Truncated 16-cell

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.

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.)

tructure

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

Projections

Centered on octahedron

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.

Centered on truncated tetrahedron

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

* The projection envelope is a truncated cube.
* The nearest truncated tetrahedron to the 4D viewpoint projects to the center of the envelope, with its triangular faces joined to 4 octahedral volumes that connect it to 4 of the triangular faces of the envelope.
* The remaining space in the envelope is filled by 4 other truncated tetrahedra.
* These volumes are the images of the cells lying on the near side of the truncated 16-cell; the other cells project onto the same layout except in the dual configuration.
* The six octagonal faces of the projection envelope are the images of the remaining 6 truncated tetrahedral cells.

Images

ee also

* 16-cell
* 24-cell
* Uniform polychoron
* Truncated octahedron

External links

* [http://members.aol.com/Polycell/section2.html 2. Convex uniform polychora based on the tesseract (8-cell) and hexadecachoron (16-cell)] - Model 17