Editing Bipyramid (section)

=== 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 />