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# Deltoidal icositetrahedron

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 Title: Deltoidal icositetrahedron Author: World Heritage Encyclopedia Language: English Subject: Collection: Catalan Solids Publisher: World Heritage Encyclopedia Publication Date:

### Deltoidal icositetrahedron

Deltoidal icositetrahedron

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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° 7' 5"
\arccos(-\frac{7 + 4\sqrt{2}}{17})
Dual polyhedron rhombicuboctahedron
Properties convex, face-transitive

Net

In geometry, a deltoidal icositetrahedron (also a trapezoidal icositetrahedron, tetragonal icosikaitetrahedron,[1] and strombic icositetrahedron) is a Catalan solid which looks a bit like an overinflated cube. Its dual polyhedron is the rhombicuboctahedron.

## Contents

• Dimensions 1
• Occurrences in nature and culture 2
• Orthogonal projections 2.1
• Related polyhedra 3
• Dyakis dodecahedron 3.1
• Related polyhedra and tilings 4
• References 6

## Dimensions

The 24 faces are kites. The short and long edges of each kite are in the ratio \scriptstyle1:(2-\frac{1}{\sqrt{2}})\approx 1:1.292893\dots

If its smallest edges have length 1, its surface area is \scriptstyle{6\sqrt{29-2\sqrt{2}}} and its volume is \scriptstyle{\sqrt{122+71\sqrt{2}}}.

## 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:

 Projective Image Dual symmetry image [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[2][3] or diploid.[4]

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.

Uniform octahedral polyhedra
Symmetry: [4,3], (*432) [4,3]+, (432) [3+,4], (3*2)
{4,3} t{4,3} r{4,3} t{3,4} {3,4} rr{4,3} tr{4,3} sr{4,3} s{3,4}
Duals to uniform polyhedra
V43 V3.82 V(3.4)2 V4.62 V34 V3.43 V4.6.8 V34.4 V35

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

## References

1. ^ Conway, Symmetries of things, p.284-286
2. ^ Isohedron 24k
3. ^ The Isometric Crystal System
4. ^ https://www.uwgb.edu/dutchs/symmetry/xlforms.htm
• (Section 3-9)
• (The thirteen semiregular convex polyhedra and their duals, Page 23, Deltoidal icositetrahedron)
• The Symmetries of Things 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, ISBN 978-1-56881-220-5 [2] (Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, page 286, tetragonal icosikaitetrahedron)