 Bipyramid

For Dipyramid, see mountain and Dipyramid (Alaska).
Set of bipyramids
(Example hexagonal form)Faces 2n triangles Edges 3n Vertices n + 2 Face configuration V4.4.n Symmetry group D_{nh}, [n,2], (*22n) Dual polyhedron ngonal prism Properties convex, facetransitive Net An ngonal bipyramid or dipyramid is a polyhedron formed by joining an ngonal pyramid and its mirror image basetobase.
The referenced ngon 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 facetransitive bipyramids are the dual polyhedra of the uniform prisms and will generally have isosceles triangle faces.
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}.
Contents
Volume
The volume of a bipyramid is where B is the area of the base and h the height from the base to the apex. This works for any location of the apex, provided that h is measured as the perpendicular distance from the plane which contains the base.
The volume of a bipyramid whose base is a regular nsided polygon with side length s and whose height is h is therefore:
Equilateral triangle bipyramids
Only three kinds of bipyramids can have all edges of the same length (which implies that all faces are equilateral triangles, and thus the bipyramid is a deltahedron): the triangular, tetragonal, and pentagonal bipyramids. The tetragonal bipyramid with identical edges, or regular octahedron, counts among the Platonic solids, while the triangular and pentagonal bipyramids with identical edges count among the Johnson solids (J12 and J13).
Triangular bipyramid Square bipyramid
(Octahedron)Pentagonal bipyramid Forms
 Triangular bipyramid  6 faces  dual triangular prism
 Square bipyramid (the regular 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
 ...ngonal bipyramid  2n faces  dual ngonal prism
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10Symmetry groups
If the base is regular and the line through the apexes intersects the base at its center, the symmetry group of the nagonal 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 D_{4} as subgroups.
Star bipyramids
Selfintersecting bipyramids exist with a star polygon central figure, defined by triangular faces connecting each polygon edge to these two points.
For example, a pentagrammic dipyramid is an isohedral star polyhedron composed of 10 intersecting isoceles triangles. It is the dual to the pentagrammic prism.
Higher dimensions
In general, a bipyramid can be seen as an npolytope constructed with a (n1)polytope in a hyperplane with two points in opposite directions, equal distance perpendicular from the hyperplane. If the (n1)polytope is a regular polytope, it will have identical pyramids facets.
See also
External links
 Weisstein, Eric W., "Dipyramid" from MathWorld.
 Olshevsky, George, Bipyramid at Glossary for Hyperspace.
 The Uniform Polyhedra
 Virtual Reality Polyhedra The Encyclopedia of Polyhedra
Polyhedron navigator Platonic solids (regular) Archimedean solids
(Semiregular/Uniform)Catalan solids
(Dual semiregular)triakis tetrahedron · rhombic dodecahedron · triakis octahedron · tetrakis cube · deltoidal icositetrahedron · disdyakis dodecahedron · pentagonal icositetrahedron · rhombic triacontahedron · triakis icosahedron · pentakis dodecahedron · deltoidal hexecontahedron · disdyakis triacontahedron · pentagonal hexecontahedronDihedral regular Dihedral uniform Duals of dihedral uniform bipyramids · trapezohedraDihedral others Degenerate polyhedra are in italics.Categories: Polyhedra
 Pyramids and bipyramids
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