Trapezohedron

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Set of trapezohedra

Faces	2n kites
Edges	4n
Vertices	2n+2
Face configuration	V3.3.3.n
Symmetry group	D_nd
Dual polyhedron	antiprism
Properties	convex, face-uniform

The n-gonal trapezohedron or deltohedron is the dual polyhedron of a regular n-gonal antiprism. Its 2n faces are congruent deltoids (or kites). The faces are symmetrically staggered.

The name trapezohedron is misleading as the faces are not trapezoids, but the alternative deltohedron is sometimes confused with the unrelated term deltahedron.

The n-gon part of the name does not reference the faces here but arrangement of vertices around an axis of symmetry. The dual n-gonal antiprism has two actual n-gon faces.

An n-gonal trapezohedron can be decomposed into two equal n-gonal pyramids and an n-gonal antiprism.

In texts describing the crystal habits of minerals, the word trapezohedron is often used to refer to the polyhedron properly known as a deltoidal icositetrahedron.

1 Forms
2 Examples
3 Symmetry
4 See also
5 External links

[edit] Forms

Trigonal trapezohedron - 6 (rhombic) faces - dual octahedron
- A cube is a special case trigonal trapezohedron with square faces
- A trigonal trapezohedron is a special case rhombohedron with congruent rhombic faces
Tetragonal trapezohedron - 8 kite faces - dual square antiprism
Pentagonal trapezohedron - 10 kite faces - dual pentagonal antiprism
Hexagonal trapezohedron - 12 kite faces - dual hexagonal antiprism
Heptagonal trapezohedron - 14 kite faces - dual heptagonal antiprism
Octagonal trapezohedron - 16 kite faces - dual octagonal antiprism
Enneagonal trapezohedron - 18 kite faces - dual enneagonal antiprism
Decagonal trapezohedron - 20 kite faces - dual decagonal antiprism

...n-gonal trapezohedron - 2n kite faces - dual n-gonal antiprism

In the case of the dual of a regular triangular antiprism the kites are rhombi, hence these trapezohedra are also zonohedra. They are called rhombohedron. They are cubes scaled in the direction of a body diagonal. Also they are the parallelepipeds with congruent rhombic faces.

A special case of a rhombohedron is one of the which the rhombi which form the faces have angles of 60° and 120°. It can be decomposed into two equal regular tetrahedra and a regular octahedron. Since parallelepipeds can fill space, so can a combination of regular tetrahedra and regular octahedra.

[edit] Examples

Crystal arrangements of atoms can repeat in space with trapezohedral cells.
The pentagonal trapezohedron is the primary non-platonic solid polyhedron used as a die in roleplaying games such as Dungeons and Dragons. Having 10 sides, it can be used in repetition to generate any decimal-based uniform probability desired.

[edit] Symmetry

The symmetry group of an n-gonal trapezohedron is D_nd of order 4n, except in the case of a cube, which has the larger symmetry group O_d of order 48, which has four versions of D_3d as subgroups.

The rotation group is D_n of order 2n, except in the case of a cube, which has the larger rotation group O of order 24, which has four versions of D₃ as subgroups.