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.

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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
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.<link href="mw-data:TemplateStyles:r1154941027" rel="mw-deduplicated-inline-style"/><span class="frac" role="math"><span class="num">9</span>⁄<span class="den">2</span></span>	3.3.3.<link href="mw-data:TemplateStyles:r1154941027" rel="mw-deduplicated-inline-style"/><span class="frac" role="math"><span class="num">9</span>⁄<span class="den">4</span></span>
D_9d [2⁺,9] (2*9)	3.3.3.9	3.3.3.<link href="mw-data:TemplateStyles:r1154941027" rel="mw-deduplicated-inline-style"/><span class="frac" role="math"><span class="num">9</span>⁄<span class="den">5</span></span>
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.<link href="mw-data:TemplateStyles:r1154941027" rel="mw-deduplicated-inline-style"/><span class="frac" role="math"><span class="num">11</span>⁄<span class="den">2</span></span>	4.4.<link href="mw-data:TemplateStyles:r1154941027" rel="mw-deduplicated-inline-style"/><span class="frac" role="math"><span class="num">11</span>⁄<span class="den">3</span></span>	4.4.<link href="mw-data:TemplateStyles:r1154941027" rel="mw-deduplicated-inline-style"/><span class="frac" role="math"><span class="num">11</span>⁄<span class="den">4</span></span>	4.4.<link href="mw-data:TemplateStyles:r1154941027" rel="mw-deduplicated-inline-style"/><span class="frac" role="math"><span class="num">11</span>⁄<span class="den">5</span></span>	3.3.3.<link href="mw-data:TemplateStyles:r1154941027" rel="mw-deduplicated-inline-style"/><span class="frac" role="math"><span class="num">11</span>⁄<span class="den">2</span></span>	3.3.3.<link href="mw-data:TemplateStyles:r1154941027" rel="mw-deduplicated-inline-style"/><span class="frac" role="math"><span class="num">11</span>⁄<span class="den">4</span></span>	3.3.3.<link href="mw-data:TemplateStyles:r1154941027" rel="mw-deduplicated-inline-style"/><span class="frac" role="math"><span class="num">11</span>⁄<span class="den">6</span></span>
D_11d [2⁺,11] (2*11)	3.3.3.11	3.3.3.<link href="mw-data:TemplateStyles:r1154941027" rel="mw-deduplicated-inline-style"/><span class="frac" role="math"><span class="num">11</span>⁄<span class="den">3</span></span>	3.3.3.<link href="mw-data:TemplateStyles:r1154941027" rel="mw-deduplicated-inline-style"/><span class="frac" role="math"><span class="num">11</span>⁄<span class="den">5</span></span>	3.3.3.<link href="mw-data:TemplateStyles:r1154941027" rel="mw-deduplicated-inline-style"/><span class="frac" role="math"><span class="num">11</span>⁄<span class="den">7</span></span>
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
...

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.