Deltoidal icositetrahedron

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Deltoidal icositetrahedron
Deltoidal icositetrahedron
(Click here for rotating model)
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
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
Deltoidal icositetrahedron

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.


[3] 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

Where the short and long edges meet creates an angle of about 81.58°. The exact value is below represented by theta.

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
[2] [4] [6]
Image Dual cube t02 f4b.png Dual cube t02 B2.png Dual cube t02.png
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]

Dyakis dodecahedron in crystallography

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[4][5] or diploid.[6]

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.

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. It can also be seen that the threefold corners and the fourfold corners can be made to have the same distance to the center. In that case the resulting icositetrahedron will no longer have a rhombicuboctahedron for a dual, since for the rhombicuboctahedron the centers of its squares and its triangles are at different distances from the center.

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
Spherical Euclid. Compact hyperb. Paraco.
Spherical trigonal bipyramid.png
Spherical rhombic dodecahedron.png
Spherical deltoidal icositetrahedron.png
Spherical deltoidal hexecontahedron.png
Tiling Dual Semiregular V3-4-6-4 Deltoidal Trihexagonal.svg
Deltoidal triheptagonal tiling.svg
Deltoidal trioctagonal til.png
Deltoidal triapeirogonal til.png

See also[edit]


  1. ^ Conway, Symmetries of Things, p.284–286
  2. ^
  3. ^ "Kite". Retrieved 6 October 2019.
  4. ^ Isohedron 24k
  5. ^ The Isometric Crystal System
  6. ^ The 48 Special Crystal Forms

External links[edit]