Bipyramid
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Set of regular right bipyramids  

(Example hexagonal form)  
Coxeter diagram  
Schläfli symbol  { } + {n}^{[1]} 
Faces  2n triangles 
Edges  3n 
Vertices  2 + n 
Face configuration  V4.4.n 
Symmetry group  D_{nh}, [n,2], (*n22), order 4n 
Rotation group  D_{n}, [n,2]^{+}, (n22), order 2n 
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. An ngonal bipyramid has 2n triangle faces, 3n edges, and 2 + n vertices.
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.
Contents
Right, oblique and concave bipyramidsEdit
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 nbipyramid { } + {n}.
A concave bipyramid has a concave interior polygon.
The facetransitive 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 D_{nh}. Indeed, an ntonal bipyramid can be seen as the Kleetope of the respective ngonal dihedron.
VolumeEdit
The volume of a bipyramid is V =2/3Bh 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 bipyramidsEdit
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 (J_{12} and J_{13}).
Triangular bipyramid  Square bipyramid (Octahedron) 
Pentagonal bipyramid 
Kaleidoscopic symmetryEdit
If the base is regular and the line through the apexes intersects the base at its center, the symmetry group of the ngonal 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.
The digonal faces of a spherical 2nbipyramid represents the fundamental domains of dihedral symmetry in three dimensions: D_{nh}, [n,2], (*n22), order 4n. The reflection domains can be shown as alternately colored triangles as mirror images.
D_{1h}  D_{2h}  D_{3h}  D_{4h}  D_{5h}  D_{6h}  ... 

Right regular bipyramidsEdit
Polyhedron  

Coxeter  
Tiling  
Config.  V2.4.4  V3.4.4  V4.4.4  V5.4.4  V6.4.4  V7.4.4  V8.4.4  V9.4.4  V10.4.4 
Asymmetric right bipyramidsEdit
An asymmetric right bipyramid joins two unequal height pyramids. An inverted form can also have both pyramids on the same side. A regular ngonal asymmetry right pyramid has symmetry C_{n}v, order 2n. The dual polyhedron of an asymmetric bipyramid is a frustum.
Asymmetric  Inverted 

ScalenohedronEdit
A scalenohedron is topologically identical to a 2nbipyramid, but contains congruent scalene triangles.^{[2]}
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 2fold rotation axes midedge around the sides, reflection planes through the vertices, and nfold rotation symmetry on its axis, representing symmetry D_{nd}, [2^{+},2n], (2*n), order 2n. In crystallography, 8sided and 12sided scalenohedra exist.^{[3]} All of these forms are isohedra.
The second has symmetry D_{n}, [2,n], (*nn2), order 2n.
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 (0,0,±1), (±1,0,z), (0,±1,−z), where z is a parameter between 0 and 1, creating a regular octahedron at z = 0, and becoming a disphenoid with merged coplanar faces at z = 1. For z > 1, it becomes concave.
z = 0.1  z = 0.25  z = 0.5  z = 0.95  z = 1.5 

Star bipyramidsEdit
Selfintersecting 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 .
5/2  7/2  7/3  8/3  9/2  9/4  10/3  11/2  11/3  11/4  11/5  12/5 













isohedral evensided stars can also be made with zigzag offplane vertices, inout isotoxal forms, or both, like this {8/3} form:
Regular  Zigzag regular  Isotoxal  Zigzag isotoxal 

4polytopes with bipyramid cellsEdit
The dual of the rectification of each convex regular 4polytopes is a celltransitive 4polytope with bipyramidal cells. In the following, the apex vertex of the bipyramid is A and an equator vertex is E. The distance between adjacent vertices on the equator EE = 1, the apex to equator edge is AE and the distance between the apices is AA. The bipyramid 4polytope will have V_{A} vertices where the apices of N_{A} bipyramids meet. It will have V_{E} vertices where the type E vertices of N_{E} bipyramids meet. N_{AE} bipyramids meet along each type AE edge. N_{EE} bipyramids meet along each type EE edge. C_{AE} is the cosine of the dihedral angle along an AE edge. C_{EE} is the cosine of the dihedral angle along an EE edge. As cells must fit around an edge, N_{AA} cos^{−1}(C_{AA}) ≤ 2π, N_{AE} cos^{−1}(C_{AE}) ≤ 2π.
4polytope properties  Bipyramid properties  

Dual of  Coxeter diagram 
Cells  V_{A}  V_{E}  N_{A}  N_{E}  N_{AE}  N_{EE}  Cell  Coxeter diagram 
AA  AE**  C_{AE}  C_{EE} 
Rectified 5cell  10  5  5  4  6  3  3  Triangular bipyramid  2/3  0.667  −1/7  −1/7  
Rectified tesseract  32  16  8  4  12  3  4  Triangular bipyramid  √2/3  0.624  −2/5  −1/5  
Rectified 24cell  96  24  24  8  12  4  3  Triangular bipyramid  2√2/3  0.745  1/11  −5/11  
Rectified 120cell  1200  600  120  4  30  3  5  Triangular bipyramid  √5 − 1/3  0.613  −10 + 9√5/61  12√5 − 7/61  
Rectified 16cell  24*  8  16  6  6  3  3  Square bipyramid  √2  1  −1/3  −1/3  
Rectified cubic honeycomb  ∞  ∞  ∞  6  12  3  4  Square bipyramid  1  0.866  −1/2  0  
Rectified 600cell  720  120  600  12  6  3  3  Pentagonal bipyramid  5 + 3√5/5  1.447  −11 + 4√5/41  −11 + 4√5/41 
 * The rectified 16cell is the regular 24cell and vertices are all equivalent – octahedra are regular bipyramids.
 ** Given numerically due to more complex form.
Higher dimensionsEdit
In general, a bipyramid can be seen as an npolytope constructed with a (n − 1)polytope in a hyperplane with two points in opposite directions, equal distance perpendicular from the hyperplane. If the (n − 1)polytope is a regular polytope, it will have identical pyramids facets. An example is the 16cell, which is an octahedral bipyramid, and more generally an northoplex is an (n − 1)orthoplex bypyramid.
A twodimensional bipyramid is a square.
See alsoEdit
ReferencesEdit
 ^ N.W. Johnson: Geometries and Transformations, (2018) ISBN 9781107103405 Chapter 11: Finite symmetry groups, 11.3 Pyramids, Prisms, and Antiprisms, Figure 11.3c
 ^ The 48 Special Crystal Forms Scalenohedra and Trapezohedra
 ^ "Crystal Form, Zones, Crystal Habit". Tulane.edu. Retrieved 20170916.
BibliographyEdit
 Anthony Pugh (1976). Polyhedra: A visual approach. California: University of California Press Berkeley. ISBN 0520030567. Chapter 4: Duals of the Archimedean polyhedra, prisma and antiprisms
External linksEdit
Wikimedia Commons has media related to Bipyramids. 
 Weisstein, Eric W. "Dipyramid". MathWorld.
 Weisstein, Eric W. "Isohedron". MathWorld.
 The Uniform Polyhedra
 Virtual Reality Polyhedra The Encyclopedia of Polyhedra