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.

[2] When the two pyramids are mirror images, the bipyramid is called symmetric.

If all their edges are equal in length, these shapes consist of equilateral triangle faces, making them deltahedra;[4][5] the triangular bipyramid and the pentagonal bipyramid are Johnson solids, and the regular octahedron is a Platonic solid.

[6] The symmetric regular right bipyramids have prismatic symmetry, with dihedral symmetry group Dnh of order 4n: they are unchanged when rotated ⁠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.

[8][9] They are the dual polyhedra of 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.

[11] The regular octahedron is more symmetric still, as its base vertices and apices are indistinguishable and can be exchanged by reflections or rotations; the regular octahedron and its dual, the cube, have octahedral symmetry.

In the case of a regular n-sided polygon with side length s and whose altitude is h, the volume of such a bipyramid is:

A regular asymmetric right n-gonal bipyramid has symmetry group Cnv, of order 2n.

An isotoxal right (symmetric) di-n-gonal bipyramid is a right (symmetric) 2n-gonal bipyramid with an isotoxal flat polygon base: its 2n basal vertices are coplanar, but alternate in two radii.

It can be seen as another type of a right symmetric di-n-gonal scalenohedron, with an isotoxal flat polygon base.

An isotoxal right (symmetric) di-n-gonal bipyramid has n two-fold rotation axes through opposite basal vertices, n reflection planes through opposite apical edges, an n-fold rotation axis through apices, a reflection plane through base, and an n-fold rotation-reflection axis through apices,[13] representing symmetry group Dnh, [n,2], (*22n), of order 4n.

[13][16] A scalenohedron is similar to a bipyramid; the difference is that the scalenohedra has a zig-zag pattern in the middle edges.

[17] It has two apices and 2n basal vertices, 4n faces, and 6n edges; it is topologically identical to a 2n-gonal bipyramid, but its 2n basal vertices alternate in two rings above and below the center.

It can be seen as another type of a right symmetric di-n-gonal bipyramid, with a regular zigzag skew polygon base.

A regular right symmetric di-n-gonal scalenohedron has n two-fold rotation axes through opposite basal mid-edges, n reflection planes through opposite apical edges, an n-fold rotation axis through apices, and a 2n-fold rotation-reflection axis through apices (about which 1n rotations-reflections globally preserve the solid),[13] representing symmetry group Dnv = Dnd, [2+,2n], (2*n), of order 4n.

(If n is odd, then there is an inversion symmetry about the center, corresponding to the 180° rotation-reflection.)

Double example: In crystallography, regular right symmetric didigonal (8-faced) and ditrigonal (12-faced) scalenohedra exist.

[13][16] The smallest geometric scalenohedra have eight faces, and are topologically identical to the regular octahedron.

At z = 0, it is a regular octahedron; at z = 1, it has four pairs of coplanar faces, and merging these into four congruent isosceles triangles makes it a disphenoid; for z > 1, it is concave.

Example with five different edge lengths: For some particular values of zA = |zA'|, half the faces of such a scalenohedron may be isosceles or equilateral.

[20] A regular right symmetric star bipyramid has congruent isosceles triangle faces, and is isohedral.

The dual of the rectification of each convex regular 4-polytopes is a cell-transitive 4-polytope with bipyramidal cells.

A generalized n-dimensional "bipyramid" is any n-polytope constructed from an (n − 1)-polytope base lying in a hyperplane, with every base vertex connected by an edge to two apex vertices.

If the (n − 1)-polytope is a regular polytope and the apices are equidistant from its center along the line perpendicular to the base hyperplane, it will have identical pyramidal facets.