# 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 | O_{h}, BC_{3}, [4,3], *432 |

Rotation group | O, [4,3]^{+}, (432) |

Dihedral angle | 138°07′05″ arccos(−7 + 4√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.

## Dimensions

The 24 faces are kites. The short and long edges of each kite are in the ratio 1:(2 − 1/√2) ≈ 1:893... 1.292

If its smallest edges have length 1, its surface area is and its volume is .

## 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 symmetry |
[2] | [4] | [6] |
---|---|---|---|

Image | |||

Dual image |

## 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, O_{h}, order 24 |
Pyritohedral, T_{h}, order 12 | |||
---|---|---|---|---|

## 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{3 ^{1,1}} |
t{3,4} t{3 ^{1,1}} |
{3,4} {3 ^{1,1}} |
rr{4,3} s _{2}{3,4} |
tr{4,3} | sr{4,3} | h{4,3} {3,3} |
h_{2}{4,3}t{3,3} |
s{3,4} s{3 ^{1,1}} |

= |
= |
= |
= or |
= or |
= | |||||

Duals to uniform polyhedra | ||||||||||

V4^{3} |
V3.8^{2} |
V(3.4)^{2} |
V4.6^{2} |
V3^{4} |
V3.4^{3} |
V4.6.8 | V3^{4}.4 |
V3^{3} |
V3.6^{2} |
V3^{5} |

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 (**n*32) reflectional symmetry.

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 |

## See also

- Deltoidal hexecontahedron
- Tetrakis hexahedron, another 24-face Catalan solid which looks a bit like an overinflated cube.
- "The Haunter of the Dark", a story by H.P. Lovecraft, whose plot involves this figure

## References

- ↑ Conway,
*Symmetries of Things*, p.284–286 - ↑ Isohedron 24k
- ↑ The Isometric Crystal System
- ↑ https://www.uwgb.edu/dutchs/symmetry/xlforms.htm

- Williams, Robert (1979).
*The Geometrical Foundation of Natural Structure: A Source Book of Design*. Dover Publications, Inc. ISBN 0-486-23729-X. (Section 3-9) - Wenninger, Magnus (1983),
*Dual Models*, Cambridge University Press, ISBN 978-0-521-54325-5, MR 730208 (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 (Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, page 286, tetragonal icosikaitetrahedron)

## External links

- Eric W. Weisstein,
*Deltoidal icositetrahedron*(*Catalan solid*) at MathWorld. - Deltoidal (Trapezoidal) Icositetrahedron – Interactive Polyhedron model