# Trigonal trapezohedron

In geometry, a **trigonal trapezohedron**, or **trigonal deltohedron**[1], or **isohedral rhombohedron**[2], or **rhombic hexahedron**[3] is a three-dimensional figure formed by six congruent rhombi.

Trigonal trapezohedron | |
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Click on picture for large version. | |

Type | trapezohedron |

Conway notation | dA3 |

Coxeter diagram | |

Faces | 6 rhombi |

Edges | 12 |

Vertices | 8 |

Face configuration | 3,3,3,3 |

Symmetry group | D_{3d}, [2^{+},6], (2*3), order 12 |

Rotation group | D_{3}, [2,3]^{+}, (223), order 6 |

Dual polyhedron | trigonal antiprism |

Properties | convex, face-transitive |

Six identical rhombic faces can construct two configurations of trigonal trapezohedra. The *acute* or *prolate* form has three acute angle corners of the rhombic faces meeting at the two polar axis vertices. The *obtuse* or *oblate* or *flat* form has three obtuse angle corners of the rhombic faces meeting at the two polar axis vertices.

A *trigonal trapezohedron* is an *isohedral rhombohedron*. (A general rhombohedron allows up to three types of rhombic faces, three different rhombic angles, with symmetry order 2.)

## Geometry

Trigonal trapezohedra are a special kind of parallelepipeds, and are the only parallelepipeds with six congruent faces. Since all edges must have the same length, every trigonal trapezohedron is also a rhombohedron.

A trigonal trapezohedron with square faces is a cube. |
A rhombic dodecahedron can be dissected into 4 identical obtuse trigonal trapezohedra. |
The rhombic hexecontahedron can be dissected into 20 acute golden rhombohedra meeting at a center point. |

## Golden rhombohedron

Acute form |
Obtuse form |

The **golden rhombohedra** are the two special cases of the *trigonal trapezohedron* with golden rhombus faces. The *acute* or *prolate* form has three acute angle corners of the rhombic faces meeting at the two polar axis vertices. The *obtuse* or *oblate* or *flat* form has three obtuse angle corners of the rhombic faces meeting at the two polar axis vertices.

## Related polyhedra

### Asymmetric variation

A lower symmetry variation of the *trigonal trapezohedron* has only rotational symmetry, D_{3}, and is made from 6 identical irregular quadrilaterals.[4] These quadrilaterals necessarily have two adjacent sides of equal length.

Polar axis | Side | Net |
---|---|---|

A regular octahedron augmented by 2 regular tetrahedra creates a *trigonal trapezohedron*, with coplanar equilateral triangles merged into 60-degree rhombic faces.

It is the simplest of the trapezohedra, an infinite sequence of polyhedra which are dual to the antiprisms. The dual of a *trigonal trapezohedron* is a triangular antiprism.

Family of trapezohedra V.n.3.3.3 | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|

Polyhedron | ||||||||||

Tiling | ||||||||||

Config. | V2.3.3.3 | V3.3.3.3 | V4.3.3.3 | V5.3.3.3 | V6.3.3.3 | V7.3.3.3 | V8.3.3.3 | ...V10.3.3.3 | ...V12.3.3.3 | ...V∞.3.3.3 |

## See also

## References

- http://mathworld.wolfram.com/TrigonalTrapezohedron.html
- Lines, L (1965).
*Solid geometry: with chapters on space-lattices, sphere-packs and crystals*. Dover Publications. - http://www.origamiheaven.com/rhombicpolyhedra.htm
- Fair Dice: Trigonal Trapezohedron Asymmetrical sides