Hexagonal antiprism

Hexagonal antiprism
(3D model) (OFF file)
Rank	3
Type	Uniform
Space	Spherical
Notation
Bowers style acronym	Hap
Coxeter diagram	s2s12o
Elements
Faces	12 triangles, 2 hexagons
Edges	12+12
Vertices	12
Vertex figure	Isosceles trapezoid, edge lengths 1, 1, 1, √3
Measures (edge length 1)
Circumradius	$\frac{\sqrt{3+\sqrt3}}{2} ≈ 1.08766$
Volume	$\sqrt{2+2\sqrt3} ≈ 2.33754$
Dihedral angles	3–3: $\arccos\left(\frac{1-2\sqrt3}{3}\right) ≈ 145.22189°$
	6–3: $\arccos\left(\frac{3-2\sqrt3}{3}\right) ≈ 98.89943°$
Height	$\sqrt{\sqrt3-1} ≈ 0.85560$
Central density	1
Related polytopes
Army	Hap
Regiment	Hap
Dual	Hexagonal antitegum
Conjugate	Hexagonal retroprism
Abstract properties
Euler characteristic	2
Topological properties
Surface	Sphere
Orientable	Yes
Genus	0
Properties
Symmetry	(I2(12)×A1)/2, order 24
Convex	Yes
Nature	Tame

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.

External links

• Klitzing, Richard. "hap".

• Quickfur. "The Hexagonal Antiprism".

• Wikipedia Contributors. "Hexagonal antiprism".
• McCooey, David. "Hexagonal Antiprism"