 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.
Contents
Vertex configuration and symmetry groups
Because they are isogonal (vertextransitive), 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 pfold 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 pfold 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.
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)
3.4.4d_{3d}, [2^{+},3], (2*3)
3.3.3.3d_{4h}, [2,4], (*224)
4.4.4d_{4d}, [2^{+},4], (2*4)
3.3.3.4d_{5h}, [2,5], (*225)
4.4.5
4.4.5/2
3.3.3.5/2d_{5d}, [2^{+},5], (2*5)
3.3.3.5
3.3.3.5/3d_{6h}, [2,6], (*226)
4.4.6d_{6d}, [2^{+},6], (2*6)
3.3.3.6d_{7h}, [2,7], (*227)
4.4.7
4.4.7/2
4.4.7/3
3.3.3.7/2
3.3.3.7/4d_{7d}, [2^{+},7], (2*7)
3.3.3.7
3.3.3.7/3d_{8h}, [2,8], (*228)
4.4.8
4.4.8/3d_{8d}, [2^{+},8], (2*8)
3.3.3.8
3.3.3.8/3
3.3.3.8/5d_{9h}, [2,9], (*229)
4.4.9
4.4.9/2
4.4.9/4
3.3.3.9/2
3.3.3.9/4d_{9d}, [2^{+},9], (2*9)
3.3.3.9
3.3.3.9/5d_{10h}, [2,10], (*2.2.10)
4.4.10
4.4.10/3d_{10d}, [2^{+},10], (2*10)
3.3.3.10
3.3.3.10/3d_{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
3.3.3.11/6d_{11d}, [2^{+},11], (2*11)
3.3.3.11
3.3.3.11/3
3.3.3.11/5
3.3.3.11/7d_{12h}, [2,12], (*2.2.12)
4.4.12
4.4.12/5d_{12d}, [2^{+},12], (2*12)
3.3.3.12
3.3.3.12/5
3.3.3.12/7... See also
External links
Categories: Prismatoid polyhedra
 Uniform polyhedra
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