# Antiprism

Set of uniform antiprisms

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
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

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

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}

As spherical polyhedra

## Cartesian coordinates

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

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

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.

 Snub antiprisms J84 Icosahedron J85 ... ts{2,4} ts{2,6} ts{2,8} ts{2,10} ts{2,2n} ... ss{2,4} ss{2,6} ss{2,8} ss{2,10} ss{2,2n}

## Symmetry

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

Star Antiprisms are among the most beautiful polyhedra. They exist in two forms - arbitrarily called 'Normal' and 'Reflex'.

In the Normal form, the triangles connecting the two stars do NOT intersect the vertical axis of rotational symmetry. In the Reflex form, the triangles DO cross the axis.

NORMAL Antiprism of a 4th order 9 pointed star
REFLEX Antiprism of a 4th order 9 pointed star

A number of the star antiprisms are just compounds of simpler ones. The rules which dictate whether there is a unique (i.e. non-compound) star antiprism are as follows.
If the star has N sides and if the M is the star order (the perimeter of a star of order 2 rotates the centre twice). There is a unique Normal star antiprism is the numbers M & N have no common factors. There is a unique Reflex star antiprism if 2×M < N AND √3 sin(θ) - cos(θ) > 1, where θ = π × M/N
High order star antiprisms tend towards a hyperboloid shape.

An educational A3-sized poster showing all the non-star and star anti prisms - up to those with base and tops of 15 sides - together with those of a 29-agon. For best effect, view at maximum resolution
Symmetry group Star forms
d5h
[2,5]
(*225)

3.3.3.5/2
d5d
[2+,5]
(2*5)

3.3.3.5/3
d7h
[2,7]
(*227)

3.3.3.7/2

3.3.3.7/4
d7d
[2+,7]
(2*7)

3.3.3.7/3
d8d
[2+,8]
(2*8)

3.3.3.8/3

3.3.3.8/5
d9h
[2,9]
(*229)

3.3.3.9/2

3.3.3.9/4
d9d
[2+,9]
(2*9)

3.3.3.9/5
d10d
[2+,10]
(2*10)

3.3.3.10/3
d11h
[2,11]
(*2.2.11)

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

3.3.3.11/5

3.3.3.11/7
d12d
[2+,12]
(2*12)

3.3.3.12/5

3.3.3.12/7
...

## References

• 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.