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.

Vertex configuration and symmetry groups[edit]

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[edit]

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.

Images[edit]

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)
Linear antiprism.png
3.3.3
d3h
[2,3]
(*223)
Triangular prism.png
3.4.4
d3d
[2+,3]
(2*3)
Trigonal antiprism.png
3.3.3.3
d4h
[2,4]
(*224)
Tetragonal prism.png
4.4.4
d4d
[2+,4]
(2*4)
Square antiprism.png
3.3.3.4
d5h
[2,5]
(*225)
Pentagonal prism.png
4.4.5
Pentagrammic prism.png
4.4.5/2
Pentagrammic antiprism.png
3.3.3.5/2
d5d
[2+,5]
(2*5)
Pentagonal antiprism.png
3.3.3.5
Pentagrammic crossed antiprism.png
3.3.3.5/3
d6h
[2,6]
(*226)
Hexagonal prism.png
4.4.6
d6d
[2+,6]
(2*6)
Hexagonal antiprism.png
3.3.3.6
d7h
[2,7]
(*227)
Prism 7.png
4.4.7
Heptagrammic prism 7-2.png
4.4.7/2
Heptagrammic prism 7-3.png
4.4.7/3
Antiprism 7-2.png
3.3.3.7/2
Antiprism 7-4.png
3.3.3.7/4
d7d
[2+,7]
(2*7)
Antiprism 7.png
3.3.3.7
Antiprism 7-3.png
3.3.3.7/3
d8h
[2,8]
(*228)
Octagonal prism.png
4.4.8
Prism 8-3.png
4.4.8/3
d8d
[2+,8]
(2*8)
Octagonal antiprism.png
3.3.3.8
Antiprism 8-3.png
3.3.3.8/3
Antiprism 8-5.png
3.3.3.8/5
d9h
[2,9]
(*229)
Prism 9.png
4.4.9
Prism 9-2.png
4.4.9/2
Prism 9-4.png
4.4.9/4
Antiprism 9-2.png
3.3.3.9/2
Antiprism 9-4.png
3.3.3.9/4
d9d
[2+,9]
(2*9)
Enneagonal antiprism.png
3.3.3.9
Antiprism 9-5.png
3.3.3.9/5
d10h
[2,10]
(*2.2.10)
Decagonal prism.png
4.4.10
Prism 10-3.png
4.4.10/3
d10d
[2+,10]
(2*10)
Decagonal antiprism.png
3.3.3.10
Antiprism 10-3.png
3.3.3.10/3
d11h
[2,11]
(*2.2.11)
Hendecagonal prism.png
4.4.11
Prism 11-2.png
4.4.11/2
Prism 11-3.png
4.4.11/3
Prism 11-4.png
4.4.11/4
Prism 11-5.png
4.4.11/5
Antiprism 11-2.png
3.3.3.11/2
Antiprism 11-4.png
3.3.3.11/4
Antiprism 11-6.png
3.3.3.11/6
d11d
[2+,11]
(2*11)
Hendecagonal antiprism.png
3.3.3.11
Antiprism 11-3.png
3.3.3.11/3
Antiprism 11-5.png
3.3.3.11/5
Antiprism 11-7.png
3.3.3.11/7
d12h
[2,12]
(*2.2.12)
Dodecagonal prism.png
4.4.12
Prism 12-5.png
4.4.12/5
d12d
[2+,12]
(2*12)
Dodecagonal antiprism.png
3.3.3.12
Antiprism 12-5.png
3.3.3.12/5
Antiprism 12-7.png
3.3.3.12/7
...

See also[edit]

References[edit]

External links[edit]