Gourt Home::Gourt :: Bipyramid
| Set of bipyramids | |
|---|---|
| Faces | 2n triangles |
| Edges | 3n |
| Vertices | n+2 |
| Face configuration | V4.4.n |
| Symmetry group | Dnh |
| Dual polyhedron | Prisms |
| Properties | convex, face-uniform |
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.
Bipyramid faces, projected as spherical triangles, represent the fundamental domains in the dihedral symmetry Dnh.
Forms
- Triangular bipyramid - 6 faces - dual triangular prism
- Tetragonal bipyramid (octahedron is a special case) - 8 faces - dual cube
- Pentagonal bipyramid - 10 faces - dual pentagonal prism
- Hexagonal bipyramid - 12 faces - dual hexagonal prism
- Heptagonal bipyramid - 14 faces - dual heptagonal prism
- Octagonal bipyramid - 16 faces - dual octagonal prism
- Enneagonal bipyramid - 18 faces - dual enneagonal prism
- Decagonal bipyramid - 20 faces - dual decagonal prism
- ...n-agonal bipyramid - 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
"Bipyramid"