Editing Bipyramid

{{short description|Polyhedron formed by joining mirroring pyramids base-to-base}}
{{redirect|Dipyramid|the mountain|Dipyramid (Alaska)}}
{{Use dmy dates|date=April 2020}}

In geometry, a '''bipyramid''', '''dipyramid''', or '''double pyramid''' is a [[polyhedron]] formed by fusing two [[Pyramid (geometry)|pyramids]] together [[base (geometry)|base]]-to-base. The [[polygon]]al base of each pyramid must therefore be the same, and unless otherwise specified the base [[Vertex (geometry)|vertices]] are usually [[coplanar]] and a bipyramid is usually ''symmetric'', meaning the two pyramids are [[mirror image]]s across their common base plane. When each [[apex (geometry)|apex]] ({{plural form|apices}}, 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;{{efn|name=right pyramids}} otherwise it is ''oblique''. When the base is a [[regular polygon]], the bipyramid is also called ''regular''.

== Definition and properties ==
{{multiple image
 | total_width = 500
 | image1 = Triangular bipyramid.png
 | image2 = Square bipyramid.png
 | image3 = Pentagonale bipiramide.png
 | footer = The [[triangular bipyramid]], [[square bipyramid]], and [[pentagonal bipyramid]].
}}
A bipyramid is a polyhedron constructed by fusing two [[Pyramid (geometry)|pyramids]] which share the same [[polygon]]al [[base (geometry)|base]];{{r|aarts}} a pyramid is in turn constructed by connecting each vertex of its base to a single new [[vertex (geometry)|vertex]] (the [[apex (geometry)|apex]]) not lying in the plane of the base, for an {{nowrap|1={{mvar|n}}-}}gonal base forming {{mvar|n}} triangular faces in addition to the base face. An {{nowrap|1={{mvar|n}}-}}gonal bipyramid thus has {{math|2''n''}} faces, {{math|3''n''}} edges, and {{math|''n'' + 2}} vertices. {{anchor|1=Right and oblique bipyramid}}More 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.{{efn|name=right pyramids}} 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''.{{r|polya}}

When the two pyramids are mirror images, the bipyramid is called ''symmetric''. It is called ''regular'' if its base is a [[regular polygon]].{{r|aarts}} 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 triangle]]s; sometimes the name ''bipyramid'' refers specifically to symmetric regular right bipyramids,{{r|montroll}} 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 [[deltahedron|deltahedra]];{{r|trigg|uehara}} the triangular bipyramid and the pentagonal bipyramid are [[Johnson solid]]s, and the regular octahedron is a [[Platonic solid]].{{r|cromwell}}

[[File:Dual Cube-Octahedron.svg|thumb|180px|The octahedron is dual to the cube]]
The symmetric regular right bipyramids have [[prismatic symmetry]], with [[dihedral group|dihedral symmetry group]] {{math|D<sub>''nh''</sub>}} of order {{math|4''n''}}: they are unchanged when rotated {{math|{{sfrac|1|''n''}}}} 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.{{r|fsz}} Because their faces are transitive under these symmetry transformations, they are [[Isohedral figure|isohedral]].{{r|cpsb|mclean}} They are the [[dual polyhedron|dual polyhedra]] of [[Prism (geometry)|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.{{r|sibley}} The prisms share the same symmetry as the bipyramids.{{r|king}} The [[regular octahedron]] is more symmetric still, as its base vertices and apices are indistinguishable and can be exchanged by reflections or [[Rotation (mathematics)|rotations]]; the regular octahedron and its dual, the [[cube]], have [[octahedral symmetry]].{{r|armstrong}}

The [[volume]] of a symmetric bipyramid is
<math display=block> \frac{2}{3}Bh, </math>
where {{mvar|B}} is the area of the base and {{mvar|h}} the perpendicular distance from the base plane to either apex. In the case of a regular {{nowrap|1={{mvar|n}}-}}sided polygon with side length {{mvar|s}} and whose altitude is {{mvar|h}}, 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 ==
{{multiple image
 | total_width = 350
 | image1 = Concave quadrilateral bipyramid.png |caption1=A concave tetragonal bipyramid
 | image2 = Asymmetric hexagonal bipyramid.png |caption2= An asymmetric hexagonal 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 polyhedron|dual]] of an asymmetric right {{mvar|n}}-gonal bipyramid is an {{mvar|n}}-gonal [[frustum]].

A regular asymmetric right {{mvar|n}}-gonal bipyramid has symmetry group {{math|C<sub>''n''v</sub>}}, of order {{math|2''n''}}.

=== Scalene triangle bipyramids ===
[[File:EB1911 Crystallography Fig. 46 Ditetragonal Bipyramid.jpg|thumb|Example: ditetragonal bipyramid ({{math|1=2''n'' = 2×4}})]]
An isotoxal right (symmetric) '''di-{{mvar|n}}-gonal bipyramid''' is a right (symmetric) {{math|'''2'''''n''}}-gonal bipyramid with an [[Isotoxal figure|''isotoxal'']] flat polygon base: its {{math|2''n''}} basal vertices are coplanar, but alternate in two [[Radius|radii]].

All its faces are [[Congruence (geometry)|congruent]] [[scalene triangle]]s, and it is [[Isohedral figure|isohedral]]. It can be seen as another type of a right symmetric di-{{mvar|n}}-gonal [[#Scalenohedra|''scalenohedron'']], with an isotoxal flat polygon base.

An isotoxal right (symmetric) di-{{mvar|n}}-gonal bipyramid has {{mvar|n}} two-fold rotation axes through opposite basal vertices, {{mvar|n}} reflection planes through opposite apical edges, an {{mvar|n}}-fold rotation axis through apices, a reflection plane through base, and an {{mvar|n}}-fold [[Improper rotation|rotation-reflection]] axis through apices,<ref name=tulane /> representing symmetry group {{math|D<sub>''n''h</sub>, [''n'',2], (*22''n''),}} of order {{math|4''n''}}. (The reflection about the base plane corresponds to the {{math|0°}} rotation-reflection. If {{mvar|n}} is even, then there is an [[inversion symmetry]] about the center, corresponding to the {{math|180°}} rotation-reflection.)

Example with {{math|1=2''n'' = 2×3}}:
:An isotoxal right (symmetric) ditrigonal bipyramid has three similar vertical planes of symmetry, intersecting in a (vertical) {{math|3}}-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) {{math|2}}-fold rotation axes; there is no center of inversion symmetry,{{sfn|Spencer|1911|loc=6. Hexagonal system, ''rhombohedral division'', ditrigonal bipyramidal class, p. 581 (p. 603 on Wikisource)}} but there is a [[center of symmetry]]: the intersection point of the four axes.

Example with {{math|1=2''n'' = 2×4}}:
:An isotoxal right (symmetric) ditetragonal bipyramid has four vertical planes of symmetry of two kinds, intersecting in a (vertical) {{math|4}}-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) {{math|2}}-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.{{sfn|Spencer|1911|loc=2. Tegragonal system, holosymmetric class, fig. 46, p. 577 (p. 599 on Wikisource)}}

Double example:
*The bipyramid with isotoxal {{math|2×2}}-gon base vertices {{mvar|U, U', V, V'}} and right symmetric apices {{mvar|A, A'}}<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|thumb|Examples of rhombic bipyramids]]

In [[crystallography]], isotoxal right (symmetric) didigonal{{efn|The smallest geometric di-{{mvar|n}}-gonal bipyramids have eight faces, and are topologically identical to the regular [[octahedron]]. In this case ({{math|1=2''n'' = 2×2}}):<br>an isotoxal right (symmetric) didigonal bipyramid is called a ''rhombic bipyramid'',<ref name=tulane /><ref name=uwgb /> although all its faces are scalene triangles, because its flat polygon base is a rhombus.}} (8-faced), ditrigonal (12-faced), ditetragonal (16-faced), and dihexagonal (24-faced) bipyramids exist.<ref name=tulane>{{cite web|url=http://www.tulane.edu/~sanelson/eens211/forms_zones_habit.htm|title=Crystal Form, Zones, Crystal Habit|website=Tulane.edu|access-date=16 September 2017}}</ref><ref name=uwgb />

=== Scalenohedra ===
[[File:EB1911 Crystallography Fig. 68.—Scalenohedron.jpg|thumb|Example: ditrigonal scalenohedron ({{math|1=2''n'' = 2×3}})]]
A '''scalenohedron''' is similar to a bipyramid; the difference is that the scalenohedra has a zig-zag pattern in the middle edges.{{r|kp}}

It has two apices and {{math|2''n''}} basal vertices, {{math|4''n''}} faces, and {{math|6''n''}} edges; it is topologically identical to a {{math|2''n''}}-gonal bipyramid, but its {{math|2''n''}} basal vertices alternate in two rings above and below the center.<ref name=uwgb>{{Cite web|date=2013-09-18|title=The 48 Special Crystal Forms|url=https://www.uwgb.edu/dutchs/symmetry/xlforms.htm|access-date=2020-11-18|archive-url=https://web.archive.org/web/20130918103121/https://www.uwgb.edu/dutchs/symmetry/xlforms.htm|archive-date=18 September 2013}}</ref>

All its faces are [[Congruence (geometry)|congruent]] [[scalene triangle]]s, and it is [[Isohedral figure|isohedral]]. It can be seen as another type of a right symmetric di-{{mvar|n}}-gonal bipyramid, with a regular zigzag skew polygon base.

A regular right symmetric di-{{mvar|n}}-gonal scalenohedron has {{mvar|n}} two-fold rotation axes through opposite basal mid-edges, {{mvar|n}} reflection planes through opposite apical edges, an {{mvar|n}}-fold rotation axis through apices, and a {{math|'''2'''''n''}}-fold [[Improper rotation|rotation-reflection]] axis through apices (about which {{math|'''1'''''n''}} rotations-reflections globally preserve the solid),<ref name=tulane /> representing symmetry group {{math|1=D<sub>''n''v</sub> = D<sub>''n''d</sub>, [2<sup>+</sup>,2''n''], (2*''n''),}} of order {{math|4''n''}}. (If {{mvar|n}} is odd, then there is an [[inversion symmetry]] about the center, corresponding to the {{math|180°}} rotation-reflection.)

Example with {{math|1=2''n'' = 2×3}}:
:A regular right symmetric ditrigonal scalenohedron has three similar vertical planes of symmetry inclined to one another at {{math|60°}} and intersecting in a (vertical) {{math|3}}-fold rotation axis, three similar horizontal {{math|2}}-fold rotation axes, each perpendicular to a plane of symmetry, a center of inversion symmetry,{{sfn|Spencer|1911|loc=6. Hexagonal system, ''rhombohedral division'', holosymmetric class, fig. 68, p. 580 (p. 602 on Wikisource)}} and a vertical {{math|'''6'''}}-fold rotation-reflection axis.

Example with {{math|1=2''n'' = 2×2}}:
:A regular right symmetric didigonal scalenohedron has only one vertical and two horizontal {{math|2}}-fold rotation axes, two vertical planes of symmetry, which bisect the angles between the horizontal pair of axes, and a vertical {{math|'''4'''}}-fold rotation-reflection axis;{{sfn|Spencer|1911|p=2. Tetragonal system, scalenohedral class, fig. 51, p. 577 (p. 599 on Wikisource)}} it has no center of inversion symmetry.
[[File:EB1911 Crystallography Figs. 50 & 51.jpg|thumb|Examples of disphenoids and of an {{math|8}}-faced scalenohedron]]

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

Double example:
*The scalenohedron with regular zigzag skew {{math|2×2}}-gon base vertices {{mvar|U, U', V, V'}} and right symmetric apices {{mvar|A, A'}}<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 ({{math|8}}-faced) and ditrigonal ({{math|12}}-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 ({{math|1=2''n'' = 2×2}}), in crystallography, a regular right symmetric didigonal ({{math|8}}-faced) scalenohedron is called a ''tetragonal scalenohedron''.<ref name=tulane /><ref name=uwgb />

Let us temporarily focus on the regular right symmetric {{math|8}}-faced scalenohedra with {{math|1=''h'' = ''r'',}} 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 {{mvar|A, A'}} and their four basal vertices as {{mvar|U, U', V, V'}}: 
<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 {{mvar|z}} is a parameter between {{math|0}} and {{math|1}}.

At {{math|1=''z'' = 0}}, it is a regular octahedron; at {{math|1=''z'' = 1}}, it has four pairs of coplanar faces, and merging these into four congruent isosceles triangles makes it a [[disphenoid]]; for {{math|''z'' > 1}}, it is concave.

{| class=wikitable
|+ style="text-align:center;"|Example: geometric variations with regular right symmetric 8-faced scalenohedra:
!{{math|1=''z'' = 0.1}}
!{{math|1=''z'' = 0.25}}
!{{math|1=''z'' = 0.5}}
!{{math|1=''z'' = 0.95}}
!{{math|1=''z'' = 1.5}}
|-
|[[File:4-scalenohedron-01.png|120px]]
|[[File:4-scalenohedron-025.png|120px]]
|[[File:4-scalenohedron-05.png|120px]]
|[[File:4-scalenohedron-095.png|120px]]
|[[File:4-scalenohedron-15.png|120px]]
|}

If the {{math|2''n''}}-gon base is both [[Isotoxal figure|isotoxal]] in-out and [[Skew polygon|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 {{math|2×2}}-gon base vertices {{mvar|U, U', V, V'}} and right symmetric apices {{mvar|A, A'}} <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 {{math|1=''z{{sub|A}}'' = {{!}}''z{{sub|A'}}''{{!}}}}, half the faces of such a '''scaleno'''hedron may be [[Isosceles triangle|'''isosceles''']] or [[Equilateral triangle|'''equilateral''']].

Example with three different edge lengths:
*The scalenohedron with isotoxal in-out zigzag skew {{math|2×2}}-gon base vertices {{mvar|U, U', V, V'}} and right symmetric apices {{mvar|A, A'}} <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>{{cite journal |last= Rankin |first=John R. |year=1988 |title= Classes of polyhedra defined by jet graphics |journal=Computers & Graphics |volume=12 |issue=2 |pages=239–254 |doi=10.1016/0097-8493(88)90036-2}}</ref>

A regular right symmetric star bipyramid has [[Congruence (geometry)|congruent]] [[Isosceles triangle|isosceles]] triangle faces, and is [[Isohedral figure|isohedral]].

A {{math|''p''/''q''}}-bipyramid has [[Coxeter diagram]] {{CDD|node_f1|2x|node_f1|p|rat|q|node}}.

{| class=wikitable
|+ style="text-align:center;"|Example star bipyramids:
|- align=center
!Base
![[Pentagrammic bipyramid|5/2]]-gon
!7/2-gon
!7/3-gon
!8/3-gon
|- align=center
!Image
|[[File:Pentagram Dipyramid.png|100px]]
|[[File:7-2 dipyramid.png|125px]]
|[[File:7-3 dipyramid.png|125px]]
|[[File:8-3 dipyramid.png|125px]]
|}

== 4-polytopes with bipyramidal cells ==
The [[Dual polyhedron#Dual polytopes and tessellations|dual]] of the [[Rectification (geometry)|rectification]] of each [[convex regular 4-polytope]]s is a [[cell-transitive]] [[4-polytope]] with bipyramidal cells. In the following:
*{{mvar|A}} is the apex vertex of the bipyramid; 
*{{mvar|E}} is an equator vertex;
*{{overline|{{mvar|EE}}}} is the distance between adjacent vertices on the equator (equal to 1);
*{{overline|{{mvar|AE}}}} is the apex-to-equator edge length;
*{{overline|{{mvar|AA}}}} is the distance between the apices.
The bipyramid 4-polytope will have {{mvar|V<sub>A</sub>}} vertices where the apices of {{mvar|N<sub>A</sub>}} bipyramids meet. It will have {{mvar|V<sub>E</sub>}} vertices where the type {{mvar|E}} vertices of {{mvar|N<sub>E</sub>}} bipyramids meet. 
*{{tmath|N_\overline{AE} }} bipyramids meet along each type {{overline|{{mvar|AE}}}} edge. 
*{{tmath|N_\overline{EE} }} bipyramids meet along each type {{overline|{{mvar|EE}}}} edge.  
*{{tmath|C_\overline{AE} }} is the cosine of the [[dihedral angle]] along an {{overline|{{mvar|AE}}}} edge. 
*{{tmath|C_\overline{EE} }} is the cosine of the dihedral angle along an {{overline|{{mvar|EE}}}} 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>

{| class=wikitable
|+ style="text-align:center;"|4-polytopes with bipyramidal cells
!colspan=9| 4-polytope properties
!colspan=6| Bipyramid properties
|- align=center
!  Dual of <br> rectified <br> polytope
! [[Coxeter–Dynkin diagram|Coxeter<br>diagram]]
! Cells
! {{mvar|V<sub>A</sub>}}
! {{mvar|V<sub>E</sub>}}
! {{mvar|N<sub>A</sub>}}
! {{mvar|N<sub>E</sub>}}
! style="padding:0.2em;" | {{tmath|N_\overline{\!AE} }}
! style="padding:0.2em;" | {{tmath|N_\overline{\!EE} }}
! Bipyramid <br> cell
! Coxeter<br>diagram
! {{overline|{{mvar|AA}}}}
! {{overline|{{mvar|AE}}}}{{efn|Given numerically due to more complex form.}}
! {{tmath|C_\overline{AE} }}
! {{tmath|C_\overline{EE} }}
|- align=center
| [[Rectified 5-cell|R. 5-cell]]
| {{CDD|node|3|node_f1|3|node|3|node}}
| 10
| 5
| 5
| 4
| 6
| 3
| 3
| [[Triangular bipyramid|Triangular]]
| {{CDD|node_f1|2x|node_f1|3|node}}
| <math display=inline>\frac23</math>
| 0.667
| <math display=inline>-\frac17</math>
| <math display=inline>-\frac17</math>
|- align=center
| [[Rectified tesseract|R. tesseract]]
| {{CDD|node|4|node_f1|3|node|3|node}}
| 32
| 16
| 8
| 4
| 12
| 3
| 4
| [[Triangular bipyramid|Triangular]]
| {{CDD|node_f1|2x|node_f1|3|node}}
| <math display=inline>\frac{\sqrt{2}}{3}</math>
| 0.624
| <math display=inline>-\frac25</math>
| <math display=inline>-\frac15</math>
|- align=center
| [[Rectified 24-cell|R. 24-cell]]
| {{CDD|node|3|node_f1|4|node|3|node}}
| 96
| 24
| 24
| 8
| 12
| 4
| 3
| [[Triangular bipyramid|Triangular]]
| {{CDD|node_f1|2x|node_f1|3|node}}
| <math display=inline>\frac{2 \sqrt{2}}{3}</math>
| 0.745
| <math display=inline>\frac1{11}</math>
| <math display=inline>-\frac5{11}</math>
|- align=center
| [[Rectified 120-cell|R. 120-cell]]
| {{CDD|node|5|node_f1|3|node|3|node}}
| 1200
| 600
| 120
| 4
| 30
| 3
| 5
| [[Triangular bipyramid|Triangular]]
| {{CDD|node_f1|2x|node_f1|3|node}}
| <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>
|- align=center
| [[24-cell|R. 16-cell]]
| {{CDD|node|3|node_f1|3|node|4|node}}
| 24{{efn|The rectified 16-cell is the regular 24-cell and vertices are all equivalent &ndash; octahedra are regular bipyramids.}}
| 8
| 16
| 6
| 6
| 3
| 3
| [[Square bipyramid|Square]]
| {{CDD|node_f1|2x|node_f1|4|node}}
| <math display=inline>\sqrt{2}</math>
| 1
| <math display=inline>-\frac13</math>
| <math display=inline>-\frac13</math>
|- align=center
| [[Rectified cubic honeycomb|R. cubic <br> honeycomb]]
| {{CDD|node|4|node_f1|3|node|4|node}}
| ∞
| ∞
| ∞
| 6
| 12
| 3
| 4
| [[Square bipyramid|Square]]
| {{CDD|node_f1|2x|node_f1|4|node}}
| <math display=inline>1</math>
| 0.866
| <math display=inline>-\frac12</math>
| <math display=inline>0</math>
|- align=center
| [[Rectified 600-cell|R. 600-cell]]
| {{CDD|node|3|node_f1|3|node|5|node}}
| 720
| 120
| 600
| 12
| 6
| 3
| 3
| [[Pentagonal bipyramid|Pentagonal]]
| {{CDD|node_f1|2x|node_f1|5|node}}
| <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|thumb|upright=0.8|A rhombus is a 2-dimensional analog of a right symmetric bipyramid]]
A generalized {{mvar|n}}-dimensional "bipyramid" is any {{mvar|n}}-[[polytope]] constructed from an {{math|(''n'' − 1)}}-polytope ''base'' lying in a [[hyperplane]], with every base vertex connected by an edge to two ''apex'' vertices. If the {{math|(''n'' − 1)}}-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 [[Pyramid (geometry)|pyramidal]] [[Facet (geometry)|facets]].

A 2-dimensional analog of a right symmetric bipyramid is formed by joining two [[Congruence (geometry)|congruent]] [[isosceles triangle]]s base-to-base to form a [[rhombus]]. More generally, a [[kite (geometry)|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]]

== Notes ==
{{notelist|refs=

{{efn|name=right pyramids|1=The center of a regular polygon is unambiguous, but for irregular polygons sources disagree. Some sources only allow a right pyramid to have a regular polygon as a base. Others define a right pyramid as having its apices on a line perpendicular to the base and passing through its [[centroid]]. {{harvp|Polya|1954}} restricts right pyramids to those with a [[tangential polygon]] for a base, with the apices on a line perpendicular to the base and passing through the [[incenter]].}}

}}

== Citations ==
{{reflist|refs=

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<ref name="cpsb">{{cite book
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<ref name="fsz">{{cite book
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<ref name="kp">{{cite book
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}}</ref>

<ref name="mclean">{{cite journal
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 | year = 1990
 | title = Dungeons, dragons, and dice
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 | jstor = 3619822
 | s2cid = 195047512
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<ref name="montroll">{{cite book
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 | isbn = 9781439871065
 | at = [https://books.google.com/books?id=SeTqBgAAQBAJ&pg=PA6 p. 6]
 | publisher = A K Peters
 | title = Origami Polyhedra Design
 | title-link = Origami Polyhedra Design
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<ref name="polya">{{cite book
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 | title = Mathematics and Plausible Reasoning: Induction and analogy in mathematics
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<ref name="sibley">{{cite book
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<ref name="trigg">{{cite journal
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 | year = 1978
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<ref name="uehara">{{cite book
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}}

== Works cited ==

*{{cite book | author= Anthony Pugh | year= 1976 | title= Polyhedra: A visual approach | publisher= University of California Press Berkeley | location= California | isbn= 0-520-03056-7  }} Chapter 4: Duals of the Archimedean polyhedra, prisms and antiprisms
*{{cite EB1911|wstitle= Crystallography |volume= 07 |pages = 569&ndash;591 |last1= Spencer |first1= Leonard James |author-link= Leonard James Spencer}}

==External links==
{{Commons category|Bipyramids}}
*{{mathworld |urlname=Dipyramid |title=Dipyramid}}
*{{mathworld | urlname = Isohedron | title = Isohedron}}
*[http://www.mathconsult.ch/showroom/unipoly/ The Uniform Polyhedra]
*[http://www.georgehart.com/virtual-polyhedra/vp.html Virtual Reality Polyhedra] The Encyclopedia of Polyhedra
**[[VRML]] models [http://www.georgehart.com/virtual-polyhedra/alphabetic-list.html (George Hart)] [http://www.georgehart.com/virtual-polyhedra/vrml/triangular_dipyramid.wrl <3>] [http://www.georgehart.com/virtual-polyhedra/vrml/octahedron.wrl <4>] [http://www.georgehart.com/virtual-polyhedra/vrml/pentagonal_dipyramid.wrl <5>] [http://www.georgehart.com/virtual-polyhedra/vrml/hexagonal_dipyramid.wrl <6>] [https://web.archive.org/web/20150203062131/http://www.georgehart.com/virtual-polyhedra/vrml/heptagonal_dipyramid.wrl <7>] [http://www.georgehart.com/virtual-polyhedra/vrml/octagonal_dipyramid.wrl <8>] [http://www.georgehart.com/virtual-polyhedra/vrml/enneagonal_dipyramid.wrl <9>] [http://www.georgehart.com/virtual-polyhedra/vrml/decagonal_dipyramid.wrl <10>]
***[http://www.georgehart.com/virtual-polyhedra/conway_notation.html Conway Notation for Polyhedra] Try: "dP''n''", where ''n'' = 3, 4, 5, 6, ... Example: "dP4" is an octahedron.

{{Polyhedron navigator}}

[[Category:Polyhedra]]
[[Category:Bipyramids| ]]