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The n-gonal trapezohedron, antidipyramid or deltohedron is the dual polyhedron
Dual polyhedron

In geometry, polyhedron are associated into pairs called duals, where the wikt:vertex of one correspond to the face s of the other. The dual of the dual is the original polyhedron....
 of an n-gonal antiprism
Antiprism

An n-sided antiprism is a polyhedron composed of 2 parallel copies of some particular n-sided polygon, connected by an alternating band of triangles....
. Its 2n faces are congruent deltoids
Kite (geometry)

In geometry a kite, or deltoid, is a quadrilateral with two disjoint sets pairs of congruent adjacent sides, in contrast to a parallelogram, where the congruent sides are opposite....
 (or kites). The faces are symmetrically staggered.

The name trapezohedron can be misleading as the faces are not trapezoid
Trapezoid

In geometry, a trapezoid or trapezium is a quadrilateral with twoparallel sides. The term “trapezoid” is used in North America, while the term “trapezium” is prevalent in Britain....
s, but the alternative deltohedron is sometimes confused with the unrelated term deltahedron
Deltahedron

A deltahedron is a polyhedron whose face s are all equilateral triangles. The name is taken from the Greek language majuscule delta , which has the shape of an equilateral triangle....
. It is quite possible, but not necessarily true, that the name trapezohedron could be derived from trapezium (a quadrilateral with no parallel sides) as opposed to trapezoid, since the faces are kites, a type of trapezium.

The n-gon part of the name does not reference the faces here but arrangement of vertices around an axis of symmetry.






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The n-gonal trapezohedron, antidipyramid or deltohedron is the dual polyhedron
Dual polyhedron

In geometry, polyhedron are associated into pairs called duals, where the wikt:vertex of one correspond to the face s of the other. The dual of the dual is the original polyhedron....
 of an n-gonal antiprism
Antiprism

An n-sided antiprism is a polyhedron composed of 2 parallel copies of some particular n-sided polygon, connected by an alternating band of triangles....
. Its 2n faces are congruent deltoids
Kite (geometry)

In geometry a kite, or deltoid, is a quadrilateral with two disjoint sets pairs of congruent adjacent sides, in contrast to a parallelogram, where the congruent sides are opposite....
 (or kites). The faces are symmetrically staggered.

The name trapezohedron can be misleading as the faces are not trapezoid
Trapezoid

In geometry, a trapezoid or trapezium is a quadrilateral with twoparallel sides. The term “trapezoid” is used in North America, while the term “trapezium” is prevalent in Britain....
s, but the alternative deltohedron is sometimes confused with the unrelated term deltahedron
Deltahedron

A deltahedron is a polyhedron whose face s are all equilateral triangles. The name is taken from the Greek language majuscule delta , which has the shape of an equilateral triangle....
. It is quite possible, but not necessarily true, that the name trapezohedron could be derived from trapezium (a quadrilateral with no parallel sides) as opposed to trapezoid, since the faces are kites, a type of trapezium.

The n-gon part of the name does not reference the faces here but arrangement of vertices around an axis of symmetry. The dual n-gonal antiprism has two actual n-gon faces.

An n-gonal trapezohedron can be decomposed into two equal n-gonal pyramids and an n-gonal antiprism
Antiprism

An n-sided antiprism is a polyhedron composed of 2 parallel copies of some particular n-sided polygon, connected by an alternating band of triangles....
.

In texts describing the crystal habit
Crystal habit

In mineralogy, shape and size give rise to descriptive terms applied to the typical appearance, or habit of crystals.The many terms used by mineralogists to describe crystal habits are useful in communicating what specimens of a particular mineral often look like....
s of minerals, the word trapezohedron is often used to refer to the polyhedron properly known as a deltoidal icositetrahedron
Deltoidal icositetrahedron

A deltoidal icositetrahedron is a Catalan solid which looks a bit like an overinflated cube . Its dual polyhedron is the rhombicuboctahedron....
.

Forms

Trigonal Trapezohedron
Tetragonal Trapezohedron
Pentagonal Trapezohedron
Hexagonal Trapezohedron
Octagonal Trapezohedron
Decagonal Trapezohedron
# Trigonal trapezohedron
Trigonal trapezohedron

The trigonal trapezohedron or deltohedron is the first in an infinite series of face-uniform polyhedra which are dual polyhedron to the antiprisms....
 - 6 (rhombic) faces - dual octahedron
Octahedron

An octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each wikt:vertex....
    • A cube
      Cube

      A cube is a three-dimensional space solid object bounded by six square faces, facets or sides, with three meeting at each wikt:vertex. The cube can also be called a Regular polyhedron hexahedron and is one of the five Platonic solids....
       is a special case trigonal trapezohedron with square faces
    • A trigonal trapezohedron is a special case rhombohedron
      Rhombohedron

      In geometry, a rhombohedron is a three-dimensional figure like a cube, except that its faces are not squares but rhombus. It is a special case of a parallelepiped where all edges are the same length....
       with congruent rhombic faces
    1. Tetragonal trapezohedron
      Tetragonal trapezohedron

      The tetragonal trapezohedron or deltohedron is the second in an infinite series of face-uniform polyhedra which are dual polyhedron to the antiprisms. It has eight faces which are congruent kite ....
       - 8 kite faces - dual square antiprism
      Square antiprism

      In geometry, the square antiprism is the second in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps....
    2. Pentagonal trapezohedron
      Pentagonal trapezohedron

      The pentagonal trapezohedron or deltohedron is the third in an infinite series of face-transitive polyhedra which are dual polyhedra to the antiprisms....
       - 10 kite faces - dual pentagonal antiprism
      Pentagonal antiprism

      In geometry, the pentagonal antiprism is the third in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps....
    3. Hexagonal trapezohedron
      Hexagonal trapezohedron

      The hexagonal trapezohedron or deltohedron is the fourth in an infinite series of face-uniform polyhedra which are dual polyhedron to the antiprisms....
       - 12 kite faces - dual hexagonal antiprism
      Hexagonal antiprism

      In geometry, the hexagonal antiprism is the 4th in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by 2 polygon caps....
    4. Heptagonal trapezohedron - 14 kite faces - dual heptagonal antiprism
    5. Octagonal trapezohedron
      Octagonal trapezohedron

      The octagonal trapezohedron or deltohedron is the sixth in an infinite series of face-uniform polyhedra which are dual polyhedron to the antiprisms. It has sixteen faces which are congruent kite ....
       - 16 kite faces - dual octagonal antiprism
      Octagonal antiprism

      In geometry, the octagonal antiprism is the 6th in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps....
    6. Enneagonal trapezohedron - 18 kite faces - dual enneagonal antiprism
    7. Decagonal trapezohedron
      Decagonal trapezohedron

      The decagonal trapezohedron or deltohedron is the eighth in an infinite series of face-uniform polyhedra which are dual polyhedron to the antiprisms. It has twenty faces which are congruence kite ....
       - 20 kite faces - dual decagonal antiprism
      Decagonal antiprism

      In geometry, the decagonal antiprism is the eighth in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps....
    • ...n-gonal trapezohedron - 2n kite faces - dual n-gonal antiprism
      Antiprism

      An n-sided antiprism is a polyhedron composed of 2 parallel copies of some particular n-sided polygon, connected by an alternating band of triangles....


    In the case of the dual of a regular triangular antiprism the kites are rhombi, hence these trapezohedra are also zonohedra
    Zonohedron

    A zonohedron is a convex set polyhedron where every face is a polygon with point symmetry or, equivalently, symmetry under rotations through 180?....
    . They are called rhombohedron. They are cube
    Cube

    A cube is a three-dimensional space solid object bounded by six square faces, facets or sides, with three meeting at each wikt:vertex. The cube can also be called a Regular polyhedron hexahedron and is one of the five Platonic solids....
    s scaled in the direction of a body diagonal. Also they are the parallelepiped
    Parallelepiped

    In geometry, a parallelepiped is a three-dimensional figure formed by six parallelograms. It is to a parallelogram as a cube is to a square : Euclidean geometry supports all four notions but affine geometry admits only parallelograms and parallelepipeds....
    s 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
    Octahedron

    An octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each wikt:vertex....
    . Since parallelepipeds can fill space
    Tessellation

    A tessellation or tiling of the plane is a collection of plane figures that fills the plane with no overlaps and no gaps. One may also speak of tessellations of the parts of the plane or of other surfaces....
    , so can a combination of regular tetrahedra and regular octahedra
    Tetrahedral-octahedral honeycomb

    The tetrahedral-octahedral honeycomb or alternated cubic honeycomb is a space-filling tessellation in Euclidean 3-space. It is comprised of alternating octahedron and tetrahedron in a ratio of 1:2....
    .

    Examples

    • Crystal arrangements
      Crystal system

      A crystal system is a category of space groups, which characterize symmetry of structures in three dimensions with translational symmetry in three directions, having a discrete class of Point groups in three dimensions....
       of atoms can repeat in space with trapezohedral cells.
    • The pentagonal trapezohedron
      Pentagonal trapezohedron

      The pentagonal trapezohedron or deltohedron is the third in an infinite series of face-transitive polyhedra which are dual polyhedra to the antiprisms....
       is the only polyhedron other than the Platonic solid
      Platonic solid

      In geometry, a Platonic solid is a convex set polyhedron that is regular polyhedron, in the sense of a regular polygon. Specifically, the faces of a Platonic solid are congruence regular polygons, with the same number of faces meeting at each vertex....
      s commonly used as a die
      Dice

      A die is a small polyhedron object, usually cubic, used for generating Statistical randomnesss or other symbols. This makes dice suitable as gambling devices, especially for craps or sic bo, or for use in non-gambling tabletop games....
       in roleplaying games such as Dungeons and Dragons. Having 10 sides, it can be used in repetition to generate any decimal-based uniform probability
      Uniform distribution (discrete)

      In probability theory and statistics, the discrete uniform distribution is a discrete probability distribution that can be characterized by saying that all values of a finite set of possible values are equally probable....
       desired. Two dice of different colors are typically used for the two digit
      Digit

      Digit may refer to:* Digit , one of several most proximal parts of a limb* Phone number, slang as digit, as in "Let me get your digits so I can call you tonight."...
      s to represent numbers from 00 to 99.


    Symmetry

    The symmetry group
    Symmetry group

    The symmetry group of an object is the group of all isometries under which it is invariant with Function composition as the operation. It is a subgroup of the isometry group of the space concerned....
     of an n-gonal trapezohedron is Dnd of order 4n, except in the case of a cube, which has the larger symmetry group Od of order 48, which has four versions of D3d as subgroups.

    The rotation group
    Rotation group

    In classical mechanics and geometry, the rotation group is the group of all rotations about the origin of three-dimensional Euclidean space R3 under the operation of functional composition....
     is Dn 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.

    In Popular Culture

    • H.P. Lovecraft's short story "The Haunter of the Dark
      The Haunter of the Dark

      "The Haunter of the Dark" is a Horror fiction short story in the Cthulhu Mythos genre. Written by H. P. Lovecraft in November 1935 in literature, and published in the December 1936 in literature edition of Weird Tales ....
      " features a "crazily angled stone" of extraterrestrial origin called the "Shining Trapezohedron".
    • In The Sims and The Sims 2, there is a Trapezohedron over the people's heads when you select them.
    • In the PlayStation 3 game Disgaea 3: Absence of Justice
      Disgaea 3: Absence of Justice

      is the third video game in the Disgaea series by Nippon Ichi. It was released on January 31, 2008 in Japan for the PlayStation 3 video game console....
      , the most powerful emblem equipment in the game is called 'Trapezohedron'


    See also

    • Rhombic dodecahedron
      Rhombic dodecahedron

      The rhombic dodecahedron is a convex set polyhedron with 12 rhombus faces. It is an Archimedean solid solid, or a Catalan solid. Its dual is the cuboctahedron....
    • Rhombic triacontahedron
      Rhombic triacontahedron

      In geometry, the rhombic triacontahedron is a convex set polyhedron with 30 rhombus faces. It is an Archimedean solid solid, or a Catalan solid....
    • Conway Polyhedron Notation
      Conway polyhedron notation

      Conway polyhedron notation is used to describe polyhedron based on a seed polyhedron modified by various operators.The seed polyhedra are the Platonic solids, represented by their first letter of their name ; the prism s , antiprisms and pyramid s ....


    External links

    • The Encyclopedia of Polyhedra
      • VRML
        VRML

        VRML is a standard file format for representing 3-D computer graphics interactive vector graphics, designed particularly with the World Wide Web in mind....
         models
      • Try: "dAn", where n=3,4,5... example "dA5" is a pentagonal trapezohedron.