can form molecule-based magnets. Right antiprism For an antiprism with regular -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 -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 side faces are isosceles triangles. Uniform antiprism A uniform -antiprism has two congruent regular -gons as base faces, and equilateral triangles as side faces. Uniform antiprisms form an infinite class of vertex-transitive polyhedra, as do uniform prisms. For , we have the regular tetrahedron as a digonal antiprism (degenerate antiprism); for , the regular octahedron as a triangular antiprism (non-degenerate antiprism). Schlegel diagrams Cartesian coordinates Cartesian coordinates for the vertices of a right -antiprism (i.e. with regular -gon bases and isosceles triangle side faces) are: where ; if the -antiprism is uniform (i.e. if the triangles are equilateral), then: Volume and surface area Let be the edge-length of a uniform -gonal antiprism; then the volume is: and the surface area is: Related polyhedra 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 regular icosahedron. 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 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. Symmetry The symmetry group of a right -antiprism (i.e. with regular bases and isosceles side faces) is of order , except in the cases of: : the regular tetrahedron, which has the larger symmetry group of order , which has three versions of as subgroups; : the regular octahedron, which has the larger symmetry group of order , which has four versions