# Antiprism

In geometry, an ** n-gonal antiprism** or

**is a polyhedron composed of two**

*n*-antiprism*(not mirror images) of an*

**parallel direct copies***n*-sided

*, connected by an alternating band of 2*

**polygon***n*

*.*

**triangles**Antiprisms are a subclass of prismatoids, and are a (degenerate) type of snub polyhedron.

Antiprisms are similar to prisms, except that the bases are twisted relatively to each other, and that the side faces (connecting the bases) are 2*n* triangles, rather than *n* quadrilaterals.

For an antiprism with regular *n*-gon bases, one usually considers the case where these two copies are twisted by an angle of 180/*n* degrees.

The **axis** of a regular polygon is the line perpendicular to the polygon plane and lying in the polygon centre.

For an antiprism with congruent * regular* n-gon bases, twisted by an angle of 180/

*n*degrees, more regularity is obtained if the bases have the same axis: are

*; i.e. (for non-coplanar bases): if the line connecting the base centers is perpendicular to the base planes. Then, the antiprism is called a*

**coaxial****right antiprism**, and its 2

*n*side faces are

*triangles.*

**isosceles**A **uniform antiprism** has two congruent * regular* n-gon base faces, and 2

*n*

*triangles as side faces.*

**equilateral**Uniform antiprisms form an infinite class of vertex-transitive polyhedra, as do uniform prisms. For *n* = 2, we have the regular tetrahedron as a *digonal antiprism* (degenerate antiprism); for *n* = 3, the regular octahedron as a *triangular antiprism* (non-degenerate antiprism).

The existence of antiprisms was discussed and their name was coined by Johannes Kepler, though it is possible that they were previously known to Archimedes, as they satisfy the same conditions on faces and on vertices as the Archimedean solids.

Cartesian coordinates for the vertices of a right antiprism (i.e. with regular *n*-gon bases and isosceles side faces) are

There are an infinite set of truncated antiprisms, including a lower-symmetry form of the truncated octahedron (truncated triangular antiprism). These can be alternated to create snub antiprisms, two of which are Johnson solids, and the *snub triangular antiprism* is a lower symmetry form of the icosahedron.

The symmetry group of a right *n*-antiprism (i.e. with regular bases and isosceles side faces) is D_{nd} of order 4*n*, except in the cases of:

A **right star antiprism** has two congruent coaxial regular * convex* or

*polygon base faces, and 2*

**star***n*isosceles triangle side faces.

Any star antiprism with *regular* convex or star polygon bases can be made a *right* star antiprism (by translating and/or twisting one of its bases, if necessary).

In the retrograde forms but not in the prograde forms, the triangles joining the convex or star bases intersect the axis of rotational symmetry. Thus:

Also, star antiprism compounds with regular star *p*/*q*-gon bases can be constructed if *p* and *q* have common factors. Example: a star 10/4-antiprism is the compound of two star 5/2-antiprisms.