Truncated 5-cell

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Truncated 5-cell
Schlegel diagram (tetrahedron cells visible)
Type	Uniform polychoron
Cells	5 (3.3.3) 5 (3.6.6)
Faces	20 {3} 10 {6}
Edges	40
Vertices	20
Coxeter-Dynkin diagram
Schläfli symbol	t_0,1{3,3,3}
Symmetry group	A₄, [3,3,3]
Properties	convex
Vertex figure Three truncated tetrahedrons and one tetrahedron meet at each vertex in an equilateral-triangular pyramid arrangement.

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.

1 Construction
2 Structure
3 Projections
4 Images
5 Alternate names
6 See also
7 External links

[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 positioned on the original vertices.

[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 5-cell is the 4-dimensional analogue of the truncated tetrahedron.