Trapezohedron

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

The name trapezohedron can be misleading as the faces are not trapezoids, but the alternative deltohedron is sometimes confused with the unrelated term deltahedron. It is quite possible, but not necessarily true, that the name trapezohedron could be derived from trapezium (a quadrilateral with no parallel sides) as opposed to trapezoid, since the faces are kites, a type of trapezium.

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.

Forms

 * 1) Trigonal trapezohedron - 6 (rhombic) faces - dual octahedron
 * 2) * A cube is a special case trigonal trapezohedron with square faces
 * 3) * A trigonal trapezohedron is a special case rhombohedron with congruent rhombic faces
 * 4) Tetragonal trapezohedron - 8 kite faces - dual square antiprism
 * 5) Pentagonal trapezohedron - 10 kite faces - dual pentagonal antiprism
 * 6) Hexagonal trapezohedron - 12 kite faces - dual hexagonal antiprism
 * 7) Heptagonal trapezohedron - 14 kite faces - dual heptagonal antiprism
 * 8) Octagonal trapezohedron - 16 kite faces - dual octagonal antiprism
 * 9) Enneagonal trapezohedron - 18 kite faces - dual enneagonal antiprism
 * 10) 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.

Examples

 * Crystal arrangements of atoms can repeat in space with trapezohedral cells.
 * The pentagonal trapezohedron is the only polyhedron other than the Platonic solids commonly 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. Two dice of different colors are typically used for the two digits to represent numbers from 00 to 99.

Symmetry
The symmetry group of an n-gonal trapezohedron is Dnd of order 4n, except in the case of a cube, which has the larger symmetry group Od of order 48, which has four versions of D3d as subgroups.

The rotation group is Dn 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.

In Popular Culture

 * H.P. Lovecraft's short story "The Haunter of the Dark" features a "crazily angled stone" of extraterrestrial origin called the "Shining Trapezohedron".
 * In The Sims and The Sims 2, there is a Trapezohedron over the people's heads when you select them.
 * In the PlayStation 3 game Disgaea 3: Absence of Justice, the most powerful emblem equipment in the game is called 'Trapezohedron'
 * In the PlayStation game Wild Arms 2, the 'Trapezohedron' is used to imprison the Kuiper Belt but is not actually Trapezohedron shaped.