Қос параллель көпбұрыш және үшбұрыштардан құралған полиэдр

Polyhedron with parallel bases connected by triangles

In geometry, an antiprism or is a polyhedron composed of two parallel direct copies (not mirror images) of an polygon, connected by an alternating band of 2n triangles. They are represented by the Conway notation An. 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 2n triangles, rather than n quadrilaterals. The dual polyhedron of an n gonal antiprism is an n gonal trapezohedron.

In his 1619 book Harmonices Mundi, Johannes Kepler observed the existence of the infinite family of antiprisms. This has conventionally been thought of as the first discovery of these shapes, but they may have been known earlier: an unsigned printing block for the net of a hexagonal antiprism has been attributed to Hieronymus Andreae, who died in 1556. The German form of the word "antiprism" was used for these shapes in the 19th century; Karl Heinze credits its introduction to de. Although the English "anti prism" had been used earlier for an optical prism used to cancel the effects of a primary optimal element, the first use of "antiprism" in English in its geometric sense appears to be in the early 20th century in the works of H. S. M. Coxeter.

For an antiprism with regular n gon bases, one usually considers the case where these two copies are twisted by an angle of 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 degrees, more regularity is obtained if the bases have the same axis: are coaxial; 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 right antiprism, and its 2n side faces are isosceles triangles.

Four dimensional antiprisms can be defined as having two dual polyhedra as parallel opposite faces, so that each three dimensional face between them comes from two dual parts of the polyhedra: a vertex and a dual polygon, or two dual edges. Every three dimensional convex polyhedron is combinatorially equivalent to one of the two opposite faces of a four dimensional antiprism, constructed from its canonical polyhedron and its polar dual. However, there exist four dimensional polyhedra that cannot be combined with their duals to form five dimensional antiprisms.