Pentagonal bipyramid

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Pentagonal bipyramid
Pentagonale bipiramide.png
Type Bipyramid
and
Johnson
J12 - J13 - J14
Schläfli symbol { } + {5}
Coxeter diagram CDel node f1.pngCDel 2x.pngCDel node f1.pngCDel 5.pngCDel node.png
Faces 10 triangles
Edges 15
Vertices 7
Face configuration V4.4.5
Symmetry group D5h, [5,2], (*225), order 20
Rotation group D5, [5,2]+, (225), order 10
Dual pentagonal prism
Properties convex, face-transitive, (deltahedron)
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[edit]

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 have regular faces but are not uniform (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]

Spherical pentagonal bipyramid

Related polyhedra[edit]

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[edit]

Family of bipyramids
Polyhedron Triangular bipyramid.png Square bipyramid.png Pentagonale bipiramide.png Hexagonale bipiramide.png Heptagonal bipyramid.png Octagonal bipyramid.png Enneagonal bipyramid.png Decagonal bipyramid.png
Coxeter 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
Tiling Spherical digonal bipyramid.png Spherical trigonal bipyramid.png Spherical square bipyramid.png Spherical pentagonal bipyramid.png Spherical hexagonal bipyramid.png Spherical heptagonal bipyramid.png Spherical octagonal bipyramid.png Spherical enneagonal bipyramid.png Spherical decagonal bipyramid.png
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

References[edit]

  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 .

External links[edit]