An n-gonal bipyramid or dipyramid is a polyhedron formed by joining an n-gonal pyramid and its mirror image base-to-base. An n-gonal bipyramid has 2n triangle faces, 3n edges, and 2+n vertices.
The referenced n-gon 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.
A right bipyramid has two points above and below the centroid of its base. Nonright bipyramids are called oblique bipyramids. A regular bipyramid has a regular polygon internal face and is usually implied to be a right bipyramid. A right bipyramid can be represented as { }+P for internal polygon P, and a regular n-bipyramid { } + {n}.
The face-transitive regular 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 Dnh.
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.
Only three kinds of bipyramids can have all edges of the same length : 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.
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.
The digonal faces of a spherical 2n-bipyramid represents the fundamental domains of dihedral symmetry in three dimensions: Dnh,,, order 4n. The reflection domains can be shown as alternately colored triangles as mirror images.
A scalenohedron is topologically identical to a 2n-bipyramid, but contains congruent scalene triangles.
There are two types. In one type the 2n vertices around the center alternate in rings above and below the center. In the other type, the 2n vertices are on the same plane, but alternate in two radii.
The first has 2-fold rotation axes mid-edge around the sides, reflection planes through the vertices, and n-fold rotation symmetry on its axis, representing symmetry Dnd,,, order 2n. In crystallography, 8-sided and 12-sided scalenohedra exist. All of these forms are isohedrons.
The smallest scalenohedron has 8 faces and is topologically identical to the regular octahedron. The second type is a rhombic bipyramid. The first type has 6 vertices can be represented as,,, where z is a parameter between 0 and 1, creating a regular octahedron at z=0.0, and becomes a disphenoid with merged coplanar faces at z=1.0. Beyond 1, it becomes concave.
Self-intersecting bipyramids exist with a star polygon central figure, defined by triangular faces connecting each polygon edge to these two points. A {p/q} bipyramid has Coxeter diagram.
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