Omnitruncated tesseract
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| Omnitruncated tesseract | |
|---|---|
 Schlegel diagram, centered on truncated cuboctahedron, truncated octahedral cells shown |
|
| Type | Uniform polychoron |
| Cells | 8 4.6.8  16 4.6.6  24 4.4.8  32 4.4.6  |
| Faces | 288 {4} 128 {6} 48 {8} |
| Edges | 768 |
| Vertices | 384 |
| Vertex figure | Chiral scalene tetrahedron |
| Schläfli symbol | t0,1,2,3{3,3,4} |
| Coxeter-Dynkin diagrams |    |
| Symmetry group | [3,3,4] |
| Properties | convex |
In geometry, the omnitruncated tesseract is a uniform polychoron (or uniform 4-dimensional polytope) bounded by 80 cells: 8 great rhombicuboctahedra, 16 truncated octahedra, 24 octagonal prisms, and 32 hexagonal prisms. It is also known as the omnitruncated 16-cell.
Contents |
[edit] Construction
The omnitruncated tesseract can be constructed from the cantitruncated tesseract by radially displacing the great rhombicuboctahedral cells so that octagonal prisms can be inserted between their octagonal faces. As a result, the triangular prisms expand into hexagonal prisms, and the truncated tetrahedra expand into truncated octahedra.
[edit] Structure
The great rhombicuboctahedral cells are joined to the octagonal prisms via their octagonal faces, the truncated octahedra via their hexagonal faces, and the hexagonal prisms via their square faces. The octagonal prisms are joined to the hexagonal prisms and the truncated octahedra via their square faces, and the hexagonal prisms are joined to the truncated octahedra via their hexagonal faces.
[edit] Projections
In the great rhombicuboctahedron first parallel projection of the omnitruncated tesseract into 3 dimensions, the images of its cells are laid out as follows:
- The projection envelope is in the shape of a non-uniform great rhombicuboctahedron.
- Two of the great rhombicuboctahedra project to the center of the projection envelope.
- The remaining 6 great rhombicuboctahedra project to the (non-regular) octagonal faces of the envelope. These are connected to the central great rhombicuboctahedron via 6 octagonal prisms, which are the images of the octagonal prism cells, a pair to each image.
- The 8 hexagonal faces of the envelope are the images of 8 of the hexagonal prisms.
- The remaining hexagonal prisms are projected to 12 non-regular hexagonal prism images, lying where a cube's edges would be. Each image corresponds to two cells.
- Finally, the 8 volumes between the hexagonal faces of the projection envelope and the hexagonal faces of the central great rhombicuboctahedron are the images of the 16 truncated octahedra, two cells to each image.
This layout of cells in projection is similar to that of the runcitruncated 16-cell, which is analogous to the layout of faces in the octagon-first projection of the great rhombicuboctahedron into 2 dimensions. Thus, the omnitruncated tesseract may be thought of as another analogue of the great rhombicuboctahedron in 4 dimensions.
[edit] Images
.png/200px-Omnitruncated_tesseract_stereographic_(tCO).png) stereographic projection Centered on great rhombicuboctahedron |
.png/200px-Omnitruncated_tesseract_stereographic_(tO).png) stereographic projection Centered on truncated octahedron |
