# trapezohedron

Set of trapezohedra
Faces2n kites
Edges4n
Vertices2n+2
Face configurationV3.3.3.n
Symmetry groupDnd
Dual polyhedronantiprism
Propertiesconvex, face-transitive
The n-gonal trapezohedron, antidipyramid or deltohedron is the dual polyhedron of an n-gonal antiprism. Its 2n faces are congruent deltoids (or kites). The faces are symmetrically staggered.

The name trapezohedron is misleading as the faces are not trapezoids, but the alternative deltohedron is sometimes confused with the unrelated term deltahedron.

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.

In texts describing the crystal habits of minerals, the word trapezohedron is often used to refer to the polyhedron properly known as a deltoidal icositetrahedron.

## Forms

1. Trigonal trapezohedron - 6 (rhombic) faces - dual octahedron
2. * A cube is a special case trigonal trapezohedron with square faces
3. * A trigonal trapezohedron is a special case rhombohedron with congruent rhombic faces
4. Tetragonal trapezohedron - 8 kite faces - dual square antiprism
5. Pentagonal trapezohedron - 10 kite faces - dual pentagonal antiprism
6. Hexagonal trapezohedron - 12 kite faces - dual hexagonal antiprism
7. Heptagonal trapezohedron - 14 kite faces - dual heptagonal antiprism
8. Octagonal trapezohedron - 16 kite faces - dual octagonal antiprism
9. Enneagonal trapezohedron - 18 kite faces - dual enneagonal antiprism
10. Decagonal trapezohedron - 20 kite faces - dual decagonal antiprism
• ...n-gonal trapezohedron - 2n kite faces - dual n-gonal antiprism
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.

## 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.

kite, or deltoid, is a quadrilateral with two disjoint pairs of congruent adjacent sides, in contrast to a parallelogram, where the congruent sides are opposite. The geometric object is named for the wind-blown, flying kite (which is itself named for a bird), which in its
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For example, V3.4.3.4 represents the rhombic dodecahedron which is face-transitive: every face is a rhombus, and alternating vertices of the rhombus contain 3 or 4 faces each.
rotation (symmetry) group of the figure.]]

The symmetry group of an object (image, signal, etc., e.g. in 1D, 2D or 3D) is the group of all isometries under which it is invariant with composition as the operation.
polyhedra are associated into pairs called duals, where the of one correspond to the faces of the other. The dual of the dual is the original polyhedron. The dual of a polyhedron with equivalent vertices is one with equivalent faces, and of one with equivalent edges
An n-sided antiprism is a polyhedron composed of two parallel copies of some particular n-sided polygon, connected by an alternating band of triangles.

Antiprisms are a subclass of the prismatoids.
In geometry, a polyhedron is isohedral or face-transitive when all its faces are the same. More specifically, all faces must be not merely congruent but must be transitive, i.e. must lie within the same symmetry orbit.
polyhedra are associated into pairs called duals, where the of one correspond to the faces of the other. The dual of the dual is the original polyhedron. The dual of a polyhedron with equivalent vertices is one with equivalent faces, and of one with equivalent edges
An n-sided antiprism is a polyhedron composed of two parallel copies of some particular n-sided polygon, connected by an alternating band of triangles.

Antiprisms are a subclass of the prismatoids.
As an abstract term, congruence means similarity between objects. Congruence, as opposed to equivalence or approximation, is a relation which implies a kind of equivalence, though not complete equivalence.
kite, or deltoid, is a quadrilateral with two disjoint pairs of congruent adjacent sides, in contrast to a parallelogram, where the congruent sides are opposite. The geometric object is named for the wind-blown, flying kite (which is itself named for a bird), which in its
trapezoid (in North America) or trapezium (in Britain and elsewhere) is a quadrilateral, which is defined as a shape with four sides, which has one set of parallel sides.
deltahedron (plural deltahedra) is a polyhedron whose faces are all equilateral triangles. The name is taken from the Greek majuscule delta (Δ), which has the shape of an equilateral triangle.
An n-sided antiprism is a polyhedron composed of two parallel copies of some particular n-sided polygon, connected by an alternating band of triangles.

Antiprisms are a subclass of the prismatoids.
habit of crystals.

The many terms used by mineralogists to describe crystal habits are useful in communicating what specimens of a particular mineral often look like. Recognizing numerous habits helps a mineralogist to identify a large number of minerals.
A mineral is a naturally occurring substance formed through geological processes that has a characteristic chemical composition, a highly ordered atomic structure and specific physical properties.
A deltoidal icositetrahedron (or trapezoidal icositetrahedron) is a Catalan solid which looks a bit like an overinflated cube. Its dual polyhedron is the rhombicuboctahedron.

The 24 faces are deltoids or kites, not trapezoids; the trapezohedron is similarly misnamed.
The trigonal trapezohedron or deltohedron is the first in an infinite series of face-uniform polyhedra which are dual polyhedron to the antiprisms. It has six faces which are congruent rhombi.

Trigonal trapezohedrons are a subset of the rhombohedrons.
An octahedron (plural: octahedra) is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each .

The octahedron's symmetry group is Oh, of order 48.
cube[1] is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each . The cube can also be called a regular hexahedron and is one of the five Platonic solids.
In geometry, a rhombohedron is a three-dimensional figure like a cube, except that its faces are not squares but rhombi. It is a special case of a parallelepiped where all edges are the same length.
The tetragonal trapezohedron or deltohedron is the second in an infinite series of face-uniform polyhedra which are dual polyhedron to the antiprisms. It has eight faces which are congruent kites.
In geometry, the square antiprism is the second in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps.

If all its faces are regular, it is a semiregular polyhedron.
The pentagonal trapezohedron or deltohedron is the third in an infinite series of face-transitive polyhedra which are dual polyhedron to the antiprisms. It has ten faces (i.e., it is a decahedron) which are congruent kites.
In geometry, the pentagonal antiprism is the third in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps.

If faces are all regular, it is a semiregular polyhedron.
The hexagonal trapezohedron or deltohedron is the fourth in an infinite series of face-uniform polyhedra which are dual polyhedron to the antiprisms. It has twelve faces which are congruent kites.
In geometry, the hexagonal antiprism is the 4th in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps.

If faces are all regular, it is a semiregular polyhedron.
The octagonal trapezohedron or deltohedron is the sixth in an infinite series of face-uniform polyhedra which are dual polyhedron to the antiprisms. It has sixteen faces which are congruent kites.