Bipyramid

From Wikipedia, the free encyclopedia

Set of bipyramids
(Example hexagonal form)
Faces	2n triangles
Edges	3n
Vertices	n+2
Face configuration	V4.4.n
Symmetry group	D_nh
Dual polyhedron	Prisms
Properties	convex, face-transitive

Regular octahedron

triangular dipyramid J12

Pentagonal dipyramid J13

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-transitive 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 D_nh.

[edit] 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

[edit] 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 D_nh of order 4n, except in the case of a regular octahedron, which has the larger octahedral symmetry group O_h of order 48, which has three versions of D_4h as subgroups. The rotation group is D_n 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.