Antiprism

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Set of uniform antiprisms
Hexagonal antiprism
Type uniform polyhedron
Faces 2 n-gons, 2n triangles
Edges 4n
Vertices 2n
Vertex configuration 3.3.3.n
Schläfli symbol s{2,2n}
sr{2,n}
{ } ⨂ {n}
Coxeter–Dynkin diagrams CDel node h.pngCDel 2x.pngCDel node h.pngCDel 2x.pngCDel n.pngCDel node.png
CDel node h.pngCDel 2x.pngCDel node h.pngCDel n.pngCDel node h.png
Symmetry group Dnd, [2+,2n], (2*n), order 4n
Rotation group Dn, [2,n]+, (22n), order 2n
Dual polyhedron trapezohedron
Properties convex, semi-regular vertex-transitive
Net Generalized antiprisim net.svg

In geometry, an n-sided antiprism is a polyhedron composed of two parallel copies of some particular n-sided polygon, connected by an alternating band of triangles. Antiprisms are a subclass of the prismatoids.

Antiprisms are similar to prisms except the bases are twisted relative to each other, and that the side faces are triangles, rather than quadrilaterals.

In the case of a regular n-sided base, one usually considers the case where its copy is twisted by an angle 180°/n. Extra regularity is obtained by the line connecting the base centers being perpendicular to the base planes, making it a right antiprism. As faces, it has the two n-gonal bases and, connecting those bases, 2n isosceles triangles.

Uniform antiprism[edit]

A uniform antiprism has, apart from the base faces, 2n equilateral triangles as faces. As a class, the uniform antiprisms form an infinite series of vertex-uniform polyhedra, as do the uniform prisms. For n = 2 we have as degenerate case the regular tetrahedron as a digonal antiprism, and for n = 3 the non-degenerate regular octahedron as a triangular antiprism.

The dual polyhedra of the antiprisms are the trapezohedra. Their existence was first discussed and their name was coined by Johannes Kepler.

Family of uniform antiprisms
2 3 4 5 6 7 8 9 10 11 12 n
s{2,4}
sr{2,2}
s{2,6}
sr{2,3}
s{2,8}
sr{2,4}
s{2,10}
sr{2,5}
s{2,12}
sr{2,6}
s{2,14}
sr{2,7}
s{2,16}
sr{2,8}
s{2,18}
sr{2,9}
s{2,20}
sr{2,10}
s{2,22}
sr{2,11}
s{2,24}
sr{2,12}
s{2,2n}
sr{2,n}
CDel node h.pngCDel 2x.pngCDel node h.pngCDel 4.pngCDel node.png
CDel node h.pngCDel 2x.pngCDel node h.pngCDel 2x.pngCDel node h.png
CDel node h.pngCDel 2x.pngCDel node h.pngCDel 6.pngCDel node.png
CDel node h.pngCDel 2x.pngCDel node h.pngCDel 3.pngCDel node h.png
CDel node h.pngCDel 2x.pngCDel node h.pngCDel 8.pngCDel node.png
CDel node h.pngCDel 2x.pngCDel node h.pngCDel 4.pngCDel node h.png
CDel node h.pngCDel 2x.pngCDel node h.pngCDel 1x.pngCDel 0x.pngCDel node.png
CDel node h.pngCDel 2x.pngCDel node h.pngCDel 5.pngCDel node h.png
CDel node h.pngCDel 2x.pngCDel node h.pngCDel 1x.pngCDel 2x.pngCDel node.png
CDel node h.pngCDel 2x.pngCDel node h.pngCDel 6.pngCDel node h.png
CDel node h.pngCDel 2x.pngCDel node h.pngCDel 1x.pngCDel 4.pngCDel node.png
CDel node h.pngCDel 2x.pngCDel node h.pngCDel 7.pngCDel node h.png
CDel node h.pngCDel 2x.pngCDel node h.pngCDel 1x.pngCDel 6.pngCDel node.png
CDel node h.pngCDel 2x.pngCDel node h.pngCDel 8.pngCDel node h.png
CDel node h.pngCDel 2x.pngCDel node h.pngCDel 1x.pngCDel 8.pngCDel node.png
CDel node h.pngCDel 2x.pngCDel node h.pngCDel 9.pngCDel node h.png
CDel node h.pngCDel 2x.pngCDel node h.pngCDel 2x.pngCDel 0x.pngCDel node.png
CDel node h.pngCDel 2x.pngCDel node h.pngCDel 10.pngCDel node h.png
CDel node h.pngCDel 2x.pngCDel node h.pngCDel 2x.pngCDel 2x.pngCDel node.png
CDel node h.pngCDel 2x.pngCDel node h.pngCDel 11.pngCDel node h.png
CDel node h.pngCDel 2x.pngCDel node h.pngCDel 2x.pngCDel 4.pngCDel node.png
CDel node h.pngCDel 2x.pngCDel node h.pngCDel 12.pngCDel node h.png
CDel node h.pngCDel 2x.pngCDel node h.pngCDel 2x.pngCDel n.pngCDel node.png
CDel node h.pngCDel 2x.pngCDel node h.pngCDel n.pngCDel node h.png
Digonal antiprism.png Trigonal antiprism.png Square antiprism.png Pentagonal antiprism.png Hexagonal antiprism.png Antiprism 7.png Octagonal antiprism.png Enneagonal antiprism.png Decagonal antiprism.png Hendecagonal antiprism.png Dodecagonal antiprism.png
As spherical polyhedra
Spherical digonal antiprism.png Spherical trigonal antiprism.png Spherical square antiprism.png Spherical pentagonal antiprism.png Spherical hexagonal antiprism.png Spherical heptagonal antiprism.png Spherical octagonal antiprism.png

Cartesian coordinates[edit]

Cartesian coordinates for the vertices of a right antiprism with n-gonal bases and isosceles triangles are

\left( \cos\frac{k\pi}{n}, \sin\frac{k\pi}{n}, (-1)^k h \right)

with k ranging from 0 to 2n−1; if the triangles are equilateral,

2h^2=\cos\frac{\pi}{n}-\cos\frac{2\pi}{n}.

Volume and surface area[edit]

Let a be the edge-length of a uniform antiprism. Then the volume is

V = \frac{n \sqrt{4\cos^2\frac{\pi}{2n}-1}\sin \frac{3\pi}{2n} }{12\sin^2\frac{\pi}{n}} \; a^3

and the surface area is

A = \frac{n}{2} ( \cot{\frac{\pi}{n}} + \sqrt{3}) a^2.

Related polyhedra[edit]

There are an infinite set of truncated antiprisms, with the one a lower symmetry form of the truncated octahedron (truncated triangular antiprism). These can be alternated to create snub antiprisms, two of which are Johnson solids, and the snub triangular antiprism is a lower symmetry form of the icosahedron.

Truncated antiprisms
Truncated octahedron prismatic symmetry.png Truncated square antiprism.png Truncated pentagonal antiprism.png ...
ts{2,4} ts{2,6} ts{2,8} ts{2,10} ts{2,2n}
Snub antiprisms
J84 Icosahedron J85 Irregular...
Snub digonal antiprism.png Snub triangular antiprism.png Snub square antiprism colored.png Snub pentagonal antiprism.png ...
ss{2,4} ss{2,6} ss{2,8} ss{2,10} ss{2,2n}

Symmetry[edit]

The symmetry group of a right n-sided antiprism with regular base and isosceles side faces is Dnd of order 4n, except in the case of a tetrahedron, which has the larger symmetry group Td of order 24, which has three versions of D2d as subgroups, and the octahedron, which has the larger symmetry group Oh of order 48, which has four versions of D3d as subgroups.

The symmetry group contains inversion if and only if n is odd.

The rotation group is Dn of order 2n, except in the case of a tetrahedron, which has the larger rotation group T of order 12, which has three versions of D2 as subgroups, and the octahedron, which has the larger rotation group O of order 24, which has four versions of D3 as subgroups.

Star antiprism[edit]

Symmetry group Star forms
d5h
[2,5]
(*225)
Pentagrammic antiprism.png
3.3.3.5/2
d5d
[2+,5]
(2*5)
Pentagrammic crossed antiprism.png
3.3.3.5/3
d7h
[2,7]
(*227)
Antiprism 7-2.png
3.3.3.7/2
Antiprism 7-4.png
3.3.3.7/4
d7d
[2+,7]
(2*7)
Antiprism 7-3.png
3.3.3.7/3
d8d
[2+,8]
(2*8)
Antiprism 8-3.png
3.3.3.8/3
Antiprism 8-5.png
3.3.3.8/5
d9h
[2,9]
(*229)
Antiprism 9-2.png
3.3.3.9/2
Antiprism 9-4.png
3.3.3.9/4
d9d
[2+,9]
(2*9)
Antiprism 9-5.png
3.3.3.9/5
d10d
[2+,10]
(2*10)
Antiprism 10-3.png
3.3.3.10/3
d11h
[2,11]
(*2.2.11)
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)
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
d12d
[2+,12]
(2*12)
Antiprism 12-5.png
3.3.3.12/5
Antiprism 12-7.png
3.3.3.12/7
...

References[edit]

  • Anthony Pugh (1976). Polyhedra: A visual approach. California: University of California Press Berkeley. ISBN 0-520-03056-7.  Chapter 2: Archimedean polyhedra, prisma and antiprisms
  1. ^ Kabai, Sándor. "One World Trade Center Antiprism". Wolfram Demonstrations Project. Retrieved 8 October 2013. 

External links[edit]