PRISMATIC UNIFORM POLYHEDRON

A 'prismatic uniform polyhedron' is a uniform polyhedron with dihedral symmetry. They exist in two infinite families, the uniform prisms and the uniform antiprisms. All have their vertices in parallel planes and are therefore prismatoids.

Contents

Vertex configuration and symmetry groups

Enumeration

Images

See also

External links

Vertex configuration and symmetry groups

Because they are isogonal (vertex-transitive), their vertex arrangement uniquely corresponds to a symmetry group.
The difference between the prismatic and antiprismatic symmetry groups is that 'D_''p''h' has the vertices lined up in both planes, which gives it a reflection plane perpendicular to its ''p''-fold axis (parallel to the {p/q} polygon); while 'D_''p''d' has the vertices twisted relative to the other plane, which gives it a rotatory reflection. Each has ''p'' reflection planes which contain the ''p''-fold axis.

Enumeration

There are:

★ prisms, for each rational number ''p/q'' > 2, with symmetry group 'D_''p''h';

★ antiprisms, for each rational number ''p/q'' > 3/2, with symmetry group 'D_''p''d' if ''q'' is odd, 'D_''p''h' if ''q'' is even.
If ''p/q'' is an integer, i.e. if ''q'' = 1, the prism or antiprism is convex. (The fraction is always assumed to be stated in lowest terms.)
An antiprism with ''p/q'' < 2 is ''crossed'' or ''retrograde''; its vertex figure resembles a bowtie. If ''p/q'' ≤ 3/2 no uniform antiprism can exist, as its vertex figure would have to violate the triangle inequality.

Images

Note: The cube and octahedron are listed here with dihedral symmetry (as a ''square prism'' and ''triangular antiprism'' respectively), although if uniformly colored, they also have octahedral symmetry.

Symmetry group	Convex	Star forms
d3h	3.3.4
d3d	3.3.3.3
d4h	4.4.4
d4d	3.3.3.4
d5h	4.4.5	4.4.5/2	3.3.3.5/2
d5d	3.3.3.5	3.3.3.5/3
d6h	4.4.6
d6d	3.3.3.6
d7h	4.4.7	4.4.7/2	4.4.7/3	3.3.3.7/2	3.3.3.7/4
d7d	3.3.3.7	3.3.3.7/3
d8h	4.4.8	4.4.4.8/3
d8d	3.3.3.8	3.3.3.8/3	3.3.3.8/5
d9h	4.4.9	4.4.9/2	3.3.3.9/2	3.3.3.9/4
d9d	3.3.3.9	3.3.3.9/5
d10h	4.4.10	4.4.10/3
d10d	3.3.3.10	3.3.3.10/3
d11h	4.4.11	4.4.11/2	4.4.11/3	4.4.11/4	4.4.11/5	3.3.3.11/2	3.3.3.11/4	3.3.3.11/6
d11d	3.3.3.11	3.3.3.11/3	3.3.3.11/5	3.3.3.11/7
d12h	4.4.12	4.4.12/5
d12d	3.3.3.12	3.3.3.12/5	3.3.3.12/7
...

See also

★ Uniform polyhedron

★ Prism (geometry)

★ Antiprism

External links

Prisms and Antiprisms