Trapezohedron

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Set of dual-uniform n-gonal trapezohedra
Pentagonal trapezohedron.svg
Example dual-uniform pentagonal trapezohedron
Type dual-uniform in the sense of dual-semiregular polyhedron
Conway notation dAn
Schläfli symbol { } ⨁ {n}[1]
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 congruent 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 (convex) uniform n-gonal antiprism
Properties convex, face-transitive, regular vertices[2]

An n-gonal trapezohedron, antidipyramid, antibipyramid, or 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 trapezoids, or deltoids).

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.

Name[edit]

These figures, sometimes called deltohedra, must not be confused with deltahedra, 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.[5]

Also in crystallography, the word trapezohedron is often used for the polyhedron with 24 trapezoidal faces properly known as a (deltoidal) icositetrahedron.[6] Another polyhedron, with 12 trapezoidal faces, is known as a deltoid dodecahedron.[7]

Symmetry[edit]

The symmetry group of an n-gonal trapezohedron is Dnd of order 4n, except in the case of n = 3: a cube has the larger symmetry group Od of order 48 = 4×(4×3), which has four versions of D3d as subgroups.

The rotation group is Dn of order 2n, 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 D3 as subgroups.

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

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

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

Example variations for n = 6
Trapezohedron type Twisted trapezohedron Unequal trapezohedron Unequal twisted trapezohedron
Symmetry group Dn, (nn2), [n,2]+ Cnv, (*nn), [n] Cn, (nn), [n]+
Polyhedron image 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]

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

Family of n-gonal trapezohedra
Trapezohedron name Digonal trapezohedron
(Tetrahedron)
Trigonal trapezohedron Tetragonal trapezohedron Pentagonal trapezohedron Hexagonal trapezohedron Heptagonal trapezohedron Octagonal trapezohedron Decagonal trapezohedron Dodecagonal trapezohedron ... Apeirogonal trapezohedron
Polyhedron image 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 ...
Spherical tiling image 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 Plane tiling image Apeirogonal trapezohedron.svg
Face configuration 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 of trapezohedron: 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. 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. They are also the parallelepipeds with congruent rhombic faces.
    A 60° rhombohedron, dissected into a central regular octahedron and two regular tetrahedra

Examples[edit]

Star trapezohedron[edit]

A face-transitive star p/q-trapezohedron is defined by a regular zig-zag skew star 2p/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 2p/q-gon base; but if p/q < 3/2, then pq < 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 pq = q/2, so the dual star antiprism must be flat, thus degenerate, to be uniform.

A dual-uniform star 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.

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

See also[edit]

References[edit]

  1. ^ N.W. Johnson: Geometries and Transformations, (2018) ISBN 978-1-107-10340-5 Chapter 11: Finite symmetry groups, 11.3 Pyramids, Prisms, and Antiprisms, Figure 11.3c
  2. ^ "duality". maths.ac-noumea.nc. Retrieved 2020-10-19.
  3. ^ "1911 Encyclopædia Britannica/Crystallography - Wikisource, the free online library". en.m.wikisource.org. Retrieved 2020-11-16.
  4. ^ 1911 Encyclopædia Britannica/Crystallography, HEXAGONAL SYSTEM, Rhombohedral Division, TRAPEZOHEDRAL CLASS, FIG. 74.
  5. ^ 1911 Encyclopædia Britannica/Crystallography, TETRAGONAL SYSTEM, TRAPEZOHEDRAL CLASS.
  6. ^ 1911 Encyclopædia Britannica/Crystallography, FIG. 17.
  7. ^ 1911 Encyclopædia Britannica/Crystallography, FIG. 27.
  8. ^ Trigonal-trapezohedric Class, 3 2 and Hexagonal-trapezohedric Class, 6 2 2
  • 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]