# Deltoidal icositetrahedron

Deltoidal icositetrahedron

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Type Catalan
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 and tetragonal icosikaitetrahedron) is a Catalan solid which looks a bit like an overinflated cube. Its dual polyhedron is the rhombicuboctahedron.

## Dimensions

The 24 faces are deltoids or kites, also called trapezia in the US and trapezoids in Britain. The short and long edges of each kite are in the ratio 1:1.292893...

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.

## Related polyhedra

The deltoidal icositetrahedron is topologically equivalent to a cube whose faces are divided in quadrants.

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

## Related polyhedra and tilings

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)
[1+,4,3] = [3,3]
(*332)
[3+,4]
(3*2)
{4,3} t{4,3} r{4,3}
r{31,1}
t{3,4}
t{31,1}
{3,4}
{31,1}
rr{4,3}
s2{3,4}
tr{4,3} sr{4,3} h{4,3}
{3,3}
h2{4,3}
t{3,3}
s{4,3}
s{31,1}

=

=

=
=
or
=
or
=

Duals to uniform polyhedra
V43 V3.82 V(3.4)2 V4.62 V34 V3.43 V4.6.8 V34.4 V33 V3.62 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.

Dimensional family of expanded polyhedra and tilings: 3.4.n.4
Symmetry
*n32
[n,3]
Spherical Planar Hyperbolic...
*232
[2,3]
D3h
*332
[3,3]
Td
*432
[4,3]
Oh
*532
[5,3]
Ih
*632
[6,3]
P6m
*732
[7,3]

*832
[8,3]...

*∞32
[∞,3]

Expanded
figure

3.4.2.4

3.4.3.4

3.4.4.4

3.4.5.4

3.4.6.4

3.4.7.4

3.4.8.4

3.4.∞.4
Coxeter
Schläfli

rr{2,3}

rr{3,3}

rr{4,3}

rr{5,3}

rr{6,3}

rr{7,3}

rr{8,3}

rr{∞,3}
Deltoidal figure
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
Coxeter