Antiprism

Set of uniform n-gonal antiprisms

Example uniform hexagonal antiprism
Type uniform in the sense of semiregular polyhedron
Faces 2{n} + 2n{3}
Edges 4n
Vertices 2n
Conway polyhedron notation An
Vertex configuration 3.3.3.n
Schläfli symbol { }⊗{n}[1]
s{2,2n}
sr{2,n}
Coxeter diagrams
Symmetry group Dnd, [2+,2n], (2*n), order 4n
Rotation group Dn, [2,n]+, (22n), order 2n
Dual polyhedron convex dual-uniform n-gonal trapezohedron
Properties convex, vertex-transitive, regular polygon faces, congruent & coaxial bases
Net
Example uniform enneagonal antiprism net (n = 9)

In geometry, an n-gonal antiprism or n-antiprism is a polyhedron composed of two parallel direct copies (not mirror images) of an n-sided polygon, connected by an alternating band of 2n triangles.

Antiprisms are a subclass of prismatoids, and are a (degenerate) type of snub polyhedron.

Antiprisms are similar to prisms, except that the bases are twisted relatively to each other, and that the side faces (connecting the bases) are 2n triangles, rather than n quadrilaterals.

Right 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 antiprism has two congruent regular n-gon 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 regular tetrahedron as a digonal antiprism (degenerate antiprism); for n = 3, the regular octahedron as a triangular antiprism (non-degenerate antiprism).

The dual polyhedra of the antiprisms are the trapezohedra.

The existence of antiprisms 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 faces and on vertices as the Archimedean solids.

Family of uniform n-gonal antiprisms
Antiprism name Digonal antiprism (Trigonal)
Triangular antiprism
(Tetragonal)
Square antiprism
Pentagonal antiprism Hexagonal antiprism Heptagonal antiprism Octagonal antiprism Enneagonal antiprism Decagonal antiprism Hendecagonal antiprism Dodecagonal 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 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

Cartesian coordinates for the vertices of a right antiprism (i.e. with regular n-gon bases and isosceles side faces) 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, 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 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

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

Related polyhedra

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}

Symmetry

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

• n = 2: the regular tetrahedron, which has the larger symmetry group Td of order 24 = 3×(4×2), which has three versions of D2d as subgroups;
• n = 3: the regular octahedron, which has the larger symmetry group Oh of order 48 = 4×(4×3), 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 cases of:

• n = 2: the regular tetrahedron, which has the larger rotation group T of order 12 = 3×(2×2), which has three versions of D2 as subgroups;
• n = 3: the regular octahedron, which has the larger rotation group O of order 24 = 4×(2×3), which has four versions of D3 as subgroups.

Star antiprism

 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 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, 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, 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.