Pentagonal bipyramid

Short description: 13th Johnson solid; two pentagonal pyramids joined at the bases

Pentagonal bipyramid

Type| Bipyramid,
Johnson
J12 – J13 – J14
Faces| 10 triangles
Edges| 15
Vertices| 7
Vertex configuration| V4.4.5
Schläfli symbol| { } + {5}
Coxeter diagram|
Symmetry group| D5h, [5,2], (*225), order 20
Rotation group| D5, [5,2]+, (225), order 10
Dual polyhedron| pentagonal prism
Properties| convex, face-transitive, (deltahedron)
Net

File:Bipirámide pentagonal 3D.stl

In geometry, the pentagonal bipyramid (or dipyramid) is third of the infinite set of face-transitive bipyramids, and the 13th Johnson solid (J13). Each bipyramid is the dual of a uniform prism.

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

## Contents

* 1 Properties
* 2 Formulae
* 3 Related polyhedra
* 4 See also
* 5 References
* 6 External links

## Properties

If the faces are equilateral triangles, it is a deltahedron and a Johnson solid (J13). It can be seen as two pentagonal pyramids (J2) connected by their bases.

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]

The pentagonal dipyramid is 4-connected, meaning that it takes the removal of four vertices to disconnect the remaining vertices. It is one of only four 4-connected simplicial well-covered polyhedra, meaning that all of the maximal independent sets of its vertices have the same size. The other three polyhedra with this property are the regular octahedron, the snub disphenoid, and an irregular polyhedron with 12 vertices and 20 triangular faces.[2]

## 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]:[3]

[math]\displaystyle{ H = L\cdot \sqrt{2 - \frac{2}{\sqrt{5}}} \approx L\cdot 1.0514622242 }[/math]
[math]\displaystyle{ A = L^2 \cdot \frac{5\sqrt{3}}{2} \approx L^2\cdot 4.330127019 }[/math]
[math]\displaystyle{ V = L^3 \cdot \frac{5 + \sqrt{5}}{12} \approx L^3\cdot 0.6030056648 }[/math]

Spherical pentagonal bipyramid

## Related polyhedra

The pentagonal bipyramid, dt{2,5}, can be in sequence rectified, rdt{2,5}, truncated, trdt{2,5} and alternated (snubbed), srdt{2,5}:

The dual of the Johnson solid pentagonal bipyramid is the pentagonal prism, with 7 faces: 5 rectangular faces and 2 pentagons.

Dual pentagonal bipyramid  | Net of dual

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## See also

* Pentagonal bipyramidal molecular geometry

## 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. ↑ Finbow, Arthur S.; Hartnell, Bert L.; Nowakowski, Richard J. (2010), "On well-covered triangulations. III", Discrete Applied Mathematics 158 (8): 894–912, doi:10.1016/j.dam.2009.08.002 .
3. ↑ Sapiña, R.. "Area and volume of the Johnson solid J₁₃" (in es). Problemas y ecuaciones. ISSN 2659-9899. https://www.problemasyecuaciones.com/geometria3D/volumen/Johnson/J13/calculadora-area-volumen-formulas.html. Retrieved 2020-09-04.

## External links

* Eric W. Weisstein, Pentagonal dipyramid (Dipyramid) at MathWorld.
* Conway Notation for Polyhedra Try: dP5

* 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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