# Bipyramid

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 2*n* triangle faces, 3*n* edges, and 2 + *n* vertices.

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 | n-gonal prism |

Properties | convex, face-transitive |

Net |

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.

## Right, oblique and concave bipyramids

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*}.

A **concave bipyramid** has a concave interior polygon.

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 D_{nh}. Indeed, an n-tonal bipyramid can be seen as the Kleetope of the respective n-gonal dihedron.

## Volume

The volume of a bipyramid is *V* =2/3*Bh* 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 *n*-sided polygon with side length *s* and whose height is *h* is therefore:

## 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 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 symmetry

If the base is regular and the line through the apexes intersects the base at its center, the symmetry group of the *n*-gonal bipyramid has dihedral symmetry D_{nh} of order 4*n*, 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 2*n*, 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 2*n*-bipyramid represents the fundamental domains of dihedral symmetry in three dimensions: D_{nh}, [*n*,2], (**n*22), order 4*n*. 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 bipyramids

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 bipyramids

An **asymmetric right bipyramid** joins two unequal height pyramids. An *inverted* form can also have both pyramids on the same side. A regular *n*-gonal asymmetry right pyramid has symmetry C_{n}v, order 2*n*. The dual polyhedron of an asymmetric bipyramid is a frustum.

Asymmetric | Inverted |
---|---|

## Scalenohedron

A **scalenohedron** is topologically identical to a 2*n*-bipyramid, but contains congruent scalene triangles.[2]

There are two types. In one type the 2*n* vertices around the center alternate in rings above and below the center. In the other type, the 2*n* 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 D_{nd}, [2^{+},2*n*], (2**n*), order 2*n*. In crystallography, 8-sided and 12-sided scalenohedra exist.[3] All of these forms are isohedra.

The second has symmetry D_{n}, [2,*n*], (**nn*2), order 2*n*.

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 bipyramids

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

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 even-sided stars can also be made with zig-zag offplane vertices, in-out isotoxal forms, or both, like this {8/3} form:

Regular | Zig-zag regular | Isotoxal | Zig-zag isotoxal |
---|---|---|---|

## 4-polytopes with bipyramid cells

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

4-polytope 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 5-cell | 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 24-cell | 96 | 24 | 24 | 8 | 12 | 4 | 3 | Triangular bipyramid | 2√2/3 | 0.745 | 1/11 | −5/11 | ||

Rectified 120-cell | 1200 | 600 | 120 | 4 | 30 | 3 | 5 | Triangular bipyramid | √5 − 1/3 | 0.613 | −10 + 9√5/61 | 12√5 − 7/61 | ||

Rectified 16-cell | 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 600-cell | 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 16-cell is the regular 24-cell and vertices are all equivalent – octahedra are regular bipyramids.
- ** Given numerically due to more complex form.

## Higher dimensions

In general, a *bipyramid* can be seen as an *n*-polytope 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 16-cell, which is an octahedral bipyramid, and more generally an *n*-orthoplex is an (*n* − 1)-orthoplex bypyramid.

A two-dimensional bipyramid is a square.

## See also

## References

- N.W. Johnson:
*Geometries and Transformations*, (2018) ISBN 978-1-107-10340-5 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 2017-09-16.

## Bibliography

- Anthony Pugh (1976).
*Polyhedra: A visual approach*. California: University of California Press Berkeley. ISBN 0-520-03056-7. Chapter 4: Duals of the Archimedean polyhedra, prisma and antiprisms

## External links

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