Bipyramid

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For Dipyramid, see mountain and Dipyramid (Alaska).
Set of regular right bipyramids
hexagonal bipyramid
(Example hexagonal form)
Coxeter diagram CDel node f1.pngCDel 2x.pngCDel node f1.pngCDel n.pngCDel node.png
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 n-gonal bipyramid net, in this example a pentagonal bipyramid
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[edit]

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.

Concave quadrilateral bipyramid.png

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.

Volume[edit]

The volume of a bipyramid is \scriptstyle{V =} \tfrac{2}{3} \scriptstyle{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:

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

Equilateral triangle bipyramids[edit]

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 dipyramid.png Octahedron.svg Pentagonal dipyramid.png
Triangular bipyramid Square bipyramid
(Octahedron)
Pentagonal bipyramid

Kalidescopic symmetry[edit]

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, [n,2], (*n22), order 4n. The reflection domains can be shown as alternately colored triangles as mirror images.

D1h D2h D3h D4h D5h D6h ...
Spherical digonal bipyramid2.png Spherical square bipyramid2.png Spherical hexagonal bipyramid2.png Spherical octagonal bipyramid2.png Spherical decagonal bipyramid2.png Spherical dodecagonal bipyramid2.png

Right regular bipyramids[edit]

Family of bipyramids
Polyhedron Triangular bipyramid.png Square bipyramid.png Pentagonale bipiramide.png Hexagonale bipiramide.png Heptagonal bipyramid.png Octagonal bipyramid.png Enneagonal bipyramid.png Decagonal bipyramid.png
Tiling Spherical digonal bipyramid.png Spherical trigonal bipyramid.png Spherical square bipyramid.png Spherical pentagonal bipyramid.png Spherical hexagonal bipyramid.png Spherical heptagonal bipyramid.png Spherical octagonal bipyramid.png Spherical enneagonal bipyramid.png Spherical decagonal bipyramid.png E2 tiling 22i-2 dual.png
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 ...V∞.4.4

Scalenohedron[edit]

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, [2+,2n], (2*n) , order 2n. In crystallography, 8-sided and 12-sided scalenohedra exist.[1] All of these forms are isohedrons.

The second has symmetry Dn, [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=0.0, and becomes a disphenoid with merged coplanar faces at z=1.0. Beyond 1, it becomes concave.

4-scalenohedron geometric variations
z=0.10 z=0.25 z=0.50 z=0.95 z=1.5
4-scalenohedron-01.png 4-scalenohedron-025.png 4-scalenohedron-05.png 4-scalenohedron-095.png 4-scalenohedron-15.png

Star bipyramids[edit]

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 CDel node f1.pngCDel 2x.pngCDel node f1.pngCDel p.pngCDel rat.pngCDel q.pngCDel node.png.

5/2 7/2 7/3 8/3 9/2 9/4 10/3 11/2 11/3 11/4 11/5 12/5
Pentagram Dipyramid.png
CDel node f1.pngCDel 2x.pngCDel node f1.pngCDel 5.pngCDel rat.pngCDel 2x.pngCDel node.png
7-2 dipyramid.png
CDel node f1.pngCDel 2x.pngCDel node f1.pngCDel 7.pngCDel rat.pngCDel 2x.pngCDel node.png
7-3 dipyramid.png
CDel node f1.pngCDel 2x.pngCDel node f1.pngCDel 7.pngCDel rat.pngCDel 3x.pngCDel node.png
8-3 dipyramid.png
CDel node f1.pngCDel 2x.pngCDel node f1.pngCDel 8.pngCDel rat.pngCDel 3x.pngCDel node.png
9-2 dipyramid.png
CDel node f1.pngCDel 2x.pngCDel node f1.pngCDel 9.pngCDel rat.pngCDel 2x.pngCDel node.png
9-4 dipyramid.png
CDel node f1.pngCDel 2x.pngCDel node f1.pngCDel 9.pngCDel rat.pngCDel 4.pngCDel node.png
10-3 dipyramid.png
CDel node f1.pngCDel 2x.pngCDel node f1.pngCDel 10.pngCDel rat.pngCDel 3x.pngCDel node.png
11-2 dipyramid.png
CDel node f1.pngCDel 2x.pngCDel node f1.pngCDel 11.pngCDel rat.pngCDel 2x.pngCDel node.png
11-3 dipyramid.png
CDel node f1.pngCDel 2x.pngCDel node f1.pngCDel 11.pngCDel rat.pngCDel 3x.pngCDel node.png
11-4 dipyramid.png
CDel node f1.pngCDel 2x.pngCDel node f1.pngCDel 11.pngCDel rat.pngCDel 4.pngCDel node.png
11-5 dipyramid.png
CDel node f1.pngCDel 2x.pngCDel node f1.pngCDel 11.pngCDel rat.pngCDel 5.pngCDel node.png
12-5 dipyramid.png
CDel node f1.pngCDel 2x.pngCDel node f1.pngCDel 12.pngCDel rat.pngCDel 5.pngCDel node.png

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
8-3 dipyramid.png 8-3-bipyramid zigzag.png 8-3-bipyramid-inout.png 8-3-dipyramid zigzag inout.png

4-polytopes with bipyramid cells[edit]

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 VA vertices where the apices of NA bipyramids meet. It will have VE vertices where the type E vertices of NE bipyramids meet. NAE bipyramids meet along each type AE edge. NEE bipyramids meet along each type EE edge. CAE is the cosine of the dihedral angle along an AE edge. CEE is the cosine of the dihedral angle along an EE edge. As cells must fit around an edge, NAA cos−1(CAA) ≤ 2π, NAE cos−1(CAE) ≤ 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 CDel node.pngCDel 3.pngCDel node f1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png 10 5 5 4 6 3 3 Triangular bipyramid CDel node f1.pngCDel 2x.pngCDel node f1.pngCDel 3.pngCDel node.png \scriptstyle \frac{2}{3} 0.667 \scriptstyle - \frac{1}{7} \scriptstyle - \frac{1}{7}
Rectified tesseract CDel node.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png 32 16 8 4 12 3 4 Triangular bipyramid CDel node f1.pngCDel 2x.pngCDel node f1.pngCDel 3.pngCDel node.png \scriptstyle  \frac{\sqrt{2}}{3} 0.624 \scriptstyle  - \frac{2}{5} \scriptstyle  \frac{1}{5}
Rectified 24-cell CDel node.pngCDel 3.pngCDel node f1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png 96 24 24 8 12 4 3 Triangular bipyramid CDel node f1.pngCDel 2x.pngCDel node f1.pngCDel 3.pngCDel node.png \scriptstyle   \frac{2\sqrt{2}}{3} 0.745 \scriptstyle  \frac{1}{11} \scriptstyle  - \frac{5}{11}
Rectified 120-cell CDel node.pngCDel 5.pngCDel node f1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png 1200 600 120 4 30 3 5 Triangular bipyramid CDel node f1.pngCDel 2x.pngCDel node f1.pngCDel 3.pngCDel node.png \scriptstyle   \frac{\sqrt{5} - 1 }{3} 0.613 \scriptstyle  - \frac{10+9\sqrt{5}}{61} \scriptstyle   \frac{12\sqrt{5} - 7}{61}
Rectified 16-cell CDel node.pngCDel 3.pngCDel node f1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png 24* 8 16 6 6 3 3 Square bipyramid CDel node f1.pngCDel 2x.pngCDel node f1.pngCDel 4.pngCDel node.png \scriptstyle \sqrt{2} 1 \scriptstyle - \frac{1}{3} \scriptstyle - \frac{1}{3}
Rectified cubic honeycomb CDel node.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png 6 12 3 4 Square bipyramid CDel node f1.pngCDel 2x.pngCDel node f1.pngCDel 4.pngCDel node.png 1 0.866 \scriptstyle - \frac{1}{2} 0
Rectified 600-cell CDel node.pngCDel 3.pngCDel node f1.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png 720 120 600 12 6 3 3 Pentagonal bipyramid CDel node f1.pngCDel 2x.pngCDel node f1.pngCDel 5.pngCDel node.png \scriptstyle   \frac{5+3\sqrt{5}}{5} 1.447 \scriptstyle  - \frac{11+4\sqrt{5}}{41} \scriptstyle   - \frac{11+4\sqrt{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[edit]

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.

See also[edit]

References[edit]

  • 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[edit]