Editing Bipyramid (section)

== Definition and properties ==
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 | footer = The [[triangular bipyramid]], [[square bipyramid]], and [[pentagonal bipyramid]].
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A bipyramid is a polyhedron constructed by fusing two [[Pyramid (geometry)|pyramids]] which share the same [[polygon]]al [[base (geometry)|base]];{{r|aarts}} a pyramid is in turn constructed by connecting each vertex of its base to a single new [[vertex (geometry)|vertex]] (the [[apex (geometry)|apex]]) not lying in the plane of the base, for an {{nowrap|1={{mvar|n}}-}}gonal base forming {{mvar|n}} triangular faces in addition to the base face. An {{nowrap|1={{mvar|n}}-}}gonal bipyramid thus has {{math|2''n''}} faces, {{math|3''n''}} edges, and {{math|''n'' + 2}} vertices. {{anchor|1=Right and oblique bipyramid}}More generally, a right pyramid is a pyramid where the apices are on the perpendicular line through the [[centroid]] of an arbitrary polygon or the [[incenter]] of a [[tangential polygon]], depending on the source.{{efn|name=right pyramids}} Likewise, a ''right bipyramid'' is a polyhedron constructed by attaching two symmetrical right bipyramid bases; bipyramids whose apices are not on this line are called ''oblique bipyramids''.{{r|polya}}

When the two pyramids are mirror images, the bipyramid is called ''symmetric''. It is called ''regular'' if its base is a [[regular polygon]].{{r|aarts}} When the base is a regular polygon and the apices are on the perpendicular line through its center (a ''regular right bipyramid'') then all of its faces are [[isosceles triangle]]s; sometimes the name ''bipyramid'' refers specifically to symmetric regular right bipyramids,{{r|montroll}} Examples of such bipyramids are the [[triangular bipyramid]], [[octahedron]] (square bipyramid) and [[pentagonal bipyramid]]. If all their edges are equal in length, these shapes consist of [[equilateral triangle]] faces, making them [[deltahedron|deltahedra]];{{r|trigg|uehara}} the triangular bipyramid and the pentagonal bipyramid are [[Johnson solid]]s, and the regular octahedron is a [[Platonic solid]].{{r|cromwell}}

[[File:Dual Cube-Octahedron.svg|thumb|180px|The octahedron is dual to the cube]]
The symmetric regular right bipyramids have [[prismatic symmetry]], with [[dihedral group|dihedral symmetry group]] {{math|D<sub>''nh''</sub>}} of order {{math|4''n''}}: they are unchanged when rotated {{math|{{sfrac|1|''n''}}}} of a turn around the [[axis of symmetry]], reflected across any plane passing through both apices and a base vertex or both apices and the center of a base edge, or reflected across the mirror plane.{{r|fsz}} Because their faces are transitive under these symmetry transformations, they are [[Isohedral figure|isohedral]].{{r|cpsb|mclean}} They are the [[dual polyhedron|dual polyhedra]] of [[Prism (geometry)|prisms]] and the prisms are the dual of bipyramids as well; the bipyramids vertices correspond to the faces of the prism, and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other, and vice versa.{{r|sibley}} The prisms share the same symmetry as the bipyramids.{{r|king}} The [[regular octahedron]] is more symmetric still, as its base vertices and apices are indistinguishable and can be exchanged by reflections or [[Rotation (mathematics)|rotations]]; the regular octahedron and its dual, the [[cube]], have [[octahedral symmetry]].{{r|armstrong}}

The [[volume]] of a symmetric bipyramid is
<math display=block> \frac{2}{3}Bh, </math>
where {{mvar|B}} is the area of the base and {{mvar|h}} the perpendicular distance from the base plane to either apex. In the case of a regular {{nowrap|1={{mvar|n}}-}}sided polygon with side length {{mvar|s}} and whose altitude is {{mvar|h}}, the volume of such a bipyramid is:
<math display=block> \frac{n}{6}hs^2 \cot \frac{\pi}{n}. </math>