Pentagonal bipyramid

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

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Pentagonal bipyramid

Type| Bipyramid,
Johnson
J12 – J13 – J14
Faces| 10 triangles
Edges| 15
Vertices| 7
Vertex configuration|  5 × ( 3 4 ) + 2 × ( 3 5 ) {\displaystyle 5\times (3^{4})+2\times (3^{5})}
Symmetry group|  D 5 h {\displaystyle D_{5h}}
Dual polyhedron| pentagonal prism
Properties| convex, face-transitive
Net
Johnson solid J13

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.

## Construction[edit]

Like other bipyramids, the pentagonal bipyramid can be constructed by attaching the base of two pentagonal pyramids.[1] These pyramids cover their pentagonal base, such that the resulting polyhedron has 10 triangles as its faces.[2] The pentagonal bipyramid is said to be right if the pyramids are symmetrically regular and both of their apices are on the line passing through the base's center; otherwise, it is oblique. If the pyramids are regular, then all edges of the triangular bipyramid are equal in length, making up the faces equilateral triangles. A polyhedron with only equilateral triangles as faces is called a deltahedron.[3] There are only eight different convex deltahedra, one of which is the pentagonal bipyramid with regular faces. More generally, the convex polyhedron in which all faces are regular is the Johnson solid, and every convex deltahedra is a Johnson solid. The pentagonal bipyramid with the regular faces is among numbered the Johnson solids as  J 13 {\displaystyle J_{13}} , the thirteenth Johnson solid.[4]

## Properties[edit]

A pentagonal bipyramid's surface area is 10 times that of all triangles. In the case of edge length  a {\displaystyle a} , its surface area is:[2]

5 3 2 a 2 ≈ 4.3301 a 2 . {\displaystyle {\frac {5{\sqrt {3}}}{2}}a^{2}\approx 4.3301a^{2}.}

Its volume can be calculated by slicing it into two pentagonal pyramids and adding their volume. In the case of edge length  a {\displaystyle a} , this is:[2]

1 12 ( 5 + 5 ) a 3 ≈ 0.603 a 3 . {\displaystyle {\frac {1}{12}}\left(5+{\sqrt {5}}\right)a^{3}\approx 0.603a^{3}.}

The pentagonal bipyramid has three-dimensional symmetry group of dihedral group D 5 h {\displaystyle D_{5h}} of order 20: the appearance is symmetrical by rotating around the axis of symmetry that passing through apices and base's center vertically, and it has mirror symmetry relative to any bisector of the base; it is also symmetrical by reflecting it across a horizontal plane. The dihedral angle of a pentagonal bipyramid with regular faces can be calculated by adding the angle of pentagonal pyramids. The dihedral angle of a pentagonal pyramid between two adjacent triangles is approximately  138.2 ∘ {\displaystyle 138.2^{\circ }} , and that between the triangular face and the base is  37.4 ∘ {\displaystyle 37.4^{\circ }} . Therefore, the dihedral angle of a pentagonal pyramid with regular faces between two adjacent triangular faces, on the edge where two pyramids are attached, is  74.8 ∘ {\displaystyle 74.8^{\circ }} .[5]

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.[6]

The dual polyhedron of a pentagonal bipyramid is the pentagonal prism. This prism has 7 faces: 2 pentagonal faces are the base, and the rest are 5 rectangular faces. More generally, the dual polyhedron of every bipyramid is the prism.

## See also[edit]

* Pentagonal bipyramidal molecular geometry

## References[edit]

1. ^ Rajwade, A. R. (2001), Convex Polyhedra with Regularity Conditions and Hilbert's Third Problem, Texts and Readings in Mathematics, Hindustan Book Agency, p. 84, doi:10.1007/978-93-86279-06-4, ISBN 978-93-86279-06-4.
2. ^ a b c Berman, Martin (1971), "Regular-faced convex polyhedra", Journal of the Franklin Institute, 291 (5): 329–352, doi:10.1016/0016-0032(71)90071-8, MR 0290245.
3. ^ Trigg, Charles W. (1978), "An infinite class of deltahedra", Mathematics Magazine, 51 (1): 55–57, doi:10.1080/0025570X.1978.11976675, JSTOR 2689647, MR 1572246.
4. ^ Uehara, Ryuhei (2020), Introduction to Computational Origami: The World of New Computational Geometry, Springer, doi:10.1007/978-981-15-4470-5, ISBN 978-981-15-4470-5, S2CID 220150682.
5. ^ Johnson, Norman W. (1966), "Convex polyhedra with regular faces", Canadian Journal of Mathematics, 18: 169–200, doi:10.4153/cjm-1966-021-8, MR 0185507, S2CID 122006114, Zbl 0132.14603.
6. ^ Finbow, Arthur S.; Hartnell, Bert L.; Nowakowski, Richard J.; Plummer, Michael D. (2010), "On well-covered triangulations. III", Discrete Applied Mathematics, 158 (8): 894–912, doi:10.1016/j.dam.2009.08.002, MR 2602814.

## External links[edit]

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

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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
* 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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