Runcitruncated tesseract
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| Runcitruncated tesseract | |
|---|---|
 Schlegel diagram centered on a truncated cube, with cuboctahedral cells shown |
|
| Type | Uniform polychoron |
| Cells | 8 3.4.4  16 3.4.3.4  24 4.4.8  32 3.4.4  |
| Faces | 128 {3} 192 {4} 48 {8} |
| Edges | 480 |
| Vertices | 192 |
| Vertex configuration | Rectangular pyramid |
| Schläfli symbol | t0,1,3{4,3,3} |
| Coxeter-Dynkin diagrams |     |
| Symmetry group | B4, [3,3,4] |
| Properties | convex |
In geometry, the runcitruncated tesseract is a uniform polychoron (uniform 4-dimensional polytope) bounded by 80 cells: 8 truncated cubes, 16 cuboctahedra, 24 octagonal prisms, and 32 triangular prisms.
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[edit] Construction
The runcitruncated tesseract may be constructed from the truncated tesseract by expanding the truncated cube cells outward radially, and inserting octagonal prisms between them. In the process, the tetrahedra expand into cuboctahedra, and triangular prisms fill in the remaining gaps.
[edit] Projections
In the truncated cube first parallel projection of the runcitruncated tesseract into 3-dimensional space, the projection image is laid out as follows:
- The projection envelope is a non-uniform (small) rhombicuboctahedron, with 6 square faces and 12 rectangular faces.
- Two of the truncated cube cells project to a truncated cube in the center of the projection envelope.
- Six octagonal prisms connect this central truncated cube to the square faces of the envelope. These are the images of 12 of the octahedral prism cells, two cells to each image.
- The remaining 12 octahedral prisms are projected to the rectangular faces of the envelope.
- The 6 square faces of the envelope are the images of the remaining 6 truncated cube cells.
- Twelve right-angle triangular prisms connect the inner octagonal prisms. These are the images of 24 of the triangular prism cells. The remaining 8 triangular prisms project onto the triangular faces of the envelope.
- The 8 remaining volumes lying between the triangular faces of the envelope and the inner truncated cube are the images of the 16 cuboctahedral cells, a pair of cells to each image.
[edit] Images
Stereographic projection with its 128 blue triangular faces and its 192 green quad faces.
