# Prismatic uniform polyhedron

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Prismatic uniform polyhedron  A pentagrammic antiprism is made of two regular pentagrams and 10 equilateral triangles.

In geometry, 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.

## 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 Dph 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 Dpd 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.

The Dph symmetry group contains inversion if and only if p is even, while Dpd contains inversion symmetry if and only if p is odd.

## Enumeration

There are:

• prisms, for each rational number p/q > 2, with symmetry group Dph;
• antiprisms, for each rational number p/q > 3/2, with symmetry group Dpd if q is odd, Dph 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.

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