Editing Bipyramid (section)

== 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>
|}