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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.

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

The **D _{ph}** symmetry group contains inversion if and only if

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. If *p/q* = 3/2 the uniform antiprism is degenerate (has zero height).

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 | |||||

... |

- 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*.**246**(916). The Royal Society: 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.

- Prisms and Antiprisms George W. Hart