Triangular bipyramid

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For the related molecular geometrical structure, see Trigonal bipyramid molecular geometry.
Triangular bipyramid
Triangular bipyramid
Type Bipyramid
and
Johnson
J11 - J12 - J13
Schläfli symbol { } + {3}
Coxeter diagram CDel node f1.pngCDel 2.pngCDel node f1.pngCDel 3.pngCDel node.png
Faces 6 triangles
Edges 9
Vertices 5
Face configuration V3.4.4
Symmetry group D3h, [3,2], (*223) order 12
Rotation group D3, [3,2]+, (223), order 6
Dual Triangular prism
Properties Convex, face-transitive

In geometry, the triangular bipyramid (or dipyramid) is a type of hexahedron, being the first in the infinite set of face-transitive bipyramids. It is the dual of the triangular prism with 6 isosceles triangle faces.

It is also one of the Johnson solids, (J12) with equilateral triangle faces. As the name suggests, it can be constructed by joining two tetrahedra along one face. Although all its faces are congruent and the solid is face-transitive, it is not a Platonic solid because some vertices adjoin three faces and others adjoin four. As a Johnson solid, with 6 equilateral triangles, it is also in the set of deltahedra.

A Johnson solid is one of 92 strictly convex polyhedra that have regular faces but are not uniform (that is, they are not Platonic solids, Archimedean solids, prisms or antiprisms). They were named by Norman Johnson, who first listed these polyhedra in 1966.[1]

Triangular dipyramid.png

Dual polyhedron[edit]

The dual of the Johnson solid triangular bipyramid is the triangular prism, with 5 faces: 3 rectangular faces and 2 triangular.

Dual triangular bipyramid Net of dual
Dual triangular dipyramid.png Dual triangular dipyramid net.png

Related polyhedra and honeycombs[edit]

The triangular bipyramid, dt{2,3}, can be in sequence rectified, rdt{2,3}, truncated, trdt{2,3} and alternated (snubbed), srdt{2,3}:

Snub rectified triangular bipyramid sequence.png

The triangular bipyramid can be constructed by augmentation of smaller ones, specifically two stacked regular octahedra with 4 triangular bipyramids added around the sides, and 1 tetrahedron above and below. This polyhedron has 24 equilateral triangle faces, but it is not a Johnson solid because it has coplanar faces. It is a coplanar 24 triangle deltahedron. This polyhedron exists as the augmentation of cells in a gyrated alternated cubic honeycomb. Larger triangular polyhedra can be generated similarly, like 9, 16 or 25 triangles per larger triangle face, seen as a section of a triangular tiling.

Triangulated bipyramid.png

The triangular bipyramid fill the space with octahedron or truncated tetrahedron.[2]

Tetrahedral-truncated tetrahedral honeycomb slab.png
Layers of the uniform quarter cubic honeycomb can be shifted to pair up regular tetrahedral cells which combined into triangular bipyramids.
Tetroctahedric semicheck.png
The gyrated tetrahedral-octahedral honeycomb has pairs of adjacent regular tetrahedra that can be seen as triangular bipyramids.

See also[edit]

Family of bipyramids
2 3 4 5 6 7 8 9 10 11 12 ...
CDel node f1.pngCDel 2.pngCDel node f1.pngCDel 2.pngCDel node.png CDel node f1.pngCDel 2.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node f1.pngCDel 2.pngCDel node f1.pngCDel 4.pngCDel node.png CDel node f1.pngCDel 2.pngCDel node f1.pngCDel 5.pngCDel node.png CDel node f1.pngCDel 2.pngCDel node f1.pngCDel 6.pngCDel node.png CDel node f1.pngCDel 2.pngCDel node f1.pngCDel 7.pngCDel node.png CDel node f1.pngCDel 2.pngCDel node f1.pngCDel 8.pngCDel node.png CDel node f1.pngCDel 2.pngCDel node f1.pngCDel 9.pngCDel node.png CDel node f1.pngCDel 2.pngCDel node f1.pngCDel 1x.pngCDel 0x.pngCDel node.png CDel node f1.pngCDel 2.pngCDel node f1.pngCDel 1x.pngCDel 1x.pngCDel node.png CDel node f1.pngCDel 2.pngCDel node f1.pngCDel 1x.pngCDel 2x.pngCDel node.png CDel node f1.pngCDel 2.pngCDel node f1.pngCDel infin.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 Bicone.svg
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

References[edit]

External links[edit]