# Hexagonal bipyramid

Hexagonal bipyramid
Type bipyramid
Schläfli symbol { } + {6}
Coxeter diagram
Faces 12 triangles
Edges 18
Vertices 8
Face configuration V4.4.6
Symmetry group D6h, [6,2], (*226), order 24
Rotation group D6, [6,2]+, (226), order 12
Dual hexagonal prism
Properties convex, face-transitive

A hexagonal bipyramid is a polyhedron formed from two hexagonal pyramids joined at their bases. The resulting solid has 12 triangular faces, 8 vertices and 18 edges. The 12 faces are identical isosceles triangles.

Although it is face-transitive, it is not a Platonic solid because some vertices have four faces meeting and others have six faces.

It is one of an infinite set of bipyramids. Having twelve faces, it is a type of dodecahedron, although that name is usually associated with the regular polyhedral form with pentagonal faces. The term dodecadeltahedron is sometimes used to distinguish the bipyramid from the Platonic solid, although in chemistry this more often refers to the snub disphenoid.

The hexagonal bipyramid has a plane of symmetry (which is 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 apices. These planes are rhombic and lie at 30° angles 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:

## Related polyhedra

The hexagonal bipyramid, dt{2,6}, can be in sequence truncated, tdt{2,6} and alternated (snubbed), sdt{2,6}:

The hexagonal bipyramid, dt{2,6}, can be in sequence rectified, rdt{2,6}, truncated, trdt{2,6} and alternated (snubbed), srdt{2,6}:

Uniform hexagonal dihedral spherical polyhedra
Symmetry: [6,2], (*622) [6,2]+, (622) [1+,6,2], (322) [6,2+], (2*3)
{6,2} t{6,2} r{6,2} 2t{6,2}=t{2,6} 2r{6,2}={2,6} rr{6,2} tr{6,2} sr{6,2} h{6,2} s{2,6}
Uniform duals
V62 V122 V62 V4.4.6 V26 V4.4.6 V4.4.12 V3.3.3.6 V32 V3.3.3.3

It 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 $n \ge 7.$

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.

Dimensional family of omnitruncated spherical polyhedra and tilings: 4.6.2n
Sym.
*n32
[n,3]
Spherical Euclid. Compact hyperb. Paraco. Noncompact hyperbolic
*232
[2,3]
D3h
*332
[3,3]
Td
*432
[4,3]
Oh
*532
[5,3]
Ih
*632
[6,3]
P6m
*732
[7,3]
*832
[8,3]
*∞32
[∞,3]

[12i,3]

[9i,3]

[6i,3]

[3i,3]
Figure
Schläfli tr{2,3} tr{3,3} tr{4,3} tr{5,3} tr{6,3} tr{7,3} tr{8,3} tr{∞,3} tr{12i,3} tr{9i,3} tr{6i,3} tr{3i,3}
Coxeter
Dual figures
Coxeter
Duals
Face V4.6.4 V4.6.6 V4.6.8 V4.6.10 V4.6.12 V4.6.14 V4.6.16 V4.6.∞ V4.6.24i V4.6.18i V4.6.12i V4.6.6i
Family of bipyramids
2 3 4 5 6 7 8 9 10 11 12...
As spherical polyhedra