.4

V3.4.5.4

V3.4.6.4

V3.4.7.4

V3.4.8.4

V3.4.∞.4