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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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 Dph 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 Dpd 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 Dph symmetry group contains inversion if and only if p is even, while Dpd contains inversion symmetry if and only if p is odd.
There are:
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.
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 | ||||||
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d3h, [2,3], (*223) | 3.4.4 |
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d3d, [2+,3], (2*3) | 3.3.3.3 |
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d4h, [2,4], (*224) | 4.4.4 |
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d4d, [2+,4], (2*4) | 3.3.3.4 |
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d5h, [2,5], (*225) | 4.4.5 |
4.4.5/2 |
3.3.3.5/2 |
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d5d, [2+,5], (2*5) | 3.3.3.5 |
3.3.3.5/3 |
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d6h, [2,6], (*226) | 4.4.6 |
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d6d, [2+,6], (2*6) | 3.3.3.6 |
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d7h, [2,7], (*227) | 4.4.7 |
4.4.7/2 |
4.4.7/3 |
3.3.3.7/2 |
3.3.3.7/4 |
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d7d, [2+,7], (2*7) | 3.3.3.7 |
3.3.3.7/3 |
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d8h, [2,8], (*228) | 4.4.8 |
4.4.8/3 |
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d8d, [2+,8], (2*8) | 3.3.3.8 |
3.3.3.8/3 |
3.3.3.8/5 |
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d9h, [2,9], (*229) | 4.4.9 |
4.4.9/2 |
4.4.9/4 |
3.3.3.9/2 |
3.3.3.9/4 |
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d9d, [2+,9], (2*9) | 3.3.3.9 |
3.3.3.9/5 |
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d10h, [2,10], (*2.2.10) | 4.4.10 |
4.4.10/3 |
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d10d, [2+,10], (2*10) | 3.3.3.10 |
3.3.3.10/3 |
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d11h, [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 |
d11d, [2+,11], (2*11) | 3.3.3.11 |
3.3.3.11/3 |
3.3.3.11/5 |
3.3.3.11/7 |
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d12h, [2,12], (*2.2.12) | 4.4.12 |
4.4.12/5 |
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d12d, [2+,12], (2*12) | 3.3.3.12 |
3.3.3.12/5 |
3.3.3.12/7 |
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