Prismatic uniform polyhedron

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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.

[edit] 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.

[edit] 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.

[edit] 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.

Symmetry group	Convex	Star forms
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
...