Bipyramid
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| Set of bipyramids | |
|---|---|
| Image:Triangular dipyramid.png | |
| Faces | 2n triangles |
| Edges | 3n |
| Vertices | n+2 |
| Face configuration | V4.4.n |
| Symmetry group | Dnh |
| Dual polyhedron | Prisms |
| Properties | convex, semi-regular (face-uniform) |
Image:Pentagonal dipyramid.png Image:Octahedron.jpg An n-agonal bipyramid or dipyramid is a polyhedron formed by joining an n-agonal pyramid and its mirror image base-to-base.
The referenced n-agon in the name of the bipyramids is not an external face but an internal one, existing on the primary symmetry plane which connects the two pyramid halves.
The face-uniform bipyramids are the dual polyhedra of the uniform prisms and will generally have isosceles triangle faces.
Three bipyramids can be made out of all equilateral triangles, the octahedron (tetragonal bipyramid), which counts among the Platonic solids, and the triangular and pentagonal bipyramids, which count among the Johnson solids.
A bipyramid can be projected on a sphere or globe as n equally spaced lines of longitude going from pole to pole, and bisected by a line around the equator.
Forms
- Triangular dipyramid - 6 faces - dual triangular prism
- Tetragonal dipyramid octahedron - 8 faces - dual cube
- Pentagonal dipyramid - 10 faces - dual pentagonal prism
- Hexagonal dipyramid - 12 faces - dual hexagonal prism
- Septagonal dipyramid - 14 faces - dual septagonal prism
- Octagonal dipyramid - 16 faces - dual octagonal prism
- Enneagonal dipyramid - 18 faces - dual enneagonal prism
- Decagonal dipyramid - 20 faces - dual decagonal prism
- ...n-agonal dipyramid - 2n faces - dual n-agonal prism
Symmetry groups
If the base is regular and the line through the apexes intersects the base at its center, the symmetry group of the n-agonal bipyramid has dihedral symmetry Dnh of order 4n, except in the case of a regular octahedron, which has the larger octahedral symmetry group Oh of order 48, which has three versions of D4h as subgroups. The rotation group is Dn of order 2n, except in the case of a regular octahedron, which has the larger symmetry group O of order 24, which has three versions of D4 as subgroups.
External links
- The Uniform Polyhedra
- Virtual Reality Polyhedra The Encyclopedia of Polyhedra
- VRML models (George Hart) <3> <4> <5> <6> <7> <8> <9> <10>
- Conway Notation for Polyhedra Try: "dPn", where n=3,4,5,6... example "dP4" is an octahedron.Template:Geometry-stub
- VRML models (George Hart) <3> <4> <5> <6> <7> <8> <9> <10>