 Trapezohedron

Set of trapezohedra Faces 2n kites Edges 4n Vertices 2n+2 Face configuration V3.3.3.n Symmetry group D_{nd}, [2^{+},2n], (2*n) Dual polyhedron antiprism Properties convex, facetransitive The ngonal trapezohedron, antidipyramid or deltohedron is the dual polyhedron of an ngonal antiprism. Its 2n faces are congruent kites (also called trapezia in the US, trapezoids in Britain, or deltoids). The faces are symmetrically staggered.
The ngon part of the name does not reference the faces here but arrangement of vertices around an axis of symmetry. The dual ngonal antiprism has two actual ngon faces.
An ngonal trapezohedron can be decomposed into two equal ngonal pyramids and an ngonal antiprism.
Contents
Name
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.
Forms
 Trigonal trapezohedron  6 (rhombic) faces  dual octahedron
 A cube is a special case trigonal trapezohedron with square faces
 A trigonal trapezohedron is a special case rhombohedron with congruent rhombic faces
 Tetragonal trapezohedron  8 kite faces  dual square antiprism
 Pentagonal trapezohedron  10 kite faces  dual pentagonal antiprism
 Hexagonal trapezohedron  12 kite faces  dual hexagonal antiprism
 Heptagonal trapezohedron  14 kite faces  dual heptagonal antiprism
 Octagonal trapezohedron  16 kite faces  dual octagonal antiprism
 Enneagonal trapezohedron  18 kite faces  dual enneagonal antiprism
 Decagonal trapezohedron  20 kite faces  dual decagonal antiprism
 ...ngonal trapezohedron  2n kite faces  dual ngonal antiprism
In the case of the dual of a regular triangular antiprism the kites are rhombi, hence these trapezohedra are also zonohedra. They are called rhombohedron. They are cubes scaled in the direction of a body diagonal. Also they are the parallelepipeds with congruent rhombic faces.
A special case of a rhombohedron is one of the 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.
Examples
 Crystal arrangements of atoms can repeat in space with trapezohedral cells.
 The pentagonal trapezohedron is the only polyhedron other than the Platonic solids commonly used as a die in roleplaying games such as Dungeons and Dragons. Having 10 sides, it can be used in repetition to generate any decimalbased uniform probability desired. Two dice of different colors are typically used for the two digits to represent numbers from 00 to 99.
Symmetry
The symmetry group of an ngonal 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 D_{3} as subgroups.
In popular culture
 H.P. Lovecraft's short story "The Haunter of the Dark" features a "crazily angled stone" of extraterrestrial origin called the "Shining Trapezohedron".
 In the PlayStation 3 game Disgaea 3: Absence of Justice, the most powerful emblem equipment in the game is called 'Trapezohedron'
 In Final Fantasy XIII as the catalyst required to create any of the six ultimate weapons.
 In the PlayStation game Wild Arms 2, the 'Trapezohedron' is used to imprison the Kuiper Belt but is not actually Trapezohedron shaped.
See also
External links
 Weisstein, Eric W., "Trapezohedron" from MathWorld.
 Virtual Reality Polyhedra The Encyclopedia of Polyhedra
 Paper model tetragonal (square) trapezohedron
Polyhedron navigator Platonic solids (regular) Archimedean solids
(Semiregular/Uniform)Catalan solids
(Dual semiregular)triakis tetrahedron · rhombic dodecahedron · triakis octahedron · tetrakis cube · deltoidal icositetrahedron · disdyakis dodecahedron · pentagonal icositetrahedron · rhombic triacontahedron · triakis icosahedron · pentakis dodecahedron · deltoidal hexecontahedron · disdyakis triacontahedron · pentagonal hexecontahedronDihedral regular Dihedral uniform Duals of dihedral uniform bipyramids · trapezohedraDihedral others Degenerate polyhedra are in italics.Categories:  Trigonal trapezohedron  6 (rhombic) faces  dual octahedron
Wikimedia Foundation. 2010.
Look at other dictionaries:
Trapezohedron — Trap e*zo*he dron, n. [NL., from trapezium + Gr. ? seat, base, fr. ? to sit.] (Crystalloq.) (a) A solid bounded by twenty four equal and similar trapeziums; a tetragonal trisoctahedron. See the Note under {Trisoctahedron}. (b) A tetartohedral… … The Collaborative International Dictionary of English
trapezohedron — [trə pē΄zō hē′drən, trap΄i zōhē′drən] n. [ModL: see TRAPEZIUM & HEDRON] a solid figure, esp. a crystal, all of whose faces are trapeziums … English World dictionary
trapezohedron — Trisoctahedron Tris*oc ta*he dron, n. [Gr. ? thrice + FE. octahedron.] (Crystallog.) A solid of the isometric system bounded by twenty four equal faces, three corresponding to each face of an octahedron. [1913 Webster] {Tetragonal trisoctahedron} … The Collaborative International Dictionary of English
trapezohedron — noun (plural drons or trapezohedra) Etymology: New Latin, from trapezium + o + hedron Date: circa 1822 a crystalline form whose faces are trapeziums … New Collegiate Dictionary
trapezohedron — trapezohedral, adj. /treuh pee zeuh hee dreuhn, trap euh /, n., pl. trapezohedrons, trapezohedra / dreuh/. 1. Crystall. a crystal form having all faces trapeziums. 2. Geom. a trapezoid trisoctahedron. [1810 20; TRAPEZ(OID) + O + HEDRON] * * * … Universalium
trapezohedron — noun Any of several forms of crystal with trapezia as faces … Wiktionary
trapezohedron — trape·zo·he·dron … English syllables
trapezohedron — tra•pe•zo•he•dron [[t]trəˌpi zəˈhi drən, ˌtræp ə [/t]] n. pl. drons, dra [[t] drə[/t]] cry a crystal form having all faces trapeziums • Etymology: 1810–20 … From formal English to slang
trapezohedron — /trəpizoʊˈhidrən/ (say truhpeezoh heedruhn) noun (plural trapezohedrons or trapezohedra /trəpizoʊˈhidrə/ (say truhpeezoh heedruh)) a three dimensional shape with all its faces trapeziums, as in some crystals. {trapez(ium) + o + hedron} … Australian English dictionary
trapezohedron — noun a polyhedron whose faces are trapeziums • Hypernyms: ↑polyhedron … Useful english dictionary