Hexagonal antiprism

The hexagonal antiprism, or hap, is a prismatic uniform polyhedron. It consists of 12 triangles and 2 hexagons. Each vertex joins one hexagon and three triangles. As the name suggests, it is an antiprism based on a hexagon.

Vertex coordinates
A hexagonal antiprism of edge length 1 has vertex coordinates given by:
 * $$\left(±\frac12,\,±\frac{\sqrt3}{2},\,\frac{\sqrt{\sqrt3-1}}{2}\right),$$
 * $$\left(±1,\,0,\,\frac{\sqrt{\sqrt3-1}}{2}\right),$$
 * $$\left(±\frac{\sqrt3}{2},\,±\frac12,\,-\frac{\sqrt{\sqrt3-1}}{2}\right),$$
 * $$\left(0,\,±1,\,-\frac{\sqrt{\sqrt3-1}}{2}\right).$$

Representations
A hexagonal antiprism has the following Coxeter diagrams:


 * s2s12o (alternated dodecagonal prism)
 * s2s6s (alternated dihexagonal prism)
 * xo6ox&#x (bases considered separately)

General variant
The hexagonal antiprism has a general isogonal variant of the form xo6ox&#y that maintains its full symmetry. This variant uses isosceles triangles as sides.

If the base edges are of length b and the lacing edges are of length l, its height is given by $$\sqrt{l^2-b^2(2-\sqrt3)}$$.

The bases of the pentagonal antiprism are rotated from each other by an angle of 30°. If this angle is changed the result is more properly called a hexagonal gyroprism.

A notable case occurs as the alternation of the uniform dodecagonal prism. This specific case has base edges of length $$\sqrt{2+\sqrt3}$$ and side edges of length $$\sqrt2$$.

Related polyhedra
A triangular cupola can be attached to a base of the hexagonal antiprism to form the gyroelongated triangular cupola. If a second triangular cupola is attached to the other base, the result is the gyroelongated triangular bicupola.