Trapezohedron

These figures, sometimes called deltohedra, must not be confused with deltahedra, whose faces are equilateral triangles.

In crystallography, describing the crystal habits of minerals, the word trapezohedron is often used for the polyhedron properly known as a deltoidal icositetrahedron; another polyhedron is known as a deltoid dodecahedron.[3]

The symmetry group of an n-gonal trapezohedron is D_nd of order 4n, except in the case of a cube, which has the larger symmetry group O_d of order 48, which has four versions of D_3d as subgroups.

The rotation group is D_n 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 symmetry from D_nd (order 4n) to D_n (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 C_nv (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 C_n (cyclic) symmetry, order n, and is called an unequal twisted trapezohedron.

Example variations

Type	Twisted trapezohedron		Unequal trapezohedron	Unequal twisted trapezohedron
Symmetry	Dn, (nn2), [n,2]+		Cnv, (*nn), [n]	Cn, (nn), [n]+
Image (n=6)
Net

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 n-gonal trapezohedra
Polyhedron image										...	Apeirogonal trapezohedron
Spherical tiling image										Plane tiling image
Face configuration Vn.3.3.3	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: 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
  • A special case of a rhombohedron is one in 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.

• Crystal arrangements of atoms can repeat in space with trigonal and hexagonal trapezohedral cells.[4]
• The pentagonal trapezohedron is the only polyhedron other than the Platonic solids commonly used as a die in roleplaying games such as Dungeons & Dragons. Having 10 sides, it can be used in repetition to generate any decimal-based uniform probability desired. Two dice of different colors are typically used for the two digits to represent numbers from 00 to 99.

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 .

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
10/3	11/2	11/3	11/4	11/5	11/6	11/7	12/5	12/7

Wikimedia Commons has media related to Trapezohedra.

• Diminished trapezohedron
• Rhombic dodecahedron
• Rhombic triacontahedron
• Bipyramid
• Truncated trapezohedron
• Conway polyhedron notation
• The Haunter of the Dark, a short story by H.P. Lovecraft in which a fictional ancient artifact known as The Shining Trapezohedron plays a crucial role.

• 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

• Weisstein, Eric W. "Trapezohedron". MathWorld.
• Weisstein, Eric W. "Isohedron". MathWorld.
• Virtual Reality Polyhedra The Encyclopedia of Polyhedra
  • VRML models (George Hart) <3> <4> <5> <6> <7> <8> <9> <10>
  • Conway Notation for Polyhedra Try: "dAn", where n=3,4,5... example "dA5" is a pentagonal trapezohedron.
• Paper model tetragonal (square) trapezohedron