- Antiprism
An "n"-sided

**antiprism**is apolyhedron composed of two parallel copies of some particular "n"-sidedpolygon , connected by an alternating band oftriangle s. Antiprisms are a subclass of theprismatoid s.Antiprisms are similar to prisms except the bases are twisted relative to each other, and that the side faces are triangles, rather than quadrilaterials.

In the case of a regular "n"-sided base, one usually considers the case where its copy is twisted by an angle 180°/n. Extra regularity is obtained by the line connecting the base centers being perpendicular to the base planes, making it a

**right antiprism**. It has, apart from the base faces, 2"n" isosceles triangles as faces.A

**uniform antiprism**has, apart from the base faces, 2"n" equilateral triangles as faces. They form an infinite series of vertex-uniform polyhedra, as do the uniform prisms. For "n"=2 we have as degenerate case the regulartetrahedron , and for "n"=3 the non-degenerate regularoctahedron .The dual polyhedra of the antiprisms are the trapezohedra. Their existence was first discussed and their name was coined by

Johannes Kepler .**Cartesian coordinates**Cartesian coordinates for the vertices of a right antiprism with "n"-gonal bases and isosceles triangles are: $(\; cos(kpi/n),\; sin(kpi/n),\; (-1)^k\; a\; );$with "k" ranging from 0 to 2"n"-1; if the triangles are equilateral,:$2a^2=cos(pi/n)-cos(2pi/n);$.**Symmetry**The

symmetry group of a right "n"-sided antiprism with regular base and isosceles side faces is "D_{nd}" of order 4"n", except in the case of a tetrahedron, which has the larger symmetry group**T**of order 24, which has three versions of "D_{d}_{2d}" as subgroups, and the octahedron, which has the larger symmetry group**O**of order 48, which has four versions of "D_{h}_{3d}" as subgroups.The symmetry group contains inversion

if and only if "n" is odd.The

rotation group is "D_{n}" of order 2"n", except in the case of a tetrahedron, which has the larger rotation group**T**of order 12, which has three versions of "D_{2}" as subgroups, and the octahedron, which has the larger rotation group**O**of order 24, which has four versions of "D_{3}" as subgroups.**See also***

Prismatic uniform polyhedron

****triangular antiprism**(Octahedron )

**Square antiprism

**Pentagonal antiprism

**Hexagonal antiprism

**Octagonal antiprism

**Decagonal antiprism

**Dodecagonal antiprism

*Apeirogonal antiprism **External links***

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** GlossaryForHyperspace | anchor=Prismatic | title=Prismatic polytopes

* [*http://home.comcast.net/~tpgettys/nonconvexprisms.html Nonconvex Prisms and Antiprisms*]

* [*http://www.software3d.com/Prisms.php Paper models of prisms and antiprisms*]

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**antiprism**— antiprizmė statusas T sritis chemija apibrėžtis Iškilusis daugiasienis, gaunamas prizmės pagrindus vienas kito atžvilgiu pasukus tam tikru kampu. atitikmenys: angl. antiprism rus. антипризма … Chemijos terminų aiškinamasis žodynas**antiprism**— noun A polyhedron composed of two parallel copies of some particular polygon, connected by an alternating band of triangles … Wiktionary**Grand antiprism**— (Schlegel diagram wireframe) Type Uniform polychoron Uniform index 47 Cells 100+200 (3.3.3) … Wikipedia**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. It consists of two pentagons joined to each other by a ring of 10 triangles for… … Wikipedia**Octagonal antiprism**— Uniform Octagonal antiprism Type Prismatic uniform polyhedron Elements F = 18, E = 32 V = 16 (χ = 2) Faces by sides 16{3}+2{8} … Wikipedia**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.If all its faces are regular, it is a semiregular polyhedron.When eight points are… … Wikipedia**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 two polygon caps.If faces are all regular, it is a semiregular polyhedron. See also * Set of… … Wikipedia**Decagonal antiprism**— Uniform Decagonal antiprism Type Prismatic uniform polyhedron Elements F = 22, E = 40 V = 20 (χ = 2) Faces by sides 20{3}+2{10} … Wikipedia**Heptagonal antiprism**— Infobox Polyhedron with vertfig Polyhedron Type=Semiregular polyhedron Face List=16 triangles 2 heptagons Edge Count=21 Vertex Count=14 Wythoff Symbol= 2 2 7 Symmetry Group=D7d Vertex List=3.3.3.7 Dual=Heptagonal trapezohedron Property… … Wikipedia**Dodecagonal antiprism**— Uniform Dodecagonal antiprism Type Prismatic uniform polyhedron Elements F = 26, E = 48 V = 24 (χ = 2) Faces by sides 24{3}+2{12} … Wikipedia