Pentagonal bipyramid

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.

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

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

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

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

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

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

See also

• Pentagonal bipyramidal molecular geometry

"Regular" right (symmetric) n-gonal bipyramids:

Name	Digonal bipyramid	Triangular bipyramid (J12)	Square bipyramid (O)	Pentagonal bipyramid (J13)	Hexagonal bipyramid	Heptagonal bipyramid	Octagonal bipyramid	Enneagonal bipyramid	Decagonal bipyramid	...	Apeirogonal bipyramid
Polyhedron image										...
Spherical tiling image										Plane tiling image
Face configuration	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										...

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, 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.
3. 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