Trapezohedron

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Set of trapezohedra
Decagonal trapezohedron.
Schläfli symbol { } ⨁ {n}
Coxeter diagram CDel node fh.pngCDel 2.pngCDel node fh.pngCDel 2x.pngCDel n.pngCDel node.png
CDel node fh.pngCDel 2.pngCDel node fh.pngCDel n.pngCDel node fh.png
Faces 2n kites
Edges 4n
Vertices 2n + 2
Face configuration V3.3.3.n
Symmetry group Dnd, [2+,2n], (2*n), order 4n
Rotation group Dn, [2,n]+, (22n), order 2n
Dual polyhedron antiprism
Properties convex, face-transitive

The n-gonal trapezohedron, antidipyramid 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.

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[edit]

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[edit]

Family of trapezohedra
2 3 4 5 6 7 8 9 10 11 12 ...
CDel node fh.pngCDel 2x.pngCDel node fh.pngCDel 4.pngCDel node.png
CDel node fh.pngCDel 2x.pngCDel node fh.pngCDel 2x.pngCDel node fh.png
CDel node fh.pngCDel 2x.pngCDel node fh.pngCDel 6.pngCDel node.png
CDel node fh.pngCDel 2x.pngCDel node fh.pngCDel 3.pngCDel node fh.png
CDel node fh.pngCDel 2x.pngCDel node fh.pngCDel 8.pngCDel node.png
CDel node fh.pngCDel 2x.pngCDel node fh.pngCDel 4.pngCDel node fh.png
CDel node fh.pngCDel 2x.pngCDel node fh.pngCDel 1x.pngCDel 0x.pngCDel node.png
CDel node fh.pngCDel 2x.pngCDel node fh.pngCDel 5.pngCDel node fh.png
CDel node fh.pngCDel 2x.pngCDel node fh.pngCDel 1x.pngCDel 2x.pngCDel node.png
CDel node fh.pngCDel 2x.pngCDel node fh.pngCDel 6.pngCDel node fh.png
CDel node fh.pngCDel 2x.pngCDel node fh.pngCDel 1x.pngCDel 4.pngCDel node.png
CDel node fh.pngCDel 2x.pngCDel node fh.pngCDel 7.pngCDel node fh.png
CDel node fh.pngCDel 2x.pngCDel node fh.pngCDel 1x.pngCDel 6.pngCDel node.png
CDel node fh.pngCDel 2x.pngCDel node fh.pngCDel 8.pngCDel node fh.png
CDel node fh.pngCDel 2x.pngCDel node fh.pngCDel 2x.pngCDel 0x.pngCDel node.png
CDel node fh.pngCDel 2x.pngCDel node fh.pngCDel 10.pngCDel node fh.png
CDel node fh.pngCDel 2x.pngCDel node fh.pngCDel 2x.pngCDel 4.pngCDel node.png
CDel node fh.pngCDel 2x.pngCDel node fh.pngCDel 12.pngCDel node fh.png
Digonal trapezohedron.png TrigonalTrapezohedron.svg Tetragonal trapezohedron.png Pentagonal trapezohedron.svg Hexagonal trapezohedron.png Octagonal trapezohedron.png Decagonal trapezohedron.png
As spherical polyhedra
Spherical digonal antiprism.png Spherical trigonal trapezohedron.png Spherical tetragonal trapezohedron.png Spherical pentagonal trapezohedron.png Spherical hexagonal trapezohedron.png Spherical heptagonal trapezohedron.png Spherical octagonal trapezohedron.png Spherical decagonal trapezohedron.png Spherical dodecagonal trapezohedron.png

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[edit]

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[edit]

Star trapezohedra[edit]

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 CDel node fh.pngCDel 2.pngCDel node fh.pngCDel p.pngCDel rat.pngCDel q.pngCDel node.png.

Uniform dual p/q star trapezohedra up to p=12
5/2 5/3 7/2 7/3 7/4 8/3 8/5 9/2 9/4 9/5
5-2 deltohedron.png
CDel node fh.pngCDel 2.pngCDel node fh.pngCDel 5.pngCDel rat.pngCDel 2x.pngCDel node fh.png
5-3 deltohedron.png
CDel node fh.pngCDel 2.pngCDel node fh.pngCDel 5.pngCDel rat.pngCDel 3x.pngCDel node fh.png
7-2 deltohedron.png
CDel node fh.pngCDel 2.pngCDel node fh.pngCDel 7.pngCDel rat.pngCDel 2x.pngCDel node fh.png
7-3 deltohedron.png
CDel node fh.pngCDel 2.pngCDel node fh.pngCDel 7.pngCDel rat.pngCDel 3x.pngCDel node fh.png
7-4 deltohedron.png
CDel node fh.pngCDel 2.pngCDel node fh.pngCDel 7.pngCDel rat.pngCDel 4.pngCDel node fh.png
8-3 deltohedron.png
CDel node fh.pngCDel 2.pngCDel node fh.pngCDel 8.pngCDel rat.pngCDel 3x.pngCDel node fh.png
8-5 deltohedron.png
CDel node fh.pngCDel 2.pngCDel node fh.pngCDel 8.pngCDel rat.pngCDel 5.pngCDel node fh.png
9-2 deltohedron.png
CDel node fh.pngCDel 2.pngCDel node fh.pngCDel 9.pngCDel rat.pngCDel 2x.pngCDel node fh.png
9-4 deltohedron.png
CDel node fh.pngCDel 2.pngCDel node fh.pngCDel 9.pngCDel rat.pngCDel 4.pngCDel node fh.png
9-5 deltohedron.png
CDel node fh.pngCDel 2.pngCDel node fh.pngCDel 9.pngCDel rat.pngCDel 5.pngCDel node fh.png
10/3 11/2 11/3 11/4 11/5 11/6 11/7 12/5 12/7
10-3 deltohedron.png
CDel node fh.pngCDel 2.pngCDel node fh.pngCDel 10.pngCDel rat.pngCDel 3x.pngCDel node fh.png
11-2 deltohedron.png
CDel node fh.pngCDel 2.pngCDel node fh.pngCDel 11.pngCDel rat.pngCDel 2x.pngCDel node fh.png
11-3 deltohedron.png
CDel node fh.pngCDel 2.pngCDel node fh.pngCDel 11.pngCDel rat.pngCDel 3x.pngCDel node fh.png
11-4 deltohedron.png
CDel node fh.pngCDel 2.pngCDel node fh.pngCDel 11.pngCDel rat.pngCDel 4.pngCDel node fh.png
11-5 deltohedron.png
CDel node fh.pngCDel 2.pngCDel node fh.pngCDel 11.pngCDel rat.pngCDel 5.pngCDel node fh.png
11-6 deltohedron.png
CDel node fh.pngCDel 2.pngCDel node fh.pngCDel 11.pngCDel rat.pngCDel 6.pngCDel node fh.png
11-7 deltohedron.png
CDel node fh.pngCDel 2.pngCDel node fh.pngCDel 11.pngCDel rat.pngCDel 7.pngCDel node fh.png
12-5 deltohedron.png
CDel node fh.pngCDel 2.pngCDel node fh.pngCDel 12.pngCDel rat.pngCDel 5.pngCDel node fh.png
12-7 deltohedron.png
CDel node fh.pngCDel 2.pngCDel node fh.pngCDel 12.pngCDel rat.pngCDel 7.pngCDel node fh.png

See also[edit]

References[edit]

  • Anthony Pugh (1976). Polyhedra: A visual approach. California: University of California Press Berkeley. ISBN 0-520-03056-7.  Chapter 4: Duals of the Archimedean polyhedra, prisma and antiprisms

External links[edit]