Bipyramid

For Dipyramid, see mountain and Dipyramid (Alaska).

Set of regular right bipyramids
(Example hexagonal form)
Coxeter diagram
Schläfli symbol	{ } + {n}
Faces	2n triangles
Edges	3n
Vertices	2 + n
Face configuration	V4.4.n
Symmetry group	Dnh, [n,2], (*n22), order 4n
Rotation group	Dn, [n,2]+, (n22), order 2n
Dual polyhedron	n-gonal prism
Properties	convex, face-transitive
Net

A bipyramid made with straws and elastics. An extra axial straw is added which doesn't exist in the simple polyhedron

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.

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 =⁠2/3⁠Bh 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 n-sided polygon with side length s and whose height is h is therefore:

$V={\frac {n}{6}}hs^{2}\cot {\frac {\pi }{n}}.$

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 (J₁₂ and J₁₃).

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 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 D₄ as subgroups.

The digonal faces of a spherical 2n-bipyramid 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.

D1h	D2h	D3h	D4h	D5h	D6h	...

Right regular bipyramids

Regular right symmetric n-gonal bipyramids:

Bipyramid name	Digonal bipyramid	Triangular bipyramid	Square bipyramid	Pentagonal bipyramid	Hexagonal bipyramid	...	Apeirogonal bipyramid
Polyhedron image						...
Spherical tiling image						Plane tiling image
Face config.	V2.4.4	V3.4.4	V4.4.4	V5.4.4	V6.4.4	...	V∞.4.4
Coxeter diagram						...

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_nv, order 2n. The dual polyhedron of an asymmetric bipyramid is a frustum.

Example hexagonal forms

Asymmetric	Inverted

Scalenohedron

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 D_nd, [2⁺,2n], (2*n), order 2n. In crystallography, 8-sided and 12-sided scalenohedra exist. 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.

4-scalenohedron geometric variations

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	VA	VE	NA	NE	NAE	NEE	Cell	Coxeter diagram	AA	AE**	CAE	CEE
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 pyramidal 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

• Trapezohedron

References

General references

• 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

• Weisstein, Eric W. "Dipyramid". MathWorld.
• Weisstein, Eric W. "Isohedron". MathWorld.
• The Uniform Polyhedra
• Virtual Reality Polyhedra The Encyclopedia of Polyhedra
  • VRML models (George Hart) <3> <4> <5> <6> <7> <8> <9> <10>
    • Conway Notation for Polyhedra Try: "dPn", where n = 3, 4, 5, 6, ... example "dP4" is an octahedron.