Polychora With Bipyramid Cells
The dual of the rectification of each convex regular polychoron is a cell-transitive polychoron with bipyramidal cells. In the following, the apex vertex of the bipyramid is A and an equator vertex is E. The distance between adjacent vertexes on the equator EE=1, the apex to equator edge is AE and the distance between the apexes is AA. The bipyramid polychoron will have VA vertexes where the apexes of NA bipyramids meet. It will have VE vertexes where the type E vertexes of NE bipyramids meet. NAE bipyramids meet along each type AE edge. NEE bipyramids meet along each type EE edge. CAE is the cosine of the dihedral angle along an AE edge. CEE is the cosine of the dihedral angle along an EE edge. As cells must fit around an edge, NAA cos-1(CAA) < 2, NAE cos-1(CAE) < 2
| Polychoron Properties | Bipyramid Properties | |||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Dual of | Coxeter diagram |
Cells | VA | VE | NA | NE | NAE | NEE | Cell | Coxeter diagram |
AA | AE** | CAE | CEE |
| Rectified 5-cell | 10 | 5 | 5 | 4 | 6 | 3 | 3 | Triangular bipyramid | 0.667 | |||||
| Rectified tesseract | 32 | 16 | 8 | 4 | 12 | 3 | 4 | Triangular bipyramid | 0.624 | |||||
| Rectified 24-cell | 96 | 24 | 24 | 8 | 12 | 4 | 3 | Triangular bipyramid | 0.745 | |||||
| Rectified 120-cell | 1200 | 600 | 120 | 4 | 30 | 3 | 5 | Triangular bipyramid | 0.613 | |||||
| Rectified 16-cell | 24* | 8 | 16 | 6 | 6 | 3 | 3 | Square bipyramid | 1 | |||||
| Rectified cubic honeycomb | ∞ | ∞ | ∞ | 6 | 12 | 3 | 4 | Square bipyramid | ||||||
| Rectified 600-cell | 720 | 120 | 600 | 12 | 6 | 3 | 3 | Pentagonal bipyramid | 1.447 | |||||
*The rectified 16-cell is the regular 24-cell and vertices are all equivalent – octahedra are regular bipyramids. **Given numerically due to more complex form.
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