Antiprism

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

Dual polyhedra of antiprisms are trapezohedra. Their existence was discussed and their name was coined by Johannes Kepler, though it is possible that they were previously known to Archimedes, as they satisfy the same conditions on vertices as the Archimedean solids.

Family of uniform n-gonal antiprisms
Polyhedron image												...	Apeirogonal antiprism
Spherical tiling image												Plane tiling image
Vertex configuration n.3.3.3	2.3.3.3	3.3.3.3	4.3.3.3	5.3.3.3	6.3.3.3	7.3.3.3	8.3.3.3	9.3.3.3	10.3.3.3	11.3.3.3	12.3.3.3	...	∞.3.3.3

Schlegel diagrams

A3

A4

A5

A6

A7

A8

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

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

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

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

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

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

and the surface area is

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

There are an infinite set of truncated antiprisms, including 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.

Antiprisms
				...
s{2,4}	s{2,6}	s{2,8}	s{2,10}	s{2,2n}
Truncated antiprisms
				...
ts{2,4}	ts{2,6}	ts{2,8}	ts{2,10}	ts{2,2n}
Snub antiprisms
J84	Icosahedron	J85	Irregular faces...
				...
ss{2,4}	ss{2,6}	ss{2,8}	ss{2,10}	ss{2,2n}

The symmetry group of a right n-sided antiprism with regular base and isosceles side faces is D_nd of order 4n, except in the case of a tetrahedron, which has the larger symmetry group T_d of order 24, which has three versions of D_2d as subgroups, and the octahedron, which has the larger symmetry group O_h of order 48, which has four versions of D_3d as subgroups.

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

The rotation group is D_n 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.

5/2-antiprism			5/3-antiprism
9/2-antiprism		9/4-antiprism		9/5-antiprism

This shows all the non-star and star antiprisms up to 15 sides - together with those of an icosikaienneagon.

Uniform star antiprisms are named by their star polygon bases, {p/q}, and exist in prograde and retrograde (crossed) solutions. Crossed forms have intersecting vertex figures, and are denoted by inverted fractions, p/(p - q) instead of p/q, e.g. 5/3 instead of 5/2.

In the retrograde forms but not in the prograde forms, the triangles joining the star bases intersect the axis of rotational symmetry.

Some retrograde star antiprisms with regular convex polygon bases cannot be constructed with equal edge lengths, so are not uniform polyhedra.

Star antiprism compounds also can be constructed where p and q have common factors; example: a 10/4 star antiprism is the compound of two 5/2 star antiprisms.

Star antiprisms by symmetry, up to 12
Symmetry group	Uniform stars				Other stars
D4d [2+,8] (2*5)					3.3/2.3.4
D5h [2,5] (*225)	3.3.3.5/2				3.3/2.3.5
D5d [2+,10] (2*5)	3.3.3.5/3
D6d [2+,12] (2*6)					3.3/2.3.6
D7h [2,7] (*227)	3.3.3.7/2	3.3.3.7/4
D7d [2+,14] (2*7)	3.3.3.7/3
D8d [2+,16] (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+,18] (2*9)	3.3.3.9/5
D10d [2+,12] (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+,22] (2*11)	3.3.3.11/3	3.3.3.11/5	3.3.3.11/7
D12d [2+,24] (2*12)	3.3.3.12/5	3.3.3.12/7
...

• Apeirogonal antiprism
• Rectified antiprism
• Grand antiprism – a four-dimensional polytope
• One World Trade Center, a building consisting primarily of an elongated square antiprism
• Skew polygon

• 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

Wikimedia Commons has media related to Antiprisms.

• Weisstein, Eric W. "Antiprism". MathWorld.
• Nonconvex Prisms and Antiprisms
• Paper models of prisms and antiprisms