Set of uniform antiprisms
Type	uniform polyhedron
Faces	2 n-gons, 2n triangles
Edges	4n
Vertices	2n
Vertex configuration	3.3.3.n
Symmetry group	Dnd
Dual polyhedron	trapezohedron
Properties	convex, semi-regular vertex-uniform

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: the vertices are symmetrically staggered.

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. It has, apart from the base faces, 2n isosceles triangles as faces.

A uniform antiprism has, apart from the base faces, 2n equilateral triangles as faces. They form an infinite series of vertex-uniform polyhedra, as do the uniform prisms. For n=2 we have as degenerate case the regular tetrahedron.

Forms

---

• (linear antiprism) Tetrahedron - 4 triangles - self dual
• (trigonal antiprism) Octahedron - 8 triangles - dual cube
• Square antiprism - 8 triangles, 2 squares - dual tetragonal trapezohedron
• Pentagonal antiprism - 10 triangles, 2 pentagons - dual pentagonal trapezohedron
• Hexagonal antiprism - 12 triangles, 2 hexagons - dual hexagonal trapezohedron
• Heptagonal antiprism - 14 triangles, 2 septagons - dual heptagonal trapezohedron
• Octagonal antiprism - 16 triangles, 2 octagons - dual octagonal trapezohedron
• ...
• Decagonal antiprism - 20 triangles, 2 decagons - dual decagonal trapezohedron
• ...
• Dodecagonal antiprism - 24 triangles, 2 dodecagons - dual dodecagonal trapezohedron
• ...
• n-gonal antiprism - 2n triangles, 2 n-gons - dual n-gonal trapezohedron

If n=3 then we only have triangles; we get the octahedron, a particular type of right triangular antiprism which is also edge- and face-uniform, and so counts among the Platonic solids. The dual polyhedra of the antiprisms are the trapezohedra. Their existence was first discussed and their name was coined by Johannes Kepler.

Cartesian coordinates

---

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

$(\; \backslash cos(k\backslash pi/n),\; \backslash sin(k\backslash pi/n),\; (-1)^k\; a\; )\backslash ;$

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

$2a^2=\backslash cos(\backslash pi/n)-\backslash cos(2\backslash pi/n)\backslash ;$.

Symmetry

---

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_d of order 48, which has four versions of D_3d as subgroups.

The symmetry group contains inversion iff 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.

Star antiprisms

---

Uniform antiprisms can also be constructed on star polygons: {n/m} = {5/2}, {7/3}, {7/4}, {8/3}, {9/2}, {9/4}, {10/3}...

For any coprime pair of integers n,m such that 2m < n, there are two forms:

• a normal antiprism with vertex configuration 3.3.3.n/m;
• a crossed antiprism with vertex configuration 3.3.3.n/(n-m).

3.3.3.5/2
3.3.3.5/3

External links

---

• Paper models of prisms and antiprisms
• The Uniform Polyhedra
• Virtual Reality Polyhedra The Encyclopedia of Polyhedra
  • VRML models (George Hart) <3> <4> <5> <6> <7> <8> <9> <10>

Polyhedra | Uniform polyhedra | Prismatoid polyhedra

Antiprisma | Antiprisma | 反角柱 | Antygraniastosłup | Antiprisma | Антипризма