Trapezohedron

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;{{sfn|Spencer|1911|p=581, or p. 603 on Wikisource, CRYSTALLOGRAPHY, 6. HEXAGONAL SYSTEM, Rhombohedral Division, TRAPEZOHEDRAL CLASS, FIG. 74}}^,{{sfn|Spencer|1911|p=577, or p. 599 on Wikisource, CRYSTALLOGRAPHY, 2. TETRAGONAL SYSTEM, TRAPEZOHEDRAL CLASS}} 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.{{sfn|Spencer|1911|p=581, or p. 603 on Wikisource, CRYSTALLOGRAPHY, 6. HEXAGONAL SYSTEM, Rhombohedral Division, TRAPEZOHEDRAL CLASS, FIG. 74}} 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.{{sfn|Spencer|1911|p=582, or p. 604 on Wikisource, CRYSTALLOGRAPHY, 6. HEXAGONAL SYSTEM, Hexagonal Division, TRAPEZOHEDRAL CLASS}}

Crystal arrangements of atoms can repeat in space with trigonal and hexagonal trapezohedron cells.[http://www.metafysica.nl/turing/promorph_crystals_2.html Trigonal-trapezohedric Class, 3 2 and Hexagonal-trapezohedric Class, 6 2 2]

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,{{sfn|Spencer|1911|p=574, or p. 596 on Wikisource, CRYSTALLOGRAPHY, 1. CUBIC SYSTEM, HOLOSYMMETRIC CLASS, FIG. 17}} 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 dodecahedron{{sfn|Spencer|1911|p=575, or p. 597 on Wikisource, CRYSTALLOGRAPHY, 1. CUBIC SYSTEM, TETRAHEDRAL CLASS, FIG. 27}} 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,{{sfn|Spencer|1911|p=582, or p. 604 on Wikisource, CRYSTALLOGRAPHY, 6. HEXAGONAL SYSTEM, Hexagonal Division, TRAPEZOHEDRAL CLASS}} but two order-6 apices (i.e. poles) and two rings of six order-3 vertices each.

An {{mvar|n}}-trapezohedron is defined by a regular zig-zag skew {{math|2n}}-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 {{mvar|n}}-trapezohedron has two apical vertices on its polar axis, and {{math|2n}} basal vertices in two regular {{mvar|n}}-gonal rings. It has {{math|2n}} congruent kite faces, and it is isohedral.

{{Trapezohedra}}

• {{math|1=n = 2}}. 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.

• {{math|1=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.File:Gyroelongated triangular bipyramid.png 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 {{math|60°}} and {{math|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.

• {{math|1=n = 5}}. 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 {{math|00}} to {{math|99}}.

The symmetry group of an {{mvar|n}}-gonal trapezohedron is {{math|1=D{{sub|nd}} = D{{sub|nv}}}}, of order {{math|4n}}, except in the case of {{math|1=n = 3}}: a cube has the larger symmetry group {{math|O_d}} of order {{math|1=48 = 4×(4×3)}}, which has four versions of {{math|D_3d}} as subgroups.

The rotation group of an {{mvar|n}}-trapezohedron is {{math|D_n}}, of order {{math|2n}}, except in the case of {{math|1=n = 3}}: a cube has the larger rotation group {{math|O}} of order {{math|1=24 = 4×(2×3)}}, which has four versions of {{math|D₃}} as subgroups.

Note: Every {{mvar|n}}-trapezohedron with a regular zig-zag skew {{math|2n}}-gon base and {{math|2n}} congruent non-twisted kite faces has the same (dihedral) symmetry group as the dual-uniform {{mvar|n}}-trapezohedron, for {{math|n ≥ 4}}.

One degree of freedom within symmetry from {{math|D_nd}} (order {{math|4n}}) to {{math|D_n}} (order {{math|2n}}) changes the congruent kites into congruent quadrilaterals with three edge lengths, called twisted kites, and the {{mvar|n}}-trapezohedron is called a twisted trapezohedron. (In the limit, one edge of each quadrilateral goes to zero length, and the {{mvar|n}}-trapezohedron becomes an {{mvar|n}}-bipyramid.)

If the kites surrounding the two peaks are not twisted but are of two different shapes, the {{mvar|n}}-trapezohedron can only have {{math|C_nv}} (cyclic with vertical mirrors) symmetry, order {{math|2n}}, and is called an unequal or asymmetric trapezohedron. Its dual is an unequal {{mvar|n}}-antiprism, with the top and bottom {{mvar|n}}-gons of different radii.

If the kites are twisted and are of two different shapes, the {{mvar|n}}-trapezohedron can only have {{math|C_n}} (cyclic) symmetry, order {{mvar|n}}, and is called an unequal twisted trapezohedron.

A star {{math|p/q}}-trapezohedron (where {{math|2 ≤ q < 1p}}) is defined by a regular zig-zag skew Star polygon 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 {{math|p/q}}-trapezohedron has two apical vertices on its polar axis, and {{math|2p}} basal vertices in two regular {{mvar|p}}-gonal rings. It has {{math|2p}} congruent kite faces, and it is isohedral.

Such a star {{math|p/q}}-trapezohedron is a self-intersecting, crossed, or non-convex form. It exists for any regular zig-zag skew star {{math|2p/q}}-gon base (where {{math|2 ≤ q < 1p}}).

But if {{math|{{sfrac|p|q}} < {{sfrac|3|2}}}}, then {{math|(p − q){{sfrac|360°|p}} < {{sfrac|q|2}}{{sfrac|360°|p}}}}, so the dual star antiprism (of the star trapezohedron) cannot be uniform (i.e. cannot have equal edge lengths); and if {{math|1={{sfrac|p|q}} = {{sfrac|3|2}}}}, then {{math|1=(p − q){{sfrac|360°|p}} = {{sfrac|q|2}}{{sfrac|360°|p}}}}, so the dual star antiprism must be flat, thus degenerate, to be uniform.

A dual-uniform star {{math|p/q}}-trapezohedron has Coxeter-Dynkin diagram {{CDD|node_fh|2x|node_fh|p|rat|q|node_fh}}.

{{Commonscat|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.

{{reflist}}

• {{cite book | author= Anthony Pugh | year= 1976 | title= Polyhedra: A visual approach | publisher= University of California Press Berkeley | location= California | isbn= 0-520-03056-7 }} Chapter 4: Duals of the Archimedean polyhedra, prisma and antiprisms
• {{cite EB1911|wstitle= Crystallography |volume= 07 | pages = 569–591 |last1= Spencer |first1= Leonard James |author-link= Leonard James Spencer}}

• {{mathworld |urlname=Trapezohedron |title=Trapezohedron}}
• {{mathworld | urlname = Isohedron | title = Isohedron}}
• [http://www.korthalsaltes.com/model.php?name_en=square%20trapezohedron Paper model tetragonal (square) trapezohedron]

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