| Set of trapezohedra | |
|---|---|
| Faces | 2n kites |
| Edges | 4n |
| Vertices | 2n+2 |
| Face configuration | V3.3.3.n |
| Symmetry group | Dnd, [2+,2n], (2*n) |
| 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.
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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.
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.
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.
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