Deltoidal icositetrahedron

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Deltoidal icositetrahedron
Deltoidal icositetrahedron
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Click here for spinning version.
Type Catalan
Conway notation oC or deC
Coxeter diagram CDel node f1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node f1.png
Face polygon DU10 facets.png
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
Deltoidal icositetrahedron
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.

Dimensions[edit]

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[edit]

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[edit]

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

Orthogonal projections
Projective
symmetry
[2] [4] [6]
Image Dual cube t02 f4b.png Dual cube t02 B2.png Dual cube t02.png
Dual
image
Cube t02 f4b.png 3-cube t02 B2.svg 3-cube t02.svg

Related polyhedra[edit]

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

Dyakis dodecahedron[edit]

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
Partial cubic honeycomb.png Deltoidal icositetrahedron octahedral.png Deltoidal icositetrahedron octahedral gyro.png Deltoidal icositetrahedron gyro.png Deltoidal icositetrahedron concave-gyro.png

Related polyhedra and tilings[edit]

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)
Uniform polyhedron-43-t0.svg Uniform polyhedron-43-t01.svg Uniform polyhedron-43-t1.svg Uniform polyhedron-43-t12.svg Uniform polyhedron-43-t2.svg Uniform polyhedron-43-t02.png Uniform polyhedron-43-t012.png Uniform polyhedron-43-s012.png Uniform polyhedron-43-h01.svg
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node h.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.png CDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node.png
{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
Octahedron.svg Triakisoctahedron.jpg Rhombicdodecahedron.jpg Tetrakishexahedron.jpg Hexahedron.svg Deltoidalicositetrahedron.jpg Disdyakisdodecahedron.jpg Pentagonalicositetrahedronccw.jpg Dodecahedron.svg
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.
Spherical trigonal bipyramid.png
V3.4.2.4
Spherical rhombic dodecahedron.png
V3.4.3.4
Spherical deltoidal icositetrahedron.png
V3.4.4.4
Spherical deltoidal icositetrahedron.png
V3.4.5.4
Tiling Dual Semiregular V3-4-6-4 Deltoidal Trihexagonal.svg
V3.4.6.4
Deltoidal triheptagonal til.png
V3.4.7.4
Deltoidal trioctagonal til.png
V3.4.8.4
Deltoidal triapeirogonal til.png
V3.4.∞.4

See also[edit]

References[edit]

External links[edit]