Prismatic uniform polyhedron

Short description: Uniform polyhedron with dihedral symmetry

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.

## Contents

* 1 Vertex configuration and symmetry groups
* 2 Enumeration
* 3 Forms by symmetry
* 4 See also
* 5 References
* 6 External links

## 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 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.

## Enumeration

There are:

* prisms, for each rational number p/q > 2, with symmetry group Dph;
* antiprisms, for each rational number p/q > 3/2, with symmetry group Dpd if q is odd, Dph 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

|
|

D2d
[2+,2]
(2*2)  |
3.3.3
D3h
[2,3]
(*223)  |
3.4.4
D3d
[2+,3]
(2*3)  |
3.3.3.3
D4h
[2,4]
(*224)  |
4.4.4
D4d
[2+,4]
(2*4)  |
3.3.3.4
D5h
[2,5]
(*225)  |
4.4.5 |
4.4.​5⁄2 |
3.3.3.​5⁄2
D5d
[2+,5]
(2*5)  |
3.3.3.5 |
3.3.3.​5⁄3
D6h
[2,6]
(*226)  |
4.4.6
D6d
[2+,6]
(2*6)  |
3.3.3.6
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
D7d
[2+,7]
(2*7)  |
3.3.3.7 |
3.3.3.​7⁄3
D8h
[2,8]
(*228)  |
4.4.8 |
4.4.​8⁄3
D8d
[2+,8]
(2*8)  |
3.3.3.8 |
3.3.3.​8⁄3 |
3.3.3.​8⁄5
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
D9d
[2+,9]
(2*9)  |
3.3.3.9 |
3.3.3.​9⁄5
D10h
[2,10]
(*2.2.10)  |
4.4.10 |
4.4.​10⁄3
D10d
[2+,10]
(2*10)  |
3.3.3.10 |
3.3.3.​10⁄3
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
D12h
[2,12]
(*2.2.12)  |
4.4.12 |
4.4.​12⁄5
D12d
[2+,12]
(2*12)  |
3.3.3.12  |
3.3.3.​12⁄5 |
3.3.3.​12⁄7
...

## See also

* Uniform polyhedron
* Prism (geometry)
* Antiprism

## 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.
* 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 .

## External links

* Prisms and Antiprisms George W. Hart

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* t
* e

Convex polyhedra

Platonic solids (regular)|

* tetrahedron
* cube
* octahedron
* dodecahedron
* icosahedron

Archimedean solids
(semiregular or uniform)

* truncated tetrahedron
* cuboctahedron
* truncated cube
* truncated octahedron
* rhombicuboctahedron
* truncated cuboctahedron
* snub cube
* icosidodecahedron
* truncated dodecahedron
* truncated icosahedron
* rhombicosidodecahedron
* truncated icosidodecahedron
* snub dodecahedron

Catalan solids
(duals of Archimedean)

* triakis tetrahedron
* rhombic dodecahedron
* triakis octahedron
* tetrakis hexahedron
* deltoidal icositetrahedron
* disdyakis dodecahedron
* pentagonal icositetrahedron
* rhombic triacontahedron
* triakis icosahedron
* pentakis dodecahedron
* deltoidal hexecontahedron
* disdyakis triacontahedron
* pentagonal hexecontahedron

Dihedral regular|

* dihedron
* hosohedron

Dihedral uniform|

* prisms
* antiprisms

duals:|

* bipyramids
* trapezohedra

Dihedral others|

* pyramids
* truncated trapezohedra
* gyroelongated bipyramid
* cupola
* bicupola
* pyramidal frusta
* bifrustum
* rotunda
* birotunda
* prismatoid
* scutoid

Degenerate polyhedra are in italics.

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