Set of trapezohedra
Faces	2n kites
Edges	4n
Vertices	2n+2
Face configuration	V3.3.3.n
Symmetry group	Dnd
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 kites, also called deltoids (not trapezoids, so the name trapezohedron is misleading). 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.

The name deltohedron should not be confused with deltahedron.

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

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.

See also

---

• Rhombic dodecahedron
• Rhombic triacontahedron
• Conway Polyhedron Notation

External links

---

• Virtual Reality Polyhedra The Encyclopedia of Polyhedra
  • VRML models (George Hart) <3> <4> <5> <6> <7> [http://www.georgehart.com/virtual-polyhedra/vrml/enneagonal_trapezohedron.wrl <9> <10>
  • Conway Notation for Polyhedra Try: "dAn", where n=3,4,5... example "dA5" is a pentagonal trapezohedron.

polyhedra

Trapezoedro