A (symmetric) -gonal bipyramid or dipyramid is a polyhedron formed by joining an -gonal pyramid and its mirror image base-to-base. An -gonal bipyramid has triangle faces, edges, and vertices.
The "-gonal" in the name of a bipyramid does not refer to a face but to the internal polygon base, lying in the mirror plane that connects the two pyramid halves. (If it were a face, then each of its edges would connect three faces instead of two.)
"Regular", right bipyramids
A "regular" bipyramid has a regular polygon base. It is usually implied to be also a right bipyramid.
A right bipyramid has its two apices right above and right below the center or the centroid of its polygon base.
A "regular" right (symmetric) -gonal bipyramid has Schläfli symbol }.
A right (symmetric) bipyramid has Schläfli symbol , for polygon base .
The "regular" right (thus face-transitive) -gonal bipyramid with regular vertices is the dual of the -gonal uniform (thus right) prism, and has congruent isosceles triangle faces.
A "regular" right (symmetric) -gonal bipyramid can be projected on a sphere or globe as a "regular" right (symmetric) -gonal spherical bipyramid: equally spaced lines of longitude going from pole to pole, and an equator line bisecting them.
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 "regular" right (symmetric) triangular, tetragonal, and pentagonal bipyramids. The tetragonal or square bipyramid with same length edges, or regular octahedron, counts among the Platonic solids; the triangular and pentagonal bipyramids with same length edges count among the Johnson solids and .
Kaleidoscopic symmetry
A "regular" right (symmetric) -gonal bipyramid has dihedral symmetry group , of order , except in the case of a regular octahedron, which has the larger octahedral symmetry group , of order , which has three versions of as subgroups. The rotation group is , of order , except in the case of a regular octahedron, which has the larger rotation group , of order , which has three versions of as subgroups.
Note: Every "regular" right (symmetric) -gonal bipyramid has the same (dihedral) symmetry group as the dual-uniform -gonal bipyramid, for .
The triangle faces of a "regular" right (symmetric) -gonal bipyramid, projected as the spherical triangle faces of a "regular" right (symmetric) -gonal spherical bipyramid, represent the fundamental domains of dihedral symmetry in three dimensions: , [], (), of order . These domains can be shown as alternately colored spherical triangles: across a reflection plane through cocyclic edges, mirror image domains are in different colors (indirect isometry); about an -fold or a -fold rotation axis through opposite vertices, a domain and its image are in the same color (direct isometry). An -gonal (symmetric) bipyramid can be seen as the Kleetope of the "corresponding" -gonal dihedron.
Volume
Volume of a (symmetric) bipyramid:
where is the area of the base and the height from the base plane to any apex.
This works for any shape of the base, and for any location of the apices, provided that is measured as the perpendicular distance from the base plane to any apex. Hence:
Volume of a (symmetric) bipyramid whose base is a regular -sided polygon with side length and whose height is :
Oblique bipyramids
Non-right bipyramids are called oblique bipyramids.
Concave bipyramids
A concave bipyramid has a concave polygon base.
(*) Its base has no obvious center; but if its apices are right above and right below the centroid of its base, then it is a right bipyramid. Anyway, it is a concave octahedron.
Asymmetric/inverted right bipyramids
An asymmetric right bipyramid joins two right pyramids with congruent bases but unequal heights, base-to-base.
An inverted right bipyramid joins two right pyramids with congruent bases but unequal heights, base-to-base, but on the same side of their common base.
The dual of an asymmetric/inverted right -gonal bipyramid is an -gonal frustum.
A "regular" asymmetric/inverted right -gonal bipyramid has symmetry group , of order .
Scalene triangle bipyramids
An "isotoxal" right (symmetric) di--gonal bipyramid is a right (symmetric) -gonal bipyramid with an isotoxal flat polygon base: its basal vertices are coplanar, but alternate in two radii.
All its faces are congruent scalene triangles, and it is isohedral. It can be seen as another type of a right "symmetric" di--gonal scalenohedron, with an isotoxal flat polygon base.