Prismatic homogeneous polyhedron

Since they are isogonal (vertex-transitive), their uniquely corresponds .

The difference between prismatic and antiprismatic symmetry groups is that D _{p h} has edges connecting vertices on two planes perpendicular to these planes, which defines a plane of symmetry parallel to the polygons, while D _{p d} has intersecting edges, which gives a rotational symmetry. Each body has p reflection planes that contain p- fold axes of the polygons.

A symmetry group D _{p h} contains central symmetry if and only if p is even, while D _{p d} contains central symmetry if and only if p is odd.

Exist:

• Prisms for each rational p / q > 2 with a symmetry group D _{p h} ;
• Antiprisms for each rational p / q > 3/2 with the symmetry group D _{p d} if q is odd and D _{p h} if even.

If p / q is an integer, i.e. q = 1, the prism or antiprism is convex. (A fraction is always considered irreducible.)

The antiprism with p / q <2 is self-intersecting or degenerate ; its vertex figure is like a bow tie. With p / q ≤ 3/2, homogeneous antiprisms do not exist, since their vertex figure would violate the triangle inequality .

Note: The tetrahedron , cube, and octahedron are listed below as having dihedral symmetry (such as diagonal antiprism , square prism, and triangular antiprism, respectively), although if uniformly colored, the tetrahedron also has tetrahedral symmetry, while the cube and octahedron have octahedral symmetry.

• Prism (geometry)
• Antiprism

• Prisms and Antiprisms George W. Hart