# Trapezohedron

An *n*-gonal **trapezohedron**, **antidipyramid**, **antibipyramid**, or **deltohedron** is the dual polyhedron of an *n*-gonal antiprism. The 2*n* faces of an *n*-trapezohedron are congruent and symmetrically staggered; they are called *twisted kites*. With a higher symmetry, its 2*n* faces are *kites* (also called *trapezoids*,^{[citation needed]} or *delt oids*).

The *n*-gon 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 delt**o**hedra, must not be confused with delt**a**hedra, whose faces are equilateral triangles.

A *twisted* trigonal trapezohedron (with six *twisted* trapezoidal faces) and a *twisted* tetragonal trapezohedron (with eight *twisted* trapezoidal faces) exist as crystals; in crystallography^{[3]} (describing the crystal habits of minerals), they are just called *trigonal trapezohedron* and *tetragonal trapezohedron*. They have no symmetry plane, and no symmetry center. The trigonal trapezohedron has one 3-fold symmetry axis, perpendicular to three 2-fold symmetry axes.^{[4]} The tetragonal trapezohedron has one 4-fold symmetry axis, perpendicular to four 2-fold symmetry axes.^{[4]}

Also in crystallography, the word *trapezohedron* is often used for the polyhedron with 24 trapezoidal faces properly known as a *(deltoidal) icositetrahedron*.^{[5]} Another polyhedron, with 12 trapezoidal faces, is known as a *deltoid dodecahedron*.^{[6]}

The symmetry group of an *n*-gonal trapezohedron is D_{nd}, of order 4*n*, except in the case of *n* = 3: a cube has the larger symmetry group O_{d} of order 48 = 4×(4×3), which has four versions of D_{3d} as subgroups.

The rotation group of an *n*-trapezohedron is D_{n}, of order 2*n*, except in the case of *n* = 3: a cube has the larger rotation group O of order 24 = 4×(2×3), which has four versions of D_{3} as subgroups.

One degree of freedom within symmetry from D_{nd} (order 4*n*) to D_{n} (order 2*n*) changes the congruent kites into congruent quadrilaterals with three edge lengths, called *twisted kites*, and the *n*-trapezohedron is called a *twisted trapezohedron*. (In the limit, one edge of each quadrilateral goes to zero length, and the *n*-trapezohedron becomes an *n*-bipyramid.)

If the kites surrounding the two peaks are not twisted but are of two different shapes, the *n*-trapezohedron can only have C_{nv} (cyclic with vertical mirrors) symmetry, order 2*n*, and is called an *unequal* or *asymmetric trapezohedron*. Its dual is an *unequal n-antiprism*, with the top and bottom polygons of different radii.

If the kites are twisted and are of two different shapes, the *n*-trapezohedron can only have C_{n} (cyclic) symmetry, order *n*, and is called an *unequal twisted trapezohedron*.

An *n*-trapezohedron has 2*n* quadrilateral faces, with 2*n*+2 vertices. Two apexes are on the polar axis, and the other vertices are in two regular *n*-gonal rings of vertices.

A face-transitive **star p/q-trapezohedron** is defined by a regular zig-zag skew star

**2**

*p*/

*q*-gon base, two symmetric apexes with no degree of freedom right above and right below the base, and kite faces connecting each pair of base adjacent edges to one apex.

Such a star *p*/*q*-trapezohedron is a *self-intersecting*, *crossed*, or *non-convex* form. It exists for any regular zig-zag skew star **2***p*/*q*-gon base; but if *p*/*q* < 3/2, then *p* — *q* < *q*/2, so the dual star antiprism (of the star trapezohedron) cannot be uniform (i.e.: cannot have equal edge lengths); and if *p*/*q* = 3/2, then *p* — *q* = *q*/2, so the dual star antiprism must be flat, thus degenerate, to be uniform.

A dodecagonal trapezohedron is a trapezohedron with 24 kites. It has 12-fold antiprismmic symmetry, order 48.