View source for Hexagonal bipyramid

{{Short description|Polyhedron; 2 hexagonal pyramids joined base-to-base}}
{{Infobox polyhedron
|image =Hexagonale bipiramide.png
|type=[[bipyramid]]
|coxeter={{CDD|node_f1|2x|node_f1|6|node}}<BR>{{CDD|node_f1|2x|node_f1|3|node_f1}}
|schläfli = {{math|{ } + {6} }}
|faces=12 [[triangle]]s
|edge=18
|vertices=8
|symmetry={{math|''D''<sub>6h</sub>, [6,2], (*226),}} order 24
|rotation_group  = {{math|''D''<sub>''6''</sub>, [6,2]<sup>+</sup>, (226),}} order 12
|vertex_config={{math|V4.4.6}}
|dual=[[hexagonal prism]]
|properties=[[Convex polytope|convex]], [[face-transitive]]
}}

A '''hexagonal bipyramid''' is a [[polyhedron]] formed from two hexagonal [[pyramid (geometry)|pyramids]] joined at their bases. The resulting solid has 12 triangular [[face (geometry)|faces]], 8 [[vertex (geometry)|vertices]] and 18 edges. The 12 faces are identical [[isosceles triangle]]s.

Although it is [[face-transitive]], it is not a [[Platonic solid]] because some vertices have four faces meeting and others have six faces, and it is not a [[Johnson solid]] because its faces cannot be [[equilateral triangle]]s; 6 equilateral triangles would make a flat vertex.

It is one of an infinite set of [[bipyramid]]s. Having twelve faces, it is a type of [[dodecahedron]], although that name is usually associated with the [[regular polyhedron|regular polyhedral]] form with pentagonal faces.

The hexagonal bipyramid has a [[plane of symmetry]] (which is [[Horizontal plane|horizontal]] in the figure to the right) where the bases of the two pyramids are joined. This plane is a regular [[hexagon]]. There are also six planes of symmetry crossing through the two [[apex (geometry)|apices]]. These planes are [[rhombus|rhombic]] and lie at 30° [[angle]]s to each other, [[perpendicular]] to the horizontal plane.

== Images ==
It can be drawn as a tiling on a sphere which also represents the fundamental domains of [3,2], *322 [[dihedral symmetry]]:
:[[File:Spherical hexagonal bipyramid.svg|320px]]

== Related polyhedra ==
The hexagonal bipyramid, dt{2,6}, can be in sequence [[truncation (geometry)|truncated]], tdt{2,6} and alternated ([[Snub (geometry)|snubbed]]), sdt{2,6}:
:[[File:Snub hexagonal bipyramid sequence.png|480px]]

The ''hexagonal bipyramid'', dt{2,6}, can be in sequence [[rectification (geometry)|rectified]], rdt{2,6}, [[truncation (geometry)|truncated]], trdt{2,6} and alternated ([[Snub (geometry)|snubbed]]), srdt{2,6}:
:[[File:Snub rectified hexagonal bipyramid sequence.png|540px]]

{{Hexagonal dihedral truncations}}

It is the first polyhedra in a sequence defined by the [[face configuration]] ''V4.6.2n''. This group is special for having all even number of edges per vertex and form bisecting planes through the polyhedra and infinite lines in the plane, and continuing into the hyperbolic plane for any <math>n \ge 7.</math>

With an even number of faces at every vertex, these polyhedra and tilings can be shown by alternating two colors so all adjacent faces have different colors.

Each face on these domains also corresponds to the fundamental domain of a [[symmetry group]] with order 2,3,n mirrors at each triangle face vertex.
{{Omnitruncated table}}

{{Bipyramids}}

==See also==
* [[hexagonal trapezohedron]] A similar 12-sided polyhedron with a twist and [[Kite (geometry)|kite]] faces.
* [[Snub disphenoid]] Another 12-sided polyhedron with 2-fold symmetry and only triangular faces.

==External links==
* {{mathworld | urlname = Dipyramid | title = Dipyramid}}
*[http://www.georgehart.com/virtual-polyhedra/vp.html Virtual Reality Polyhedra] The Encyclopedia of Polyhedra
** [[VRML]] model [http://www.georgehart.com/virtual-polyhedra/vrml/hexagonal_dipyramid.wrl hexagonal dipyramid] {{Webarchive|url=https://web.archive.org/web/20210414131958/http://www.georgehart.com/virtual-polyhedra/vrml/hexagonal_dipyramid.wrl |date=2021-04-14 }}
**[http://www.georgehart.com/virtual-polyhedra/conway_notation.html Conway Notation for Polyhedra] Try: dP6

[[Category:Polyhedra]]
[[Category:Bipyramids]]