Pentagonal bipyramid

Pentagonal bipyramid
Pentagonal bipyramid
Type	Bipyramid; and; Johnson; J12 - J13 - J14
Faces	10 triangles
Edges	15
Vertices	7
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
Face configuration	V4.4.5
Properties	convex, face-transitive, (deltahedron)

Johnson solid J₁₃

net

In geometry, the pentagonal bipyramid (or dipyramid) is third of the infinite set of face-transitive bipyramids. 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.

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 ( $H$ ), surface area ( $A$ ) and volume ( $V$ ) can be used if all faces are regular, with edge length $L$ :^[3]

H=L\cdot {\sqrt {2-{\frac {2}{\sqrt {5}}}}}\approx L\cdot 1.0514622242

A=L^{2}\cdot {\frac {5{\sqrt {3}}}{2}}\approx L^{2}\cdot 4.330127019

V=L^{3}\cdot {\frac {5+{\sqrt {5}}}{12}}\approx L^{3}\cdot 0.6030056648

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

References

^ 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, Zbl 0132.14603.
^ 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.
^ Sapiña, R. "Area and volume of the Johnson solid J₁₃". Problemas y ecuaciones (in Spanish). ISSN 2659-9899. Retrieved 2020-09-04.

External links

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

[1] 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, Zbl 0132.14603.

[2] 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.

[pye-3] Sapiña, R. "Area and volume of the Johnson solid J₁₃". Problemas y ecuaciones (in Spanish). ISSN 2659-9899. Retrieved 2020-09-04.

[1]

[2]

[3]

Bipyramid name	Digonal bipyramid	Triangular bipyramid (See: J₁₂)	Square bipyramid (See: O)	Pentagonal bipyramid (See: J₁₃)	Hexagonal bipyramid	Heptagonal bipyramid	Octagonal bipyramid	Enneagonal bipyramid	Decagonal bipyramid	...	Apeirogonal bipyramid
Polyhedron image										...
Spherical tiling image										Plane tiling image
Face config.	V2.4.4	V3.4.4	V4.4.4	V5.4.4	V6.4.4	V7.4.4	V8.4.4	V9.4.4	V10.4.4	...	V∞.4.4
Coxeter diagram										...

v t 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)

Pentagonal bipyramid

Type	Bipyramid and Johnson J₁₂ - J₁₃ - J₁₄
Faces	10 triangles
Edges	15
Vertices	7
Schläfli symbol	{ } + {5}
Coxeter diagram
Symmetry group	D_5h, [5,2], (*225), order 20
Rotation group	D₅, [5,2]⁺, (225), order 10
Dual polyhedron	pentagonal prism
Face configuration	V4.4.5
Properties	convex, face-transitive, (deltahedron)