Hexagonal antiprism

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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
Volume
Dihedral angles3–3:
 6–3:
Height
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[edit | edit source]

A hexagonal antiprism of edge length 1 has vertex coordinates given by:

  • ,
  • ,
  • ,
  • .

Representations[edit | edit source]

A hexagonal antiprism has the following Coxeter diagrams:

General variant[edit | edit source]

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 .

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 and side edges of length .

Related polyhedra[edit | edit source]

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.

External links[edit | edit source]