antiprism

An antiprism is a polyhedron formed by two parallel copies of a regular n-gon that are rotated with respect to each other and connected by an alternating band of triangles. The two bases serve as the top and bottom faces, while the 2n triangular side faces join corresponding vertices of the two bases. The resulting solid is convex for all n ≥ 3.

In standard notation, an n-antiprism has V = 2n vertices, E = 4n edges, and F = 2n + 2 faces (two n-gonal bases and 2n triangular side faces). The side faces are isosceles triangles in the typical construction, and the bases are regular n-gons. The triangular antiprism (n = 3) is the regular octahedron.

Special cases and examples include the square antiprism (n = 4), which has two square bases and eight triangular side faces, and the pentagonal antiprism (n = 5), with two pentagonal bases and ten triangular side faces. As n increases, the base faces become larger polygons while the side faces remain triangles, giving a family of solids with increasing belt size.

Symmetry and classification: An ordinary antiprism possesses dihedral symmetry of type Dn d (order 4n); the n = 3 case (the octahedron) has higher symmetry, Oh. Only the triangular antiprism is a regular polyhedron; for n > 3 the bases are polygons and the side faces are triangles, so the full polyhedron is not regular.

Relation to prisms: An antiprism differs from a prism in that its lateral faces are triangles produced by twisting one base relative to the other, rather than rectangular faces connecting corresponding vertices.