Bipyramid

In geometry, a bipyramid, dipyramid, or double pyramid is a polyhedron formed by fusing two pyramids together base-to-base. The polygonal base of each pyramid must therefore be the same, and unless otherwise specified the base vertices are usually coplanar and a bipyramid is usually symmetric, meaning the two pyramids are mirror images across their common base plane. When each apex (Template:Plural form, the off-base vertices) of the bipyramid is on a line perpendicular to the base and passing through its center, it is a right bipyramid;Template:Efn otherwise it is oblique. When the base is a regular polygon, the bipyramid is also called regular.

Definition and properties

Template:Multiple image A bipyramid is a polyhedron constructed by fusing two pyramids which share the same polygonal base;Template:R a pyramid is in turn constructed by connecting each vertex of its base to a single new vertex (the apex) not lying in the plane of the base, for an Template:Nowrapgonal base forming Template:Mvar triangular faces in addition to the base face. An Template:Nowrapgonal bipyramid thus has Template:Math faces, Template:Math edges, and Template:Math vertices. Template:AnchorMore 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.Template:Efn Likewise, a right bipyramid is a polyhedron constructed by attaching two symmetrical right bipyramid bases; bipyramids whose apices are not on this line are called oblique bipyramids.Template:R

When the two pyramids are mirror images, the bipyramid is called symmetric. It is called regular if its base is a regular polygon.Template:R When the base is a regular polygon and the apices are on the perpendicular line through its center (a regular right bipyramid) then all of its faces are isosceles triangles; sometimes the name bipyramid refers specifically to symmetric regular right bipyramids,Template:R Examples of such bipyramids are the triangular bipyramid, octahedron (square bipyramid) and pentagonal bipyramid. If all their edges are equal in length, these shapes consist of equilateral triangle faces, making them deltahedra;Template:R the triangular bipyramid and the pentagonal bipyramid are Johnson solids, and the regular octahedron is a Platonic solid.Template:R

File:Dual Cube-Octahedron.svg

The octahedron is dual to the cube

The symmetric regular right bipyramids have prismatic symmetry, with dihedral symmetry group Template:Math of order Template:Math: they are unchanged when rotated Template:Math 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.Template:R Because their faces are transitive under these symmetry transformations, they are isohedral.Template:R 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.Template:R The prisms share the same symmetry as the bipyramids.Template:R 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.Template:R

The volume of a symmetric bipyramid is <math display=block> \frac{2}{3}Bh, </math> where Template:Mvar is the area of the base and Template:Mvar the perpendicular distance from the base plane to either apex. In the case of a regular Template:Nowrapsided polygon with side length Template:Mvar and whose altitude is Template:Mvar, the volume of such a bipyramid is: <math display=block> \frac{n}{6}hs^2 \cot \frac{\pi}{n}. </math>

Related and other types of bipyramid

Concave bipyramids

A concave bipyramid has a concave polygon base, and one example is a concave tetragonal bipyramid or an irregular concave octahedron. A bipyramid with an arbitrary polygonal base could be considered a right bipyramid if the apices are on a line perpendicular to the base passing through the base's centroid.

Asymmetric bipyramids

An asymmetric bipyramid has apices which are not mirrored across the base plane; for a right bipyramid this only happens if each apex is a different distance from the base.

The dual of an asymmetric right Template:Mvar-gonal bipyramid is an Template:Mvar-gonal frustum.

A regular asymmetric right Template:Mvar-gonal bipyramid has symmetry group Template:Math, of order Template:Math.

Scalene triangle bipyramids

File:EB1911 Crystallography Fig. 46 Ditetragonal Bipyramid.jpg

Example: ditetragonal bipyramid (Template:Math)

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

All its faces are congruent scalene triangles, and it is isohedral. It can be seen as another type of a right symmetric di-Template:Mvar-gonal scalenohedron, with an isotoxal flat polygon base.

An isotoxal right (symmetric) di-Template:Mvar-gonal bipyramid has Template:Mvar two-fold rotation axes through opposite basal vertices, Template:Mvar reflection planes through opposite apical edges, an Template:Mvar-fold rotation axis through apices, a reflection plane through base, and an Template:Mvar-fold rotation-reflection axis through apices,<ref name=tulane /> representing symmetry group Template:Math of order Template:Math. (The reflection about the base plane corresponds to the Template:Math rotation-reflection. If Template:Mvar is even, then there is an inversion symmetry about the center, corresponding to the Template:Math rotation-reflection.)

Example with Template:Math:

An isotoxal right (symmetric) ditrigonal bipyramid has three similar vertical planes of symmetry, intersecting in a (vertical) Template:Math-fold rotation axis; perpendicular to them is a fourth plane of symmetry (horizontal); at the intersection of the three vertical planes with the horizontal plane are three similar (horizontal) Template:Math-fold rotation axes; there is no center of inversion symmetry,Template:Sfn but there is a center of symmetry: the intersection point of the four axes.

Example with Template:Math:

An isotoxal right (symmetric) ditetragonal bipyramid has four vertical planes of symmetry of two kinds, intersecting in a (vertical) Template:Math-fold rotation axis; perpendicular to them is a fifth plane of symmetry (horizontal); at the intersection of the four vertical planes with the horizontal plane are four (horizontal) Template:Math-fold rotation axes of two kinds, each perpendicular to a plane of symmetry; two vertical planes bisect the angles between two horizontal axes; and there is a centre of inversion symmetry.Template:Sfn

Double example:

• The bipyramid with isotoxal Template:Math-gon base vertices Template:Mvar and right symmetric apices Template:Mvar<math display=block>\begin{alignat}{5}

 U &= (1,0,0), & \quad V &= (0,2,0), & \quad A &= (0,0,1), \\
 U' &= (-1,0,0), & \quad V' &= (0,-2,0), & \quad A' &= (0,0,-1),

\end{alignat}</math> has its faces isosceles. Indeed:

• • Upper apical edge lengths:<math display=block>\begin{align}

 \overline{AU} &= \overline{AU'} = \sqrt{2} \,, \\[2pt]
 \overline{AV} &= \overline{AV'} = \sqrt{5} \,;

\end{align}</math>

• • Base edge lengths: <math display=block>

 \overline{UV} = \overline{VU'} = \overline{U'V'} = \overline{V'U} = \sqrt{5} \,;

</math>

• • Lower apical edge lengths (equal to upper edge lengths):<math display=block>\begin{align}

 \overline{A'U} &= \overline{A'U'} = \sqrt{2} \,, \\[2pt]
 \overline{A'V} &= \overline{A'V'} = \sqrt{5} \,.

\end{align}</math>

• The bipyramid with same base vertices, but with right symmetric apices <math display=block>\begin{align}

 A &= (0,0,2), \\
 A' &= (0,0,-2),

\end{align}</math> also has its faces isosceles. Indeed:

• • Upper apical edge lengths:<math display=block>\begin{align}

 \overline{AU} &= \overline{AU'} = \sqrt{5} \,, \\[2pt]
 \overline{AV} &= \overline{AV'} = 2\sqrt{2} \,;

\end{align}</math>

• • Base edge length (equal to previous example): <math display=block>

 \overline{UV} = \overline{VU'} = \overline{U'V'} = \overline{V'U} = \sqrt{5}\,;

</math>

• • Lower apical edge lengths (equal to upper edge lengths):<math display=block>\begin{align}

 \overline{A'U} &= \overline{A'U'} = \sqrt{5}\,, \\[2pt]
 \overline{A'V} &= \overline{A'V'} = 2\sqrt{2}\,.

\end{align}</math>

File:EB1911 Crystallography Figs. 54 & 55 Orthorhombic Bipyramids.jpg

Examples of rhombic bipyramids

In crystallography, isotoxal right (symmetric) didigonalTemplate:Efn (8-faced), ditrigonal (12-faced), ditetragonal (16-faced), and dihexagonal (24-faced) bipyramids exist.<ref name=tulane>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref><ref name=uwgb />

Scalenohedra

File:EB1911 Crystallography Fig. 68.—Scalenohedron.jpg

Example: ditrigonal scalenohedron (Template:Math)

A scalenohedron is similar to a bipyramid; the difference is that the scalenohedra has a zig-zag pattern in the middle edges.Template:R

It has two apices and Template:Math basal vertices, Template:Math faces, and Template:Math edges; it is topologically identical to a Template:Math-gonal bipyramid, but its Template:Math basal vertices alternate in two rings above and below the center.<ref name=uwgb>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

All its faces are congruent scalene triangles, and it is isohedral. It can be seen as another type of a right symmetric di-Template:Mvar-gonal bipyramid, with a regular zigzag skew polygon base.

A regular right symmetric di-Template:Mvar-gonal scalenohedron has Template:Mvar two-fold rotation axes through opposite basal mid-edges, Template:Mvar reflection planes through opposite apical edges, an Template:Mvar-fold rotation axis through apices, and a Template:Math-fold rotation-reflection axis through apices (about which Template:Math rotations-reflections globally preserve the solid),<ref name=tulane /> representing symmetry group Template:Math of order Template:Math. (If Template:Mvar is odd, then there is an inversion symmetry about the center, corresponding to the Template:Math rotation-reflection.)

Example with Template:Math:

A regular right symmetric ditrigonal scalenohedron has three similar vertical planes of symmetry inclined to one another at Template:Math and intersecting in a (vertical) Template:Math-fold rotation axis, three similar horizontal Template:Math-fold rotation axes, each perpendicular to a plane of symmetry, a center of inversion symmetry,Template:Sfn and a vertical Template:Math-fold rotation-reflection axis.

Example with Template:Math:

A regular right symmetric didigonal scalenohedron has only one vertical and two horizontal Template:Math-fold rotation axes, two vertical planes of symmetry, which bisect the angles between the horizontal pair of axes, and a vertical Template:Math-fold rotation-reflection axis;Template:Sfn it has no center of inversion symmetry.

File:EB1911 Crystallography Figs. 50 & 51.jpg

Examples of disphenoids and of an Template:Math-faced scalenohedron

For at most two particular values of <math>z_A = |z_{A'}|,</math> the faces of such a scalenohedron may be isosceles.

Double example:

• The scalenohedron with regular zigzag skew Template:Math-gon base vertices Template:Mvar and right symmetric apices Template:Mvar<math display=block>\begin{alignat}{5}

 U &= (3,0,2), & \quad V &= (0,3,-2), & \quad A &= (0,0,3), \\
 U' &= (-3,0,2), & \quad V' &= (0,-3,-2), & \quad A' &= (0,0,-3),

\end{alignat}</math> has its faces isosceles. Indeed:

• • Upper apical edge lengths:<math display=block>\begin{align}

 \overline{AU} &= \overline{AU'} = \sqrt{10} \,, \\[2pt]
 \overline{AV} &= \overline{AV'} = \sqrt{34} \,;

\end{align}</math>

• • Base edge length:<math display=block>

 \overline{UV} = \overline{VU'} = \overline{U'V'} = \overline{V'U} = \sqrt{34} \,;

</math>

• • Lower apical edge lengths (equal to upper edge lengths swapped):<math display=block>\begin{align}

 \overline{A'U} &= \overline{A'U'} = \sqrt{34} \,, \\[2pt]
 \overline{A'V} &= \overline{A'V'} = \sqrt{10} \,.

\end{align}</math>

• The scalenohedron with same base vertices, but with right symmetric apices<math display=block>\begin{align}

 A &= (0,0,7), \\
 A' &= (0,0,-7),

\end{align}</math> also has its faces isosceles. Indeed:

• • Upper apical edge lengths:<math display=block>\begin{align}

 \overline{AU} &= \overline{AU'} = \sqrt{34} \,, \\[2pt]
 \overline{AV} &= \overline{AV'} = 3\sqrt{10} \,;

\end{align}</math>

• • Base edge length (equal to previous example): <math display=block>

 \overline{UV} = \overline{VU'} = \overline{U'V'} = \overline{V'U} = \sqrt{34} \,;

</math>

• • Lower apical edge lengths (equal to upper edge lengths swapped):<math display=block>\begin{align}

 \overline{A'U} &= \overline{A'U'} = 3\sqrt{10} \,, \\[2pt]
 \overline{A'V} &= \overline{A'V'} = \sqrt{34} \,.

\end{align}</math>

In crystallography, regular right symmetric didigonal (Template:Math-faced) and ditrigonal (Template:Math-faced) scalenohedra exist.<ref name=tulane /><ref name=uwgb />

The smallest geometric scalenohedra have eight faces, and are topologically identical to the regular octahedron. In this case (Template:Math), in crystallography, a regular right symmetric didigonal (Template:Math-faced) scalenohedron is called a tetragonal scalenohedron.<ref name=tulane /><ref name=uwgb />

Let us temporarily focus on the regular right symmetric Template:Math-faced scalenohedra with Template:Math i.e. <math display=block>

 z_{A} = |z_{A'}| = x_{U} = |x_{U'}| = y_{V} = |y_{V'}|.

</math> Their two apices can be represented as Template:Mvar and their four basal vertices as Template:Mvar: <math display=block>\begin{alignat}{5}

 U &= (1,0,z), & \quad V &= (0,1,-z), & \quad A &= (0,0,1), \\
 U' &= (-1,0,z), & \quad V' &= (0,-1,-z), & \quad A' &= (0,0,-1),

\end{alignat}</math> where Template:Mvar is a parameter between Template:Math and Template:Math.

At Template:Math, it is a regular octahedron; at Template:Math, it has four pairs of coplanar faces, and merging these into four congruent isosceles triangles makes it a disphenoid; for Template:Math, it is concave.

Example: geometric variations with regular right symmetric 8-faced scalenohedra:

Template:Math	Template:Math	Template:Math	Template:Math	Template:Math
File:4-scalenohedron-01.png	File:4-scalenohedron-025.png	File:4-scalenohedron-05.png	File:4-scalenohedron-095.png	File:4-scalenohedron-15.png

If the Template:Math-gon base is both isotoxal in-out and zigzag skew, then not all faces of the isotoxal right symmetric scalenohedron are congruent.

Example with five different edge lengths:

• The scalenohedron with isotoxal in-out zigzag skew Template:Math-gon base vertices Template:Mvar and right symmetric apices Template:Mvar <math display=block>\begin{alignat}{5}

 U &= (1,0,1), & \quad V &= (0,2,-1), & \quad A &= (0,0,3), \\
 U' &= (-1,0,1), & \quad V' &= (0,-2,-1), & \quad A' &= (0,0,-3),

\end{alignat}</math> has congruent scalene upper faces, and congruent scalene lower faces, but not all its faces are congruent. Indeed:

• • Upper apical edge lengths:<math display=block>\begin{align}

 \overline{AU} &= \overline{AU'} = \sqrt{5} \,, \\[2pt]
 \overline{AV} &= \overline{AV'} = 2\sqrt{5} \,;

\end{align}</math>

• • Base edge length:<math display=block>

 \overline{UV} = \overline{VU'} = \overline{U'V'} = \overline{V'U} = 3;

</math>

• • Lower apical edge lengths:<math display=block>\begin{align}

 \overline{A'U} &= \overline{A'U'} = \sqrt{17} \,, \\[2pt]
 \overline{A'V} &= \overline{A'V'} = 2\sqrt{2} \,.

\end{align}</math>

For some particular values of Template:Math, half the faces of such a scalenohedron may be isosceles or equilateral.

Example with three different edge lengths:

• The scalenohedron with isotoxal in-out zigzag skew Template:Math-gon base vertices Template:Mvar and right symmetric apices Template:Mvar <math display=block>\begin{alignat}{5}

 U &= (3,0,2), & \quad V &= \left( 0,\sqrt{65},-2 \right), & \quad A &= (0,0,7), \\
 U' &= (-3,0,2), & \quad V' &= \left( 0,-\sqrt{65},-2 \right), & \quad A' &= (0,0,-7),

\end{alignat}</math> has congruent scalene upper faces, and congruent equilateral lower faces; thus not all its faces are congruent. Indeed:

• • Upper apical edge lengths:<math display=block>\begin{align}

 \overline{AU} &= \overline{AU'} = \sqrt{34} \,, \\[2pt]
 \overline{AV} &= \overline{AV'} = \sqrt{146} \,;

\end{align}</math>

• • Base edge length:<math display=block>

 \overline{UV} = \overline{VU'} = \overline{U'V'} = \overline{V'U} = 3\sqrt{10} \,;

</math>

• • Lower apical edge length(s): <math display=block>\begin{align}

 \overline{A'U} &= \overline{A'U'} = 3\sqrt{10} \,, \\[2pt]
 \overline{A'V} &= \overline{A'V'} = 3\sqrt{10} \,.

\end{align}</math>

Star bipyramids

A star bipyramid has a star polygon base, and is self-intersecting.<ref>Template:Cite journal</ref>

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

A Template:Math-bipyramid has Coxeter diagram Template:CDD.

Example star bipyramids:

Base	5/2-gon	7/2-gon	7/3-gon	8/3-gon
Image	File:Pentagram Dipyramid.png	File:7-2 dipyramid.png	File:7-3 dipyramid.png	File:8-3 dipyramid.png

4-polytopes with bipyramidal cells

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

• Template:Mvar is the apex vertex of the bipyramid;
• Template:Mvar is an equator vertex;
• Template:Overline is the distance between adjacent vertices on the equator (equal to 1);
• Template:Overline is the apex-to-equator edge length;
• Template:Overline is the distance between the apices.

The bipyramid 4-polytope will have Template:Mvar vertices where the apices of Template:Mvar bipyramids meet. It will have Template:Mvar vertices where the type Template:Mvar vertices of Template:Mvar bipyramids meet.

• Template:Tmath bipyramids meet along each type Template:Overline edge.
• Template:Tmath bipyramids meet along each type Template:Overline edge.
• Template:Tmath is the cosine of the dihedral angle along an Template:Overline edge.
• Template:Tmath is the cosine of the dihedral angle along an Template:Overline edge.

As cells must fit around an edge, <math display=block>\begin{align}

 N_\overline{EE} \arccos C_\overline{EE} &\le 2\pi, \\[4pt]
 N_\overline{AE} \arccos C_\overline{AE} &\le 2\pi.

\end{align}</math>

4-polytopes with bipyramidal cells

4-polytope properties									Bipyramid properties
Dual of rectified polytope	Coxeter diagram	Cells	Template:Mvar	Template:Mvar	Template:Mvar	Template:Mvar	Template:Tmath	Template:Tmath	Bipyramid cell	Coxeter diagram	Template:Overline	Template:OverlineTemplate:Efn	Template:Tmath	Template:Tmath
R. 5-cell	Template:CDD	10	5	5	4	6	3	3	Triangular	Template:CDD	<math display=inline>\frac23</math>	0.667	<math display=inline>-\frac17</math>	<math display=inline>-\frac17</math>
R. tesseract	Template:CDD	32	16	8	4	12	3	4	Triangular	Template:CDD	<math display=inline>\frac{\sqrt{2}}{3}</math>	0.624	<math display=inline>-\frac25</math>	<math display=inline>-\frac15</math>
R. 24-cell	Template:CDD	96	24	24	8	12	4	3	Triangular	Template:CDD	<math display=inline>\frac{2 \sqrt{2}}{3}</math>	0.745	<math display=inline>\frac1{11}</math>	<math display=inline>-\frac5{11}</math>
R. 120-cell	Template:CDD	1200	600	120	4	30	3	5	Triangular	Template:CDD	<math display=inline>\frac{\sqrt{5} - 1}{3}</math>	0.613	<math display=inline>- \frac{10 + 9\sqrt{5}}{61}</math>	<math display=inline>- \frac{7 - 12\sqrt{5}}{61}</math>
R. 16-cell	Template:CDD	24Template:Efn	8	16	6	6	3	3	Square	Template:CDD	<math display=inline>\sqrt{2}</math>	1	<math display=inline>-\frac13</math>	<math display=inline>-\frac13</math>
R. cubic honeycomb	Template:CDD	∞	∞	∞	6	12	3	4	Square	Template:CDD	<math display=inline>1</math>	0.866	<math display=inline>-\frac12</math>	<math display=inline>0</math>
R. 600-cell	Template:CDD	720	120	600	12	6	3	3	Pentagonal	Template:CDD	<math display=inline>\frac{5 + 3\sqrt{5}}{5}</math>	1.447	<math display=inline>- \frac{11 + 4\sqrt{5}}{41}</math>	<math display=inline>- \frac{11 + 4\sqrt{5}}{41}</math>

Other dimensions

File:Romb deltoid.svg

A rhombus is a 2-dimensional analog of a right symmetric bipyramid

A generalized Template:Mvar-dimensional "bipyramid" is any Template:Mvar-polytope constructed from an Template:Math-polytope base lying in a hyperplane, with every base vertex connected by an edge to two apex vertices. If the Template:Math-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.

A 2-dimensional analog of a right symmetric bipyramid is formed by joining two congruent isosceles triangles base-to-base to form a rhombus. More generally, a kite is a 2-dimensional analog of a (possibly asymmetric) right bipyramid, and any quadrilateral is a 2-dimensional analog of a general bipyramid.

See also

• Trapezohedron

Works cited

• Template:Cite book Chapter 4: Duals of the Archimedean polyhedra, prisms and antiprisms
• Template:Cite EB1911

External links

• Template:Mathworld
• Template: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.