Trapezohedron

Set of trapezohedra
Schläfli symbol	{ } ⨁ {n}
Coxeter diagram
Faces	2n kites
Edges	4n
Vertices	2n + 2
Face configuration	V3.3.3.n
Symmetry group	Dnd, [2+,2n], (2*n), order 4n
Rotation group	Dn, [2,n]+, (22n), order 2n
Dual polyhedron	antiprism
Properties	convex, face-transitive

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

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.

Name

These figures, sometimes called deltohedra, must not be confused with deltahedra, whose faces are equilateral triangles.

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

Forms

Family of trapezohedra

3	4	5	6	7	8	9	10	11	12	...
As spherical polyhedra

In the case of the dual of a regular triangular antiprism the kites are rhombi, hence these trapezohedra are also zonohedra. They are called rhombohedra. 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 in 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.

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.

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.

Star trapezohedra

Self-intersecting trapezohedron exist with a star polygon central figure, defined by kite faces connecting each polygon edge to these two points. A {p/q} trapezohedron has Coxeter-Dynkin diagram .

Uniform dual p/q star trapezohedra up to p=12

5/2	5/3	7/2	7/3	7/4	8/3	8/5	9/2	9/4	9/5
10/3	11/2	11/3	11/4	11/5	11/6	11/7	12/5	12/7

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 PlayStation 3 games Disgaea 3: Absence of Justice and Disgaea 4: A Promise Unforgotten, the most powerful emblem equipment in the game is called "Trapezohedron".
• In Final Fantasy XIII as the catalyst required to create any of the six ultimate weapons.
• In the PlayStation game Wild Arms 2, the "Trapezohedron" is used to imprison the Kuiper Belt but is not actually trapezohedron-shaped.

See also

• Rhombic dodecahedron
• Rhombic triacontahedron
• Bipyramid
• Conway polyhedron notation

External links

• Weisstein, Eric W., "Trapezohedron", MathWorld.
• Virtual Reality Polyhedra The Encyclopedia of Polyhedra
  • VRML models (George Hart) <3> <4> <5> <6> <7> <8> <9> <10>
  • Conway Notation for Polyhedra Try: "dAn", where n=3,4,5... example "dA5" is a pentagonal trapezohedron.
• Paper model tetragonal (square) trapezohedron