Antiprism

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Antiprism

An n-sided antiprism is a polyhedron

Polyhedron

|}A polyhedron is often defined as a geometry object with flat faces and straight edges .This definition of a polyhedron is not very precise, and to a modern mathematician is quite unsatisfactory....
composed of 2 parallel copies of some particular n-sided polygon

Polygon

In geometry a polygon is traditionally a plane Shape that is bounded by a closed curve path or circuit, composed of a finite sequence of straight line segments ....
, connected by an alternating band of triangle

Triangle

A triangle is one of the basic shapes of geometry: a polygon with three corners or wikt:vertex and three sides or edges which are line segments....
s. Antiprisms are a subclass of the prismatoid

Prismatoid

A prismatoid is a polyhedron where all vertices lie in two parallel planes. If the areas of the two parallel faces are A1 and A3, the cross-sectional area of the intersection of the prismatoid with a plane midway between the two parallel faces is A2, and the height is h, then the volume of the prismatoid i...
s.

Antiprisms are similar to prism

Prism (geometry)

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An n-sided antiprism is a polyhedron

Polyhedron

Polygon

Triangle

A triangle is one of the basic shapes of geometry: a polygon with three corners or wikt:vertex and three sides or edges which are line segments....
s. Antiprisms are a subclass of the prismatoid

Prismatoid

Prism (geometry)

In geometry, an n-sided prism is a polyhedron made of an n-sided polygon base, a Translation copy, and n faces joining corresponding sides....
s 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, 2n isosceles triangles as faces.

A uniform
Prismatic uniform polyhedron

A prismatic uniform polyhedron is a uniform polyhedron with Dihedral symmetry in three dimensions. They exist in two infinite families, the uniform Prism and the uniform antiprisms....
antiprism has, apart from the base faces, 2n 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 regular tetrahedron

Tetrahedron

A tetrahedron is a polyhedron composed of four triangle faces, three of which meet at each vertex . A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids....
, and for n=3 the non-degenerate 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....
.

The dual polyhedra

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 the antiprisms are the trapezohedra

Trapezohedron

The n-gonal trapezohedron, antidipyramid or deltohedron is the dual polyhedron of an n-gonal antiprism. Its 2n faces are congruent kite ....
. Their existence was first discussed and their name was coined by Johannes Kepler

Johannes Kepler

Johannes Kepler was a Germans mathematician, astronomer and astrologer, and key figure in the 17th century Scientific revolution. He is best known for his eponymous Kepler's laws of planetary motion, codified by later astronomers based on his works Astronomia nova, Harmonices Mundi, and Epitome of Copernican Astrononomy....
.

Cartesian coordinates

Cartesian coordinates for the vertices of a right antiprism with n-gonal bases and isosceles triangles are

with k ranging from 0 to 2n-1; if the triangles are equilateral, .

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 a right n-sided antiprism with regular base and isosceles side faces is D_nd of order 4n, except in the case of a tetrahedron, which has the larger symmetry group T_d of order 24, which has three versions of D_2d as subgroups, and the octahedron, which has the larger symmetry group O_h of order 48, which has four versions of D_3d as subgroups.

The symmetry group contains inversion

Inversion in a point

In Euclidean geometry, the inversion of a point X in respect to a point P is a point X* such that P is the midpoint of the line segment with endpoints X and X*....
if and only if

If and only if

If and only if, in logic and fields that rely on it such as mathematics and philosophy, is a biconditional logical connective between statements....
n is odd.

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 D_n of order 2n, except in the case of a tetrahedron, which has the larger rotation group T of order 12, which has 3 versions of D₂ as subgroups, and the octahedron, which has the larger rotation group O of order 24, which has 4 versions of D₃ as subgroups.

Antiprism

Cartesian coordinates

Symmetry

See also

External links