# Hexagonal antiprism

Hexagonal antiprism
Rank3
TypeUniform
Notation
Bowers style acronymHap
Coxeter diagrams2s12o ()
Conway notationA6
Elements
Faces12 triangles, 2 hexagons
Edges12+12
Vertices12
Vertex figureIsosceles trapezoid, edge lengths 1, 1, 1, 3
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {3+{\sqrt {3}}}}{2}}\approx 1.08766}$
Volume${\displaystyle {\sqrt {2+2{\sqrt {3}}}}\approx 2.33754}$
Dihedral angles3–3: ${\displaystyle \arccos \left({\frac {1-2{\sqrt {3}}}{3}}\right)\approx 145.22189^{\circ }}$
6–3: ${\displaystyle \arccos \left({\frac {3-2{\sqrt {3}}}{3}}\right)\approx 98.89943^{\circ }}$
Height${\displaystyle {\sqrt {{\sqrt {3}}-1}}\approx 0.85560}$
Central density1
Number of external pieces14
Level of complexity4
Related polytopes
ArmyHap
RegimentHap
DualHexagonal antitegum
ConjugateHexagonal retroprism
Abstract & topological properties
Flag count96
Euler characteristic2
SurfaceSphere
OrientableYes
Genus0
Properties
Symmetry(I2(12)×A1)/2, order 24
ConvexYes
NatureTame

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:

• ${\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {\sqrt {3}}{2}},\,{\frac {\sqrt {{\sqrt {3}}-1}}{2}}\right)}$,
• ${\displaystyle \left(\pm 1,\,0,\,{\frac {\sqrt {{\sqrt {3}}-1}}{2}}\right)}$,
• ${\displaystyle \left(\pm {\frac {\sqrt {3}}{2}},\,\pm {\frac {1}{2}},\,-{\frac {\sqrt {{\sqrt {3}}-1}}{2}}\right)}$,
• ${\displaystyle \left(0,\,\pm 1,\,-{\frac {\sqrt {{\sqrt {3}}-1}}{2}}\right)}$.

## Representations

A hexagonal antiprism has the following Coxeter diagrams:

## 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 ${\displaystyle {\sqrt {l^{2}-b^{2}(2-{\sqrt {3}})}}}$.

The bases of the hexagonal 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 ${\displaystyle {\sqrt {2+{\sqrt {3}}}}}$ and side edges of length ${\displaystyle {\sqrt {2}}}$.

## 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.