Trapezohedron

In geometry, an Template:Mvar-gonal trapezohedron, Template:Mvar-trapezohedron, Template:Mvar-antidipyramid, Template:Mvar-antibipyramid, or Template:Mvar-deltohedron^, is the dual polyhedron of an Template:Mvar-gonal antiprism. The Template:Math faces of an Template:Mvar-trapezohedron are congruent and symmetrically staggered; they are called twisted kites. With a higher symmetry, its Template:Math faces are kites (sometimes also called trapezoids, or deltoids).Template:Sfn

The "Template:Mvar-gonal" part of the name does not refer to faces here, but to two arrangements of each Template:Mvar vertices around an axis of Template:Mvar-fold symmetry. The dual Template:Mvar-gonal antiprism has two actual Template:Mvar-gon faces.

An Template:Mvar-gonal trapezohedron can be dissected into two equal Template:Mvar-gonal pyramids and an Template:Mvar-gonal antiprism.

Terminology

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

Twisted trigonal, tetragonal, and hexagonal trapezohedra (with six, eight, and twelve twisted congruent kite faces) exist as crystals; in crystallography (describing the crystal habits of minerals), they are just called trigonal, tetragonal, and hexagonal trapezohedra. They have no plane of symmetry, and no center of inversion symmetry;Template:Sfn^,Template:Sfn but they have a center of symmetry: the intersection point of their symmetry axes. The trigonal trapezohedron has one 3-fold symmetry axis, perpendicular to three 2-fold symmetry axes.Template:Sfn The tetragonal trapezohedron has one 4-fold symmetry axis, perpendicular to four 2-fold symmetry axes of two kinds. The hexagonal trapezohedron has one 6-fold symmetry axis, perpendicular to six 2-fold symmetry axes of two kinds.Template:Sfn

Crystal arrangements of atoms can repeat in space with trigonal and hexagonal trapezohedron cells.

Also in crystallography, the word trapezohedron is often used for the polyhedron with 24 congruent non-twisted kite faces properly known as a deltoidal icositetrahedron,Template:Sfn which has eighteen order-4 vertices and eight order-3 vertices. This is not to be confused with the dodecagonal trapezohedron, which also has 24 congruent kite faces, but two order-12 apices (i.e. poles) and two rings of twelve order-3 vertices each.

Still in crystallography, the deltoid dodecahedronTemplate:Sfn has 12 congruent non-twisted kite faces, six order-4 vertices and eight order-3 vertices (the rhombic dodecahedron is a special case). This is not to be confused with the hexagonal trapezohedron, which also has 12 congruent kite faces,Template:Sfn but two order-6 apices (i.e. poles) and two rings of six order-3 vertices each.

Forms

An Template:Mvar-trapezohedron is defined by a regular zig-zag skew Template:Math-gon base, two symmetric apices with no degree of freedom right above and right below the base, and quadrilateral faces connecting each pair of adjacent basal edges to one apex.

An Template:Mvar-trapezohedron has two apical vertices on its polar axis, and Template:Math basal vertices in two regular Template:Mvar-gonal rings. It has Template:Math congruent kite faces, and it is isohedral.

Special cases

• Template:Math. A degenerate form of trapezohedron: a geometric figure with 6 vertices, 8 edges, and 4 degenerate kite faces that are visually identical to triangles. As such, the trapezohedron itself is visually identical to the regular tetrahedron. Its dual is a degenerate form of antiprism that also resembles the regular tetrahedron.

• Template:Math. 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.

  File:Gyroelongated triangular bipyramid.png

  A Template:Math rhombohedron, dissected into a central regular octahedron and two regular tetrahedra

  • A special case of a rhombohedron is one in which the rhombi forming the faces have angles of Template:Math and Template:Math. 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.

• Template:Math. The pentagonal trapezohedron is the only polyhedron other than the Platonic solids commonly used as a die in roleplaying games such as Dungeons & Dragons. Being convex and face-transitive, it makes fair dice. Having 10 sides, it can be used in repetition to generate any decimal-based uniform probability desired. Typically, two dice of different colors are used for the two digits to represent numbers from Template:Math to Template:Math.

Symmetry

The symmetry group of an Template:Mvar-gonal trapezohedron is Template:Math, of order Template:Math, except in the case of Template:Math: a cube has the larger symmetry group Template:Math of order Template:Math, which has four versions of Template:Math as subgroups.

The rotation group of an Template:Mvar-trapezohedron is Template:Math, of order Template:Math, except in the case of Template:Math: a cube has the larger rotation group Template:Math of order Template:Math, which has four versions of Template:Math as subgroups.

Note: Every Template:Mvar-trapezohedron with a regular zig-zag skew Template:Math-gon base and Template:Math congruent non-twisted kite faces has the same (dihedral) symmetry group as the dual-uniform Template:Mvar-trapezohedron, for Template:Math.

One degree of freedom within symmetry from Template:Math (order Template:Math) to Template:Math (order Template:Math) changes the congruent kites into congruent quadrilaterals with three edge lengths, called twisted kites, and the Template:Mvar-trapezohedron is called a twisted trapezohedron. (In the limit, one edge of each quadrilateral goes to zero length, and the Template:Mvar-trapezohedron becomes an Template:Mvar-bipyramid.)

If the kites surrounding the two peaks are not twisted but are of two different shapes, the Template:Mvar-trapezohedron can only have Template:Math (cyclic with vertical mirrors) symmetry, order Template:Math, and is called an unequal or asymmetric trapezohedron. Its dual is an unequal Template:Mvar-antiprism, with the top and bottom Template:Mvar-gons of different radii.

If the kites are twisted and are of two different shapes, the Template:Mvar-trapezohedron can only have Template:Math (cyclic) symmetry, order Template:Mvar, and is called an unequal twisted trapezohedron.

Example: variations with hexagonal trapezohedra (n = 6)

Trapezohedron type	Twisted trapezohedron		Unequal trapezohedron	Unequal twisted trapezohedron
Symmetry group	D6, (662), [6,2]+		C6v, (*66), [6]	C6, (66), [6]+
Polyhedron image	File:Twisted hexagonal trapezohedron.png	File:Twisted hexagonal trapezohedron2.png	File:Unequal hexagonal trapezohedron.png	File:Unequal twisted hexagonal trapezohedron.png
Net	File:Twisted hexagonal trapezohedron net.png	File:Twisted hexagonal trapezohedron2 net.png	File:Unequal hexagonal trapezohedron net.png	File:Unequal twisted hexagonal trapezohedron net.png

Star trapezohedron

A star Template:Math-trapezohedron (where Template:Math) is defined by a regular zig-zag skew [[Star polygon|star Template:Math-gon]] base, two symmetric apices with no degree of freedom right above and right below the base, and quadrilateral faces connecting each pair of adjacent basal edges to one apex.

A star Template:Math-trapezohedron has two apical vertices on its polar axis, and Template:Math basal vertices in two regular Template:Mvar-gonal rings. It has Template:Math congruent kite faces, and it is isohedral.

Such a star Template:Math-trapezohedron is a self-intersecting, crossed, or non-convex form. It exists for any regular zig-zag skew star Template:Math-gon base (where Template:Math).

But if Template:Math, then Template:Math, so the dual star antiprism (of the star trapezohedron) cannot be uniform (i.e. cannot have equal edge lengths); and if Template:Math, then Template:Math, so the dual star antiprism must be flat, thus degenerate, to be uniform.

A dual-uniform star Template:Math-trapezohedron has Coxeter-Dynkin diagram Template:CDD.

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
File:5-2 deltohedron.png	File:5-3 deltohedron.png	File:7-2 deltohedron.png	File:7-3 deltohedron.png	File:7-4 deltohedron.png	File:8-3 deltohedron.png	File:8-5 deltohedron.png	File:9-2 deltohedron.png	File:9-4 deltohedron.png	File:9-5 deltohedron.png
Template:CDD	Template:CDD	Template:CDD	Template:CDD	Template:CDD	Template:CDD	Template:CDD	Template:CDD	Template:CDD	Template:CDD

10/3	11/2	11/3	11/4	11/5	11/6	11/7	12/5	12/7
File:10-3 deltohedron.png	File:11-2 deltohedron.png	File:11-3 deltohedron.png	File:11-4 deltohedron.png	File:11-5 deltohedron.png	File:11-6 deltohedron.png	File:11-7 deltohedron.png	File:12-5 deltohedron.png	File:12-7 deltohedron.png
Template:CDD	Template:CDD	Template:CDD	Template:CDD	Template:CDD	Template:CDD	Template:CDD	Template:CDD	Template:CDD

See also

• 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.

References

• Chapter 4: Duals of the Archimedean polyhedra, prisma and antiprisms
• Template:Cite EB1911

External links

• Paper model tetragonal (square) trapezohedron