Truncated 16-cell

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Truncated 16-cell
Schlegel diagram (octahedron cells visible)
Type	Uniform polychoron
Schläfli symbol	t_0,1{4,3,3}
Coxeter-Dynkin diagrams
Cells	8 3.3.3.3 16 3.6.6
Faces	64 {3} 32 {6}
Edges	120
Vertices	48
Symmetry group	[3,3,4]
Properties	convex
Vertex figure Three truncated tetrahedrons and one octahedron meet at each vertex in a square pyramid arrangement.

In geometry, the truncated 16-cell or cantic tesseract is a uniform polychoron (4-dimensional uniform polytope) which is bounded by 24 cells: 8 regular octahedra, and 16 truncated tetrahedra. It is related to, but not to be confused with, the 24-cell, which is a regular polychoron bounded by 24 regular octahedra.

1 Construction
2 Structure
3 Projections
4 Images
5 See also

[edit] Construction

The truncated 16-cell may be constructed from the 16-cell by truncating its vertices at 1/3 of the edge length. This results in the 16 truncated tetrahedral cells, and introduces the 8 octahedra (vertex figures).

(Truncating a 16-cell at 1/2 of the edge length results in the 24-cell, which has a greater degree of symmetry because the truncated cells become identical with the vertex figures.)

[edit] Structure

The truncated tetrahedra are joined to each other via their hexagonal faces. The octahedra are joined to the truncated tetrahedra via their triangular faces.

[edit] Projections

The octahedron-first parallel projection of the truncated 16-cell into 3-dimensional space has the following structure:

The projection envelope is a truncated octahedron.
The 6 square faces of the envelope are the images of 6 of the octahedral cells.
An octahedron lies at the center of the envelope, joined to the center of the 6 square faces by 6 edges. This is the image of the other 2 octahedral cells.
The remaining space between the envelope and the central octahedron is filled by 8 truncated tetrahedra (distorted by projection). These are the images of the 16 truncated tetrahedral cells, a pair of cells to each image.

This layout of cells in projection is analogous to the layout of faces in the projection of the truncated octahedron into 2-dimensional space. Hence, the truncated 16-cell may be thought of as the 4-dimensional analogue of the truncated octahedron.