- 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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