Pentagonal Bipyramid
Get Pentagonal Bipyramid essential facts below. View Videos or join the Pentagonal Bipyramid discussion. Add Pentagonal Bipyramid to your PopFlock.com topic list for future reference or share this resource on social media.
Pentagonal Bipyramid
Pentagonal bipyramid
Pentagonale bipiramide.png
TypeBipyramid
and
Johnson
J12 - J13 - J14
Faces10 triangles
Edges15
Vertices7
Schläfli symbol{ } + {5}
Coxeter diagramCDel node f1.pngCDel 2x.pngCDel node f1.pngCDel 5.pngCDel node.png
Symmetry groupD5h, [5,2], (*225), order 20
Rotation groupD5, [5,2]+, (225), order 10
Dual polyhedronpentagonal prism
Face configurationV4.4.5
Propertiesconvex, face-transitive, (deltahedron)
Johnson solid J
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]

Pentagonal dipyramid.png

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 (), surface area () and volume () can be used if all faces are regular, with edge length :[3]

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}:

Snub rectified pentagonal bipyramid sequence.png

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
Dual pentagonal dipyramid.png Dual pentagonal dipyramid net.png

See also

"Regular" right (symmetric) n-gonal bipyramids:
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 Triangular bipyramid.png Square bipyramid.png Pentagonale bipiramide.png Hexagonale bipiramide.png Heptagonal bipyramid.png Octagonal bipyramid.png Enneagonal bipyramid.png Decagonal bipyramid.png ...
Spherical tiling image Spherical digonal bipyramid.svg Spherical trigonal bipyramid.png Spherical square bipyramid.svg Spherical pentagonal bipyramid.png Spherical hexagonal bipyramid.png Spherical heptagonal bipyramid.png Spherical octagonal bipyramid.png Spherical enneagonal bipyramid.png Spherical decagonal bipyramid.png Plane tiling image Infinite bipyramid.svg
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 CDel node f1.pngCDel 2x.pngCDel node f1.pngCDel 2x.pngCDel node.png CDel node f1.pngCDel 2x.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node f1.pngCDel 2x.pngCDel node f1.pngCDel 4.pngCDel node.png CDel node f1.pngCDel 2x.pngCDel node f1.pngCDel 5.pngCDel node.png CDel node f1.pngCDel 2x.pngCDel node f1.pngCDel 6.pngCDel node.png CDel node f1.pngCDel 2x.pngCDel node f1.pngCDel 7.pngCDel node.png CDel node f1.pngCDel 2x.pngCDel node f1.pngCDel 8.pngCDel node.png CDel node f1.pngCDel 2x.pngCDel node f1.pngCDel 9.pngCDel node.png CDel node f1.pngCDel 2x.pngCDel node f1.pngCDel 10.pngCDel node.png ... CDel node f1.pngCDel 2x.pngCDel node f1.pngCDel infin.pngCDel node.png

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 J". Problemas y ecuaciones (in Spanish). ISSN 2659-9899. Retrieved .

External links


  This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

Pentagonal_bipyramid
 



 



 
Music Scenes