Rectified 5-cell
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| Rectified 5-cell | |
|---|---|
 Schlegel diagram with the 5 tetrahedral cells shown. |
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| Type | Uniform polychoron |
| Cells | 5 (3.3.3)  5 (3.3.3.3)  |
| Faces | 30 {3} |
| Edges | 30 |
| Vertices | 10 |
| Vertex figure | 2 (3.3.3) 3 (3.3.3.3) (triangular prism) |
| Schläfli symbol | t1{3,3,3} |
| Coxeter-Dynkin diagram |    |
| Symmetry group | A4, [3,3,3] |
| Properties | convex |
In geometry, the rectified 5-cell is a uniform polychoron composed of 5 regular tetrahedral and 5 regular octahedral cells. Each edge has one tetrahedron and two octahedra. Each vertex has two tetrahedra and three octahedra. In total it has 30 triangle faces, 30 edges, and 10 vertices.
It is one of three semiregular polychora made of two or more cells which are platonic solids, discovered by Thorold Gosset in his 1900 paper. He called it a Tetroctahedric for being made of tetrahedron and octahedron cells.
The vertex figure of the rectified 5-cell is a uniform triangular prism, formed by three octahedra around the sides, and two tetrahedra on the opposite ends.
Contents |
[edit] Alternate names
- Dispentachoron
- Rectified 5-cell (Norman W. Johnson)
- Rectified pentachoron
- Rectified 4-simplex
- Rap (Jonathan Bowers: for rectified pentachoron)
- Ambopentachoron (Neil Sloane & John Horton Conway)
[edit] Images
 stereographic projection (centered on octahedron) |
 Net (polytope) |
 Two orthographic projections |
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 An orthographic projection from a skew direction with pentagrammic symmetry. |
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[edit] See also
[edit] References
- T. Gosset: On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900
