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
TypePrismatic uniform polyhedron
ElementsF = 14, E = 24
V = 12 (χ = 2)
Faces by sides12{3}+2{6}
Schläfli symbols{2,12}
sr{2,6}
Wythoff symbol| 2 2 6
Coxeter diagram
Symmetry groupD6d, [2+,12], (2*6), order 24
Rotation groupD6, [6,2]+, (622), order 12
ReferencesU77(d)
DualHexagonal trapezohedron
Propertiesconvex

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 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 12.

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