of as subgroups. The symmetry group contains inversion if and only if is odd. The rotation group is of order , except in the cases of: : the regular tetrahedron, which has the larger rotation group of order , which has three versions of as subgroups; : the regular octahedron, which has the larger rotation group of order , which has four versions of as subgroups. Note: The right -antiprisms have congruent regular -gon bases and congruent isosceles triangle side faces, thus have the same (dihedral) symmetry group as the uniform -antiprism, for . Star antiprism Uniform star antiprisms are named by their star polygon bases, {p/q}, and exist in prograde and in retrograde (crossed) solutions. Crossed forms have intersecting vertex figures, and are denoted by "inverted" fractions: p/(p – q) instead of p/q; example: 5/3 instead of 5/2. A right star antiprism has two congruent coaxial regular convex or star polygon base faces, and 2n 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: Retrograde star antiprisms with regular convex polygon bases cannot have all equal edge lengths, so cannot be uniform. "Exception": a retrograde star antiprism with equilateral triangle bases (vertex configuration: 3.3/2.3.3) can be uniform; but then, it has the appearance of an equilateral triangle: it is a degenerate star polyhedron. Similarly, some retrograde star antiprisms with regular star polygon bases cannot have all equal edge lengths, so cannot be uniform. Example: a retrograde star antiprism with regular star 7/5-gon bases (vertex configuration: 3.3.3.7/5) cannot be uniform. 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. See also Apeirogonal antiprism Grand antiprism – a four-dimensional polytope One World Trade Center, a building consisting primarily of an elongated square antiprism Skew polygon References Bibliography Chapter 2: Archimedean polyhedra, prisms and antiprisms External links Nonconvex Prisms and Antiprisms Paper models of prisms and antiprisms Uniform polyhedra Prismatoid polyhedra Topological graph theory Graph drawing Coxeter groups Elementary geometry Polyhedra Polytopes Triangulation (geometry) Knot invariants