Hexagonal antiprism

In geometry, the hexagonal antiprism is the 4th in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps.

Uniform hexagonal antiprism
Type	Prismatic uniform polyhedron
Elements	F = 14, E = 24 V = 12 (χ = 2)
Faces by sides	12{3}+2{6}
Schläfli symbol	s{2,12} sr{2,6}
Wythoff symbol	| 2 2 6
Coxeter diagram
Symmetry group	D6d, [2+,12], (2*6), order 24
Rotation group	D6, [6,2]+, (622), order 12
References	U77(d)
Dual	Hexagonal trapezohedron
Properties	convex
Vertex figure 3.3.3.6

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.

If faces are all regular, it is a semiregular polyhedron.

Crossed antiprism

A crossed hexagonal antiprism is a star polyhedron, topologically identical to the convex hexagonal antiprism with the same vertex arrangement, but it can't be made uniform; the sides are isosceles triangles. Its vertex configuration is 3.3/2.3.6, with one triangle retrograde. It has D_6d symmetry, order 24.

Related polyhedra

The hexagonal faces can be replaced by coplanar triangles, leading to a nonconvex polyhedron with 24 equilateral triangles.

Uniform hexagonal dihedral spherical polyhedra
Symmetry: [6,2], (*622)							[6,2]+, (622)	[6,2+], (2*3)
{6,2}	t{6,2}	r{6,2}	t{2,6}	{2,6}	rr{6,2}	tr{6,2}	sr{6,2}	s{2,6}
Duals to uniforms
V62	V122	V62	V4.4.6	V26	V4.4.6	V4.4.12	V3.3.3.6	V3.3.3.3

Family of uniform n-gonal antiprisms

• v
• t
• e

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

External links

• Weisstein, Eric W. "Antiprism". MathWorld.
• Hexagonal Antiprism: Interactive Polyhedron model
• Virtual Reality Polyhedra www.georgehart.com: The Encyclopedia of Polyhedra
  • VRML model Archived 2007-02-07 at the Wayback Machine
• polyhedronisme A6