Truncated 5-cell
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| Truncated 5-cell | |
|---|---|
 stereographic projection (centered on truncated tetrahedron) |
|
| Type | Uniform polychoron |
| Cells | 5 (3.3.3) 5 (3.6.6) |
| Faces | 30 {3} |
| Edges | 40 |
| Vertices | 20 |
| Vertex figure | 1 (3.3.3) 3 (3.6.6) (Equilateral triangular pyramid) |
| Schläfli symbol | t0,1{3,3,3} |
| Symmetry group | A4, [3,3,3] |
| Properties | convex |
In geometry, the truncated 5-cell or truncated pentatope is a uniform polychoron (4-dimensional polytope) bounded by 10 cells: 5 tetrahedra, and 5 truncated tetrahedra.
Contents |
[edit] Alternative names
- Truncated 5-cell
- Truncated pentatope
- Truncated [[Simplex|4-simplex]
- Tip (Jonathan Bowers: for truncated pentachoron)
[edit] Construction
The truncated 5-cell may be constructed from the 5-cell by truncating its vertices at 1/3 the edge length. This truncates the 5 tetrahedral cells into truncated tetrahedra, and introduces 5 new tetrahedral cells (the vertex figure).
[edit] Structure
The truncated tetrahedra are joined to each other via their hexagonal faces, and to the tetrahedra via their triangular faces.
[edit] Projections
The tetrahedron-first parallel projection of the truncated 5-cell into 3-dimensional space has the following structure:
- The projection envelope is a truncated tetrahedron.
- One of the truncated tetrahedral cells project onto the entire envelope.
- One of the tetrahedral cells project onto a tetrahedron lying at the center of the envelope.
- Four flattened tetrahedra are joined to the triangular faces of the envelope, and connected to the central tetrahedron via 4 radial edges. These are the images of the remaining 4 tetrahedral cells.
- Between the central tetrahedron and the 4 hexagonal faces of the envelope are 4 irregular truncated tetrahedral volumes, which are the images of the 4 remaining trucated tetrahedral cells.
This layout of cells in projection is analogous to the layout of faces in the face-first projection of the truncated tetrahedron into 2-dimensional space. The truncated 16-cell is the 4-dimensional analogue of the truncated tetrahedron.
