Editing Bipyramid (section)

== 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]]
|}