Trapezohedron


In geometry, an trapezohedron, -trapezohedron, -antidipyramid, -antibipyramid, -deltohedron, or -'antitegum.' is the dual polyhedron of an antiprism. The faces of an are congruent and symmetrically staggered; they are called twisted kites. With a higher symmetry, its faces are kites. They are topologically related to the scalenohedra, which are half-symmetry variants with irregular trigons.
As there is no consistent analog in higher dimensions to the 3D antiprisms, there is similarly no general generalization of 3D trapezohedra either.The "" part of the name does not refer to faces here, but to two arrangements of each vertices around an axis of symmetry. The dual antiprism has two actual faces. An trapezohedron can be dissected into two equal pyramids and an antiprism.

Terminology

These figures, sometimes called deltohedra, are not to be confused with deltahedra, whose faces are equilateral triangles.
Twisted trigonal, tetragonal, and hexagonal trapezohedra exist as crystals; in crystallography, they are just called trigonal, tetragonal, and hexagonal trapezohedra. They have no plane of symmetry, and no center of inversion symmetry;, but they have a center of symmetry: the intersection point of their symmetry axes. The trigonal trapezohedron has one 3-fold symmetry axis, perpendicular to three 2-fold symmetry axes. The tetragonal trapezohedron has one 4-fold symmetry axis, perpendicular to four 2-fold symmetry axes of two kinds. The hexagonal trapezohedron has one 6-fold symmetry axis, perpendicular to six 2-fold symmetry axes of two kinds.
Crystal arrangements of atoms can repeat in space with trigonal and hexagonal trapezohedron cells.
Also in crystallography, the word trapezohedron is often used for the polyhedron with 24 congruent non-twisted kite faces properly known as a deltoidal icositetrahedron, which has eighteen order-4 vertices and eight order-3 vertices. This is not to be confused with the dodecagonal trapezohedron, which also has 24 congruent kite faces, but two order-12 apices and two rings of twelve order-3 vertices each.
Still in crystallography, the deltoid dodecahedron has 12 congruent non-twisted kite faces, six order-4 vertices and eight order-3 vertices. This is not to be confused with the hexagonal trapezohedron, which also has 12 congruent kite faces, but two order-6 apices and two rings of six order-3 vertices each.

Forms

An -trapezohedron is defined by a regular zig-zag skew -gon base, two symmetric apices with no degree of freedom right above and right below the base, and quadrilateral faces connecting each pair of adjacent basal edges to one apex.
An -trapezohedron has two apical vertices on its polar axis, and basal vertices in two regular -gonal rings. It has congruent kite faces, and it is isohedral.

Special cases

Symmetry

The symmetry group of an -gonal trapezohedron is, of order, except in the case of : a cube has the larger symmetry group of order, which has four versions of as subgroups.
The rotation group of an -trapezohedron is, of order, except in the case of : a cube has the larger rotation group of order, which has four versions of as subgroups.
Note: Every -trapezohedron with a regular zig-zag skew -gon base and congruent non-twisted kite faces has the same symmetry group as the dual-uniform -trapezohedron, for.
One degree of freedom within symmetry from to changes the congruent kites into congruent quadrilaterals with three edge lengths, called twisted kites, and the -trapezohedron is called a twisted trapezohedron.
If the kites surrounding the two peaks are not twisted but are of two different shapes, the -trapezohedron can only have symmetry, order, and is called an unequal or asymmetric trapezohedron. Its dual is an unequal -antiprism, with the top and bottom -gons of different radii.
If the kites are twisted and are of two different shapes, the -trapezohedron can only have symmetry, order, and is called an unequal twisted trapezohedron.

Star trapezohedron

A star -trapezohedron is defined by a regular zig-zag skew star -gon base, two symmetric apices with no degree of freedom right above and right below the base, and quadrilateral faces connecting each pair of adjacent basal edges to one apex.
A star -trapezohedron has two apical vertices on its polar axis, and basal vertices in two regular -gonal rings. It has congruent kite faces, and it is isohedral.
Such a star -trapezohedron is a self-intersecting, crossed, or non-convex form. It exists for any regular zig-zag skew star -gon base.
But if, then, so the dual star antiprism cannot be uniform ; and if, then, so the dual star antiprism must be flat, thus degenerate, to be uniform.
A dual-uniform star -trapezohedron has Coxeter-Dynkin diagram.
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