Set of uniform antiprisms
Type	uniform polyhedron
Faces	2 p-gons, 2p triangles
Edges	4p
Vertices	2p
Vertex configuration	3.3.3.p
Coxeter-Dynkin diagram
Symmetry group	Dpd
Dual polyhedron	trapezohedron
Properties	convex, semi-regular vertex-transitive

An n-sided antiprism is a polyhedron composed of two parallel copies of some particular n-sided polygon, connected by an alternating band of triangles. Download high resolution version (877x527, 53 KB)Heptadecagonal antiprism, made by me using POV-Ray, see image:poly. ... A uniform polyhedron is a polyhedron with regular polygons as faces and identical vertices. ... Look up polygon in Wiktionary, the free dictionary. ... A triangle. ... In polyhedral geometry a vertex configuration is a short-hand notation for representing a vertex as the sequence of faces around a vertex. ... Coxeter groups in the plane with equivalent diagrams. ... Image File history File links CDW_hole. ... Image File history File links No higher resolution available. ... Image File history File links CDW_hole. ... Image File history File links CDW_2. ... Image File history File links CDW_hole. ... The symmetry group of an object (e. ... The symmetry group of an object (e. ... In geometry, polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the other. ... The trapezohedron is the dual polyhedron of the corresponding antiprism. ... In mathematics, a vertex-transitive graph is a graph G such that, given any two vertices v1 and v2 of G, there is some automorphism f : G → G such that f ( v1 ) = v2. ... For the game magazine, see Polyhedron (magazine). ... Look up polygon in Wiktionary, the free dictionary. ... A triangle. ...

Antiprisms are a subclass of the prismatoids. A prismatoid is a polyhedron where all vertices lie in two parallel planes. ...

Antiprisms are similar to prisms except the bases are twisted relative to each other, and that the side faces are triangles, rather than quadrilaterials: the vertices are symmetrically staggered. In geometry, an n-sided prism is a polyhedron made of an n-sided polygonal base, a translated copy, and n faces joining corresponding sides. ...

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 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, and for n=3 the non-degenerate regular octahedron. A prismatic uniform polyhedron is a uniform polyhedron with dihedral symmetry. ... For the academic journal, see Tetrahedron (journal). ... An octahedron (plural: octahedra) is a polyhedron with eight faces. ...

The dual polyhedra of the antiprisms are the trapezohedra. Their existence was first discussed and their name was coined by Johannes Kepler. In geometry, polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the other. ... The trapezohedron is the dual polyhedron of the corresponding antiprism. ... Kepler redirects here. ...

Cartesian coordinates

Cartesian coordinates for the vertices of a right antiprism with n-gonal bases and isosceles triangles are Cartesian means relating to the French mathematician and philosopher Descartes, who, among other things, worked to merge algebra and Euclidean geometry. ...

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

.

Symmetry

The symmetry group 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 of an object (e. ...

The symmetry group contains inversion if and only if n is odd. 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*. In other words, the vector from X to P is the same as the vector from... ↔ ⇔ ≡ logical symbols representing iff. ...

The rotation group 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 three versions of D₂ as subgroups, and the octahedron, which has the larger rotation group O of order 24, which has four versions of D₃ as subgroups. In mechanics and geometry, the rotation group is the set of all rotations about the origin of 3-dimensional Euclidean space, R3. ...

See also

• Prismatic uniform polyhedron

A prismatic uniform polyhedron is a uniform polyhedron with dihedral symmetry. ...

External links

• Eric W. Weisstein, Antiprism at MathWorld.
• Olshevsky, George, Antiprism at Glossary for Hyperspace.
  • Olshevsky, George, Prismatic polytopes at Glossary for Hyperspace.
• Nonconvex Prisms and Antiprisms
• Paper models of prisms and antiprisms