Trapezohedron

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Set of trapezohedra
Decagonal trapezohedron.
Conway notation dAn
Schläfli symbol { } ⨁ {n}
Coxeter diagrams CDel node fh.pngCDel 2x.pngCDel node fh.pngCDel 2x.pngCDel n.pngCDel node.png
CDel node fh.pngCDel 2x.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, antibipyramid or deltohedron is the dual polyhedron of an n-gonal antiprism. With a highest symmetry, 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 dissected 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.

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.

One degree of freedom within Dn symmetry changes the kites into congruent quadrilaterals with 3 edges lengths. In the limit, one edge of each quadrilateral goes to zero length, and these become bipyramids.

If the kites surrounding the two peaks are of different shapes, it can only have Cnv symmetry, order 2n. These can be called unequal trapezohedra, the dual is an unequal antiprism, with the top and bottom polygons of different radii. If it twisted and unequal its symmetry is reduced to cyclic symmetry, Cn symmetry, order n.

Example variations
Type Twisted trapezohedra Unequal trapezohedra Unequal and twisted
Symmetry Dn, (nn2), [n,2]+ Cnv, (*nn), [n] Cn, (nn), [n]+
Image
(n=6)
Twisted hexagonal trapezohedron.png Twisted hexagonal trapezohedron2.png Unequal hexagonal trapezohedron.png Unequal twisted hexagonal trapezohedron.png
Net Twisted hexagonal trapezohedron net.png Twisted hexagonal trapezohedron2 net.png Unequal hexagonal trapezohedron net.png Unequal twisted hexagonal trapezohedron net.png

Forms[edit]

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

Family of trapezohedra V.n.3.3.3
Polyhedron Digonal trapezohedron.png TrigonalTrapezohedron.svg Tetragonal trapezohedron.png Pentagonal trapezohedron.svg Hexagonal trapezohedron.png Heptagonal trapezohedron.png Octagonal trapezohedron.png Decagonal trapezohedron.png Dodecagonal trapezohedron.png
Tiling 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 Apeirogonal deltahedron.svg
Config. V2.3.3.3 V3.3.3.3 V4.3.3.3 V5.3.3.3 V6.3.3.3 V7.3.3.3 V8.3.3.3 ...V10.3.3.3 ...V12.3.3.3 ...V∞.3.3.3

Special cases:

  • n=2: A degenerate form, 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.
  • n=3: 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 60° rhombohedron, dissected into a central regular octahedron and two regular tetrahedra

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 2x.pngCDel node fh.pngCDel p.pngCDel rat.pngCDel q.pngCDel node fh.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 2x.pngCDel node fh.pngCDel 5.pngCDel rat.pngCDel 2x.pngCDel node fh.png
5-3 deltohedron.png
CDel node fh.pngCDel 2x.pngCDel node fh.pngCDel 5.pngCDel rat.pngCDel 3x.pngCDel node fh.png
7-2 deltohedron.png
CDel node fh.pngCDel 2x.pngCDel node fh.pngCDel 7.pngCDel rat.pngCDel 2x.pngCDel node fh.png
7-3 deltohedron.png
CDel node fh.pngCDel 2x.pngCDel node fh.pngCDel 7.pngCDel rat.pngCDel 3x.pngCDel node fh.png
7-4 deltohedron.png
CDel node fh.pngCDel 2x.pngCDel node fh.pngCDel 7.pngCDel rat.pngCDel 4.pngCDel node fh.png
8-3 deltohedron.png
CDel node fh.pngCDel 2x.pngCDel node fh.pngCDel 8.pngCDel rat.pngCDel 3x.pngCDel node fh.png
8-5 deltohedron.png
CDel node fh.pngCDel 2x.pngCDel node fh.pngCDel 8.pngCDel rat.pngCDel 5.pngCDel node fh.png
9-2 deltohedron.png
CDel node fh.pngCDel 2x.pngCDel node fh.pngCDel 9.pngCDel rat.pngCDel 2x.pngCDel node fh.png
9-4 deltohedron.png
CDel node fh.pngCDel 2x.pngCDel node fh.pngCDel 9.pngCDel rat.pngCDel 4.pngCDel node fh.png
9-5 deltohedron.png
CDel node fh.pngCDel 2x.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 2x.pngCDel node fh.pngCDel 10.pngCDel rat.pngCDel 3x.pngCDel node fh.png
11-2 deltohedron.png
CDel node fh.pngCDel 2x.pngCDel node fh.pngCDel 11.pngCDel rat.pngCDel 2x.pngCDel node fh.png
11-3 deltohedron.png
CDel node fh.pngCDel 2x.pngCDel node fh.pngCDel 11.pngCDel rat.pngCDel 3x.pngCDel node fh.png
11-4 deltohedron.png
CDel node fh.pngCDel 2x.pngCDel node fh.pngCDel 11.pngCDel rat.pngCDel 4.pngCDel node fh.png
11-5 deltohedron.png
CDel node fh.pngCDel 2x.pngCDel node fh.pngCDel 11.pngCDel rat.pngCDel 5.pngCDel node fh.png
11-6 deltohedron.png
CDel node fh.pngCDel 2x.pngCDel node fh.pngCDel 11.pngCDel rat.pngCDel 6.pngCDel node fh.png
11-7 deltohedron.png
CDel node fh.pngCDel 2x.pngCDel node fh.pngCDel 11.pngCDel rat.pngCDel 7.pngCDel node fh.png
12-5 deltohedron.png
CDel node fh.pngCDel 2x.pngCDel node fh.pngCDel 12.pngCDel rat.pngCDel 5.pngCDel node fh.png
12-7 deltohedron.png
CDel node fh.pngCDel 2x.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]