Triangular bipyramid

Short description: 12th Johnson solid; two tetrahedra joined along one face

Triangular bipyramid

Type| Bipyramid,
Johnson
J11 – J12 – J13
Faces| 6 triangles
Edges| 9
Vertices| 5
Schläfli symbol| { } + {3}
Coxeter diagram|
Symmetry group| D3h, [3,2], (*223), order 12
Rotation group| D3, [3,2]+, (223), order 6
Dual polyhedron| Triangular prism
Face configuration| V3.4.4
Properties| Convex, face-transitive

File:J12 triangular bipyramid.stl

Net

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.

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.

The bipyramid whose six faces are all equilateral triangles is one of the Johnson solids, (J12). A Johnson solid is one of 92 strictly convex polyhedra that is composed of regular polygon faces but are not uniform polyhedra (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] As a Johnson solid with all faces equilateral triangles, it is also a deltahedron.

## Contents

* 1 Formulae
* 2 Dual polyhedron
* 3 Related polyhedra and honeycombs
* 4 See also
* 5 References
* 6 External links

## Formulae

The following formulae for the height ([math]\displaystyle{ H }[/math]), surface area ([math]\displaystyle{ A }[/math]) and volume ([math]\displaystyle{ V }[/math]) can be used if all faces are regular, with edge length [math]\displaystyle{ L }[/math]:[2]

[math]\displaystyle{ H = L\cdot \frac{2\sqrt{6}}{3} \approx L\cdot 1.632993162 }[/math]
[math]\displaystyle{ A = L^2 \cdot \frac{3\sqrt{3}}{2} \approx L^2\cdot 2.598076211 }[/math]
[math]\displaystyle{ V = L^3 \cdot \frac{\sqrt{2}}{6} \approx L^3\cdot 0.235702260 }[/math]

## Dual polyhedron

The dual polyhedron of the triangular bipyramid is the triangular prism, with five faces: two parallel equilateral triangles linked by a chain of three rectangles. Although the triangular prism has a form that is a uniform polyhedron (with square faces), the dual of the Johnson solid form of the bipyramid has rectangular rather than square faces, and is not uniform.

Triangular prism  | Net

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## Related polyhedra and honeycombs

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

The triangular bipyramid can be constructed by augmentation of smaller ones, specifically two stacked regular octahedra with 3 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.

The triangular bipyramid can form a tessellation of space with octahedra or with truncated tetrahedra.[3]

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

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When projected onto a sphere, it resembles a compound of a trigonal hosohedron and trigonal dihedron. It is part of an infinite series of dual pair compounds of regular polyhedra projected onto spheres. The triangular bipyramid can be referred to as a deltoidal hexahedron for consistency with the other solids in the series, although the "deltoids" are triangles instead of kites in this case, as the angle from the dihedron is 180 degrees.

## See also

* Trigonal bipyramidal molecular geometry
* Boerdijk–Coxeter helix, an extension of the triangular bipyramid by adding more tetrahedrons

## References

1. ↑ Johnson, Norman W. (1966), "Convex polyhedra with regular faces", Canadian Journal of Mathematics 18: 169–200, doi:10.4153/cjm-1966-021-8 .
2. ↑ Sapiña, R.. "Area and volume of the Johnson solid J₁₂" (in es). Problemas y Ecuaciones. ISSN 2659-9899. https://www.problemasyecuaciones.com/geometria3D/volumen/Johnson/J12/calculadora-area-volumen-formulas.html. Retrieved 2020-09-01.
3. ↑ "J12 honeycomb". http://woodenpolyhedra.web.fc2.com/J12.html.

## External links

* Eric W. Weisstein, Triangular dipyramid (Johnson solid) at MathWorld.
* Conway Notation for Polyhedra Try: dP3

* v
* t
* e

Johnson solids

Pyramids, cupolae and rotundae|

* square pyramid
* pentagonal pyramid
* triangular cupola
* square cupola
* pentagonal cupola
* pentagonal rotunda

Modified pyramids|

* elongated triangular pyramid
* elongated square pyramid
* elongated pentagonal pyramid
* gyroelongated square pyramid
* gyroelongated pentagonal pyramid
* triangular bipyramid
* square bipyramid
* pentagonal bipyramid
* elongated triangular bipyramid
* elongated square bipyramid
* elongated pentagonal bipyramid
* gyroelongated square bipyramid

Modified cupolae and rotundae|

* elongated triangular cupola
* elongated square cupola
* elongated pentagonal cupola
* elongated pentagonal rotunda
* gyroelongated triangular cupola
* gyroelongated square cupola
* gyroelongated pentagonal cupola
* gyroelongated pentagonal rotunda
* gyrobifastigium
* triangular orthobicupola
* square orthobicupola
* square gyrobicupola
* pentagonal orthobicupola
* pentagonal gyrobicupola
* pentagonal orthocupolarotunda
* pentagonal gyrocupolarotunda
* pentagonal orthobirotunda
* elongated triangular orthobicupola
* elongated triangular gyrobicupola
* elongated square gyrobicupola
* elongated pentagonal orthobicupola
* elongated pentagonal gyrobicupola
* elongated pentagonal orthocupolarotunda
* elongated pentagonal gyrocupolarotunda
* elongated pentagonal orthobirotunda
* elongated pentagonal gyrobirotunda
* gyroelongated triangular bicupola
* gyroelongated square bicupola
* gyroelongated pentagonal bicupola
* gyroelongated pentagonal cupolarotunda
* gyroelongated pentagonal birotunda

Augmented prisms|

* augmented triangular prism
* biaugmented triangular prism
* triaugmented triangular prism
* augmented pentagonal prism
* biaugmented pentagonal prism
* augmented hexagonal prism
* parabiaugmented hexagonal prism
* metabiaugmented hexagonal prism
* triaugmented hexagonal prism

Modified Platonic solids|

* augmented dodecahedron
* parabiaugmented dodecahedron
* metabiaugmented dodecahedron
* triaugmented dodecahedron
* metabidiminished icosahedron
* tridiminished icosahedron
* augmented tridiminished icosahedron

Modified Archimedean solids|

* augmented truncated tetrahedron
* augmented truncated cube
* biaugmented truncated cube
* augmented truncated dodecahedron
* parabiaugmented truncated dodecahedron
* metabiaugmented truncated dodecahedron
* triaugmented truncated dodecahedron
* gyrate rhombicosidodecahedron
* parabigyrate rhombicosidodecahedron
* metabigyrate rhombicosidodecahedron
* trigyrate rhombicosidodecahedron
* diminished rhombicosidodecahedron
* paragyrate diminished rhombicosidodecahedron
* metagyrate diminished rhombicosidodecahedron
* bigyrate diminished rhombicosidodecahedron
* parabidiminished rhombicosidodecahedron
* metabidiminished rhombicosidodecahedron
* gyrate bidiminished rhombicosidodecahedron
* tridiminished rhombicosidodecahedron

Elementary solids|

* snub disphenoid
* snub square antiprism
* sphenocorona
* augmented sphenocorona
* sphenomegacorona
* hebesphenomegacorona
* disphenocingulum
* bilunabirotunda
* triangular hebesphenorotunda

(See also List of Johnson solids, a sortable table)

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