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Concept
Trapezohedron
Summary
In geometry, an n-gonal trapezohedron, n-trapezohedron, n-antidipyramid, n-antibipyramid, or n-deltohedron is the dual polyhedron of an n-gonal antiprism. The 2n faces of an n-trapezohedron are congruent and symmetrically staggered; they are called twisted kites. With a higher symmetry, its 2n faces are kites (also called deltoids).
The "n-gonal" part of the name does not refer to faces here, but to two arrangements of each n vertices around an axis of n-fold symmetry. The dual n-gonal antiprism has two actual n-gon faces.
An n-gonal trapezohedron can be dissected into two equal n-gonal pyramids and an n-gonal antiprism.
These figures, sometimes called deltohedra, must not be confused with deltahedra, whose faces are equilateral triangles.
Twisted trigonal, tetragonal, and hexagonal trapezohedra (with six, eight, and twelve twisted congruent kite faces) exist as crystals; in crystallography (describing the crystal habits of minerals), 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 (i.e. poles) and two rings of twelve order-3 vertices each.
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