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**has the vertices twisted relative to the other plane, which gives it a rotatory reflection. Each has

_{pd}*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**contains inversion symmetry if and only if

_{pd}*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**if_{pd}*q*is odd,**D**if_{ph}*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 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 | ||||||
---|---|---|---|---|---|---|---|---|

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 | |||||

... |

## See also

## References

- Coxeter, Harold Scott MacDonald; Longuet-Higgins, M. S.; Miller, J. C. P. (1954). "Uniform polyhedra".
*Philosophical Transactions of the Royal Society of London. Series A. Mathematical and Physical Sciences*. The Royal Society.**246**(916): 401–450. doi:10.1098/rsta.1954.0003. ISSN 0080-4614. JSTOR 91532. MR 0062446. S2CID 202575183. - Cromwell, P.;
*Polyhedra*, CUP, Hbk. 1997, ISBN 0-521-66432-2. Pbk. (1999), ISBN 0-521-66405-5. p.175 - Skilling, John (1976), "Uniform Compounds of Uniform Polyhedra",
*Mathematical Proceedings of the Cambridge Philosophical Society*,**79**(3): 447–457, doi:10.1017/S0305004100052440, MR 0397554.