Hexagonal bipyramid

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Hexagonal bipyramid
Hexagonale bipiramide.png
Type bipyramid
Schläfli symbol { } + {6}
Coxeter diagram CDel node f1.pngCDel 2.pngCDel node f1.pngCDel 6.pngCDel node.png
CDel node f1.pngCDel 2.pngCDel node f1.pngCDel 3.pngCDel node f1.png
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[edit]

It can be drawn as a tiling on a sphere which also represents the fundamental domains of [3,2], *322 dihedral symmetry:

Spherical hexagonal bipyramid.png

Related polyhedra[edit]

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

Snub hexagonal bipyramid sequence.png

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

Snub rectified hexagonal bipyramid sequence.png
Uniform hexagonal dihedral spherical polyhedra
Symmetry: [6,2], (*622) [6,2]+, (622) [1+,6,2], (322) [6,2+], (2*3)
CDel node 1.pngCDel 6.pngCDel node.pngCDel 2.pngCDel node.png CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 2.pngCDel node.png CDel node.pngCDel 6.pngCDel node 1.pngCDel 2.pngCDel node.png CDel node.pngCDel 6.pngCDel node 1.pngCDel 2.pngCDel node 1.png CDel node.pngCDel 6.pngCDel node.pngCDel 2.pngCDel node 1.png CDel node 1.pngCDel 6.pngCDel node.pngCDel 2.pngCDel node 1.png CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 2.pngCDel node 1.png CDel node h.pngCDel 6.pngCDel node h.pngCDel 2x.pngCDel node h.png CDel node h1.pngCDel 6.pngCDel node.pngCDel 2.pngCDel node.png CDel node.pngCDel 6.pngCDel node h.pngCDel 2x.pngCDel node h.png
Hexagonal dihedron.png Dodecagonal dihedron.png Hexagonal dihedron.png Spherical hexagonal prism.png Spherical hexagonal hosohedron.png Spherical truncated trigonal prism.png Spherical dodecagonal prism2.png Spherical hexagonal antiprism.png Trigonal dihedron.png Spherical trigonal antiprism.png
{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
CDel node f1.pngCDel 6.pngCDel node.pngCDel 2.pngCDel node.png CDel node f1.pngCDel 6.pngCDel node f1.pngCDel 2.pngCDel node.png CDel node.pngCDel 6.pngCDel node f1.pngCDel 2.pngCDel node.png CDel node.pngCDel 6.pngCDel node f1.pngCDel 2.pngCDel node f1.png CDel node.pngCDel 6.pngCDel node.pngCDel 2.pngCDel node f1.png CDel node f1.pngCDel 6.pngCDel node.pngCDel 2.pngCDel node f1.png CDel node f1.pngCDel 6.pngCDel node f1.pngCDel 2.pngCDel node f1.png CDel node fh.pngCDel 6.pngCDel node fh.pngCDel 2x.pngCDel node fh.png CDel node fh.pngCDel 6.pngCDel node.pngCDel 2.pngCDel node.png CDel node.pngCDel 6.pngCDel node fh.pngCDel 2x.pngCDel node fh.png
Spherical hexagonal hosohedron.png Spherical dodecagonal hosohedron.png Spherical hexagonal hosohedron.png Spherical hexagonal bipyramid.png Hexagonal dihedron.png Spherical hexagonal bipyramid.png Spherical dodecagonal bipyramid.png Spherical hexagonal trapezohedron.png Spherical trigonal hosohedron.png Spherical trigonal trapezohedron.png
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 Spherical truncated trigonal prism.png Uniform tiling 332-t012.png Uniform tiling 432-t012.png Uniform tiling 532-t012.png Uniform polyhedron-63-t012.png H2 tiling 237-7.png H2 tiling 238-7.png H2 tiling 23i-7.png H2 tiling 23j12-7.png H2 tiling 23j9-7.png H2 tiling 23j6-7.png H2 tiling 23j3-7.png
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 CDel node 1.pngCDel 2.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 7.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 8.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel infin.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel ultra.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Dual figures
Coxeter CDel node f1.pngCDel 2.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 3.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 5.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 6.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 7.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 8.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel infin.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel ultra.pngCDel node f1.pngCDel 3.pngCDel node f1.png
Duals Spherical hexagonal bipyramid.png Spherical tetrakis hexahedron.png Spherical disdyakis dodecahedron.png Spherical disdyakis triacontahedron.png Tiling Dual Semiregular V4-6-12 Bisected Hexagonal.svg H2checkers 237.png H2checkers 238.png H2checkers 23i.png H2 checkers 23j12.png H2 checkers 23j9.png H2 checkers 23j6.png H2 checkers 23j3.png
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...
CDel node f1.pngCDel 2x.pngCDel node f1.pngCDel 2.pngCDel node.png CDel node f1.pngCDel 2x.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node f1.pngCDel 2x.pngCDel node f1.pngCDel 4.pngCDel node.png CDel node f1.pngCDel 2x.pngCDel node f1.pngCDel 5.pngCDel node.png CDel node f1.pngCDel 2x.pngCDel node f1.pngCDel 6.pngCDel node.png CDel node f1.pngCDel 2x.pngCDel node f1.pngCDel 7.pngCDel node.png CDel node f1.pngCDel 2x.pngCDel node f1.pngCDel 8.pngCDel node.png CDel node f1.pngCDel 2x.pngCDel node f1.pngCDel 9.pngCDel node.png CDel node f1.pngCDel 2x.pngCDel node f1.pngCDel 1x.pngCDel 0x.pngCDel node.png CDel node f1.pngCDel 2x.pngCDel node f1.pngCDel 1x.pngCDel 1x.pngCDel node.png CDel node f1.pngCDel 2x.pngCDel node f1.pngCDel 1x.pngCDel 2x.pngCDel node.png
Triangular bipyramid.png Square bipyramid.png Pentagonale bipiramide.png Hexagonale bipiramide.png Heptagonal bipyramid.png Octagonal bipyramid.png Enneagonal bipyramid.png Decagonal bipyramid.png
As spherical polyhedra
Spherical digonal bipyramid.png Spherical trigonal bipyramid.png Spherical square bipyramid.png Spherical pentagonal bipyramid.png Spherical hexagonal bipyramid.png Spherical heptagonal bipyramid.png Spherical octagonal bipyramid.png Spherical enneagonal bipyramid.png Spherical decagonal bipyramid.png Spherical hendecagonal bipyramid.png Spherical dodecagonal bipyramid.png

See also[edit]

External links[edit]