Antiprism

For an antiprism with regular n-gon bases, one usually considers the case where these two copies are twisted by an angle of 180/n degrees.

The axis of a regular polygon is the line perpendicular to the polygon plane and lying in the polygon centre.

For an antiprism with congruent regular n-gon bases, twisted by an angle of 180/n degrees, more regularity is obtained if the bases have the same axis: are coaxial; i.e. (for non-coplanar bases): if the line connecting the base centers is perpendicular to the base planes. Then the antiprism is called a right antiprism, and its 2n side faces are isosceles triangles.

Uniform antiprism

A uniform n-antiprism has two congruent regular n-gons as base faces, and 2n equilateral triangles as side faces.

Uniform antiprisms form an infinite class of vertex-transitive polyhedra, as do uniform prisms. For n = 2, we have the digonal antiprism (degenerate antiprism), which is visually identical to the regular tetrahedron; for n = 3, the regular octahedron as a triangular antiprism (non-degenerate antiprism).

Family of uniform n-gonal antiprisms

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Antiprism name	Digonal antiprism	(Trigonal) Triangular antiprism	(Tetragonal) Square antiprism	Pentagonal antiprism	Hexagonal antiprism	Heptagonal antiprism	...	Apeirogonal antiprism
Polyhedron image							...
Spherical tiling image							Plane tiling image
Vertex config.	2.3.3.3	3.3.3.3	4.3.3.3	5.3.3.3	6.3.3.3	7.3.3.3	...	∞.3.3.3

The Schlegel diagrams of these semiregular antiprisms are as follows:

A3

A4

A5

A6

A7

A8

Cartesian coordinates

n-antiprism (i.e. with regular n-gon bases and 2n isosceles triangle side faces) are:

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

where 0 ≤ k ≤ 2n – 1;

if the n-antiprism is uniform (i.e. if the triangles are equilateral), then:

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

Volume and surface area

Let a be the edge-length of a uniform n-gonal 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:

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

Furthermore, the volume of a regular right n-gonal antiprism with side length of its bases l and height h is given by:

$${\displaystyle V={\frac {nhl^{2}}{12}}\left(\csc {\frac {\pi }{n}}+2\cot {\frac {\pi }{n}}\right).}$$

Note that the volume of a right n-gonal prism with the same l and h is:

$${\displaystyle V_{\mathrm {prism} }={\frac {nhl^{2}}{4}}\cot {\frac {\pi }{n}}}$$

Symmetry

The symmetry group of a right n-antiprism (i.e. with regular bases and isosceles side faces) is D_nd = D_nv of order 4n, except in the cases of:

• n = 2: the regular tetrahedron, which has the larger symmetry group T_d of order 24 = 3 × (4 × 2), which has three versions of D_2d as subgroups;

• n = 3: the regular octahedron, which has the larger symmetry group O_h of order 48 = 4 × (4 × 3), which has four versions of D_3d as subgroups.

The symmetry group contains

Self-crossing polyhedra

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

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

A right star antiprism has two congruent coaxial regular convex or star polygon base faces, and 2n isosceles triangle side faces.

Any star antiprism with regular convex or star polygon bases can be made a right star antiprism (by translating and/or twisting one of its bases, if necessary).

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

• Retrograde star antiprisms with regular convex polygon bases cannot have all equal edge lengths, and so cannot be uniform. "Exception": a retrograde star antiprism with equilateral triangle bases (vertex configuration: 3.3/2.3.3) can be uniform; but then, it has the appearance of an equilateral triangle: it is a degenerate star polyhedron.

• Similarly, some retrograde star antiprisms with regular star polygon bases cannot have all equal edge lengths, and so cannot be uniform. Example: a retrograde star antiprism with regular star 7/5-gon bases (vertex configuration: 3.3.3.7/5) cannot be uniform.

Also, star antiprism compounds with regular star p/q-gon bases can be constructed if p and q have common factors. Example: a star 10/4-antiprism is the compound of two star 5/2-antiprisms.

Star p/q-antiprisms by symmetry, for p ≤ 12

Symmetry group	Uniform stars				Right stars
D4d [2+,8] (2*4)					Crossed square antiprism
D5h [2,5] (*225)	3.3.3.5/2 Pentagrammic antiprism				crossed pentagonal antiprism
D5d [2+,10] (2*5)	3.3.3.5/3 Pentagrammic crossed-antiprism
D6d [2+,12] (2*6)					crossed hexagonal antiprism
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 Octagrammic antiprism	3.3.3.8/5 Octagrammic crossed-antiprism
D9h [2,9] (*229)	3.3.3.9/2 Enneagrammic antiprism (9/2)	3.3.3.9/4 Enneagrammic antiprism (9/4)
D9d [2+,18] (2*9)	3.3.3.9/5 Enneagrammic crossed-antiprism
D10d [2+,20] (2*10)	3.3.3.10/3 Decagrammic antiprism
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
...	...

See also

• Grand antiprism, a four-dimensional polytope
• Skew polygon, a three-dimensional polygon whose convex hull is an antiprism

References