# Hexagonal antiprism

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

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.

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 6-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 D6d 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
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