The n-gonal trapezohedron, antidipyramid, antibipyramid or deltohedron is the dual polyhedron of an n-gonal antiprism. Its 2n faces are congruent kites (also called trapezia or deltoids). The faces are symmetrically staggered.
Contents
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
In the case of the dual of a triangular antiprism the kites are rhombi (or squares), 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.
A degenerate form, n = 2, form a geometric tetrahedron with 6 vertices, 8 edges, and 4 degenerate kite faces that are degenerated into triangles. Its dual is a degenerate form of antiprism, also a tetrahedron.
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.
If the kites surrounding the two peaks are of different shapes, it can only have Cnv symmetry, order 2n.
Examples
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 .