From formulasearchengine
Revision as of 18:00, 17 June 2012 by en>Tamfang (→‎Volume and surface area: I didn't notice before: this long derivation is valid only (if at all) for *hexagonal* antiprism)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to navigation Jump to search
Set of uniform antiprisms
Hexagonal antiprism
Type uniform polyhedron
Faces 2 n-gons, 2n triangles
Edges 4n
Vertices 2n
Vertex configuration 3.3.3.n
Schläfli symbol h0,1{2,2n}
Coxeter–Dynkin diagrams Template:CDD
Symmetry group Dnd, [2+,2n], (2*n), order 4n
Rotation group Dn, [2,n]+, (22n), order 2n
Dual polyhedron trapezohedron
Properties convex, semi-regular vertex-transitive
Net Generalized antiprisim net.svg

In geometry, 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. Antiprisms are a subclass of the prismatoids.

Antiprisms are similar to prisms except the bases are twisted relative to each other, and that the side faces are triangles, rather than quadrilaterals.

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. As faces, it has the two n-gonal bases and, connecting those bases, 2n isosceles triangles.

Uniform antiprism

A uniform antiprism has, apart from the base faces, 2n equilateral triangles as faces. As a class, the uniform antiprisms 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.

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

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

Volume and surface area

Let a be the edge-length of a uniform antiprism. Then the volume is

and the surface area is


The symmetry group of a right n-sided antiprism with regular base and isosceles side faces is Dnd of order 4n, except in the case of a tetrahedron, which has the larger symmetry group Td of order 24, which has three versions of D2d as subgroups, and the octahedron, which has the larger symmetry group Oh of order 48, which has four versions of D3d as subgroups.

The symmetry group contains inversion if and only if n is odd.

The rotation group is Dn 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.

See also


External links

Template:Polyhedron navigator

ca:Antiprisma el:Αντιπρίσμα es:Antiprisma eo:Kontraŭprismo fr:Antiprisme ko:엇각기둥 it:Antiprisma nl:Antiprisma ja:反角柱 no:Antiprisme pl:Antygraniastosłup pt:Antiprisma ru:Антипризма sl:Antiprizma th:แอนติปริซึม zh:反棱柱