Prismatic uniform polyhedron
# Prismatic uniform polyhedron

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 $D_{ph}$ 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 $D_{pd}$ 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 $D_{ph}$ symmetry group contains inversion if and only if p is even, while $D_{pd}$ contains inversion symmetry if and only if p is odd.

## Enumeration

There are:

* prisms, for each rational number p/q > 2, with symmetry group $D_{ph};$
* antiprisms, for each rational number p/q > 3/2, with symmetry group $D_{pd}$ if q is odd, $D_{ph}$ 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. If p/q = 3/2 the uniform antiprism is degenerate (has zero height).

## Forms by symmetry

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

| Symmetry group | Convex | Star forms | Star forms | Star forms | Star forms | Star forms | Star forms | Star forms |
| - | - | - | - | - | - | - | - | - |
| $D_{2d}$ $[2^+,2]$ (2*2) | 3.3.3 |
| $D_{3h}$ [2,3] (*223) | 3.4.4 |
| $D_{3d}$ $[2^+,3]$ (2*3) | 3.3.3.3 |
| $D_{4h}$ [2,4] (*224) | 4.4.4 |
| $D_{4d}$ $[2^+,4]$ (2*4) | 3.3.3.4 |
| $D_{5h}$ [2,5] (*225) | 4.4.5 | 4.4.5⁄2 | 3.3.3.5⁄2 |
| $D_{5d}$ $[2^+,5]$ (2*5) | 3.3.3.5 | 3.3.3.5⁄3 |
| $D_{6h}$ [2,6] (*226) | 4.4.6 |
| $D_{6d}$ $[2^+,6]$ (2*6) | 3.3.3.6 |
| $D_{7h}$ [2,7] (*227) | 4.4.7 | 4.4.7⁄2 | 4.4.7⁄3 | 3.3.3.7⁄2 | 3.3.3.7⁄4 |
| $D_{7d}$ $[2^+,7]$ (2*7) | 3.3.3.7 | 3.3.3.7⁄3 |
| $D_{8h}$ [2,8] (*228) | 4.4.8 | 4.4.8⁄3 |
| $D_{8d}$ $[2^+,8]$ (2*8) | 3.3.3.8 | 3.3.3.8⁄3 | 3.3.3.8⁄5 |
| $D_{9h}$ [2,9] (*229) | 4.4.9 | 4.4.9⁄2 | 4.4.9⁄4 | 3.3.3.9⁄2 | 3.3.3.9⁄4 |
| $D_{9d}$ $[2^+,9]$ (2*9) | 3.3.3.9 | 3.3.3.9⁄5 |
| $D_{10h}$ [2,10] (*2.2.10) | 4.4.10 | 4.4.10⁄3 |
| $D_{10d}$ $[2^+,10]$ (2*10) | 3.3.3.10 | 3.3.3.10⁄3 |
| $D_{11h}$ [2,11] (*2.2.11) | 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 |
| $D_{11d}$ $[2^+,11]$ (2*11) | 3.3.3.11 | 3.3.3.11⁄3 | 3.3.3.11⁄5 | 3.3.3.11⁄7 |
| $D_{12h}$ [2,12] (*2.2.12) | 4.4.12 | 4.4.12⁄5 |
| $D_{12d}$ $[2^+,12]$ (2*12) | 3.3.3.12 | 3.3.3.12⁄5 | 3.3.3.12⁄7 |
