Cantitruncated tesseract
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| Cantitruncated tesseract | |
|---|---|
 Schlegel diagram centered on great rhombicuboctahedron cell with octagonal faces hidden. |
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| Type | Uniform polychoron |
| Cells | 8 4.6.8  16 3.6.6  32 3.4.4  |
| Faces | 64 {3} 96 {4} 64 {6} 24 {8} |
| Edges | 384 |
| Vertices | 192 |
| Vertex figure | Sphenoid |
| Schläfli symbol | t0,1,2{4,3,3} |
| Coxeter-Dynkin diagrams |     |
| Symmetry group | B4, [3,3,4] |
| Properties | convex |
In geometry, the cantitruncated tesseract is a uniform polychoron (or uniform 4-dimensional polytope) that is bounded by 56 cells: 8 great rhombicuboctahedra, 16 truncated tetrahedra, and 32 triangular prisms.
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[edit] Construction
The cantitruncated tesseract is constructed by the cantitruncation of the tesseract. Cantitruncation is often thought of as rectification followed by truncation. However, the result of this construction would be a polytope which, while its structure would be very similar to that given by cantitruncation, not all of its faces would be regular.
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[edit] Structure
The 8 great rhombicuboctahedra are joined to each other via their octagonal faces, in an arrangement corresponding to the 8 cubical cells of the tesseract. They are joined to the 16 truncated tetrahedra via their hexagonal faces, and their square faces are joined to the square faces of the 32 triangular prisms. The triangular faces of the triangular prisms are joined to the truncated tetrahedra.
The truncated tetrahedra correspond with the tesseract's vertices, and the triangular prisms correspond with the tesseract's edges.
[edit] Projections
In the great rhombicuboctahedron first parallel projection into 3 dimensions, the cells of the cantitruncated tesseract are laid out as follows:
- The projection envelope is a non-uniform truncated cube, with longer edges between octagons and shorter edges in the 8 triangles.
- The irregular octagonal faces of the envelope correspond with the images of 6 of the 8 great rhombicuboctahedral cells.
- The other two great rhombicuboctahedral cells project to a great rhombicubotahedron inscribed in the projection envelope. The octagonal faces touch the irregular octagons of the envelope.
- In the spaces corresponding to a cube's edges lie 12 volumes in the shape of irregular triangular prisms. These are the images, one per pair, of 24 of the triangular prism cells.
- The remaining 8 triangular prisms project onto the triangular faces of the projection envelope.
- The remaining 8 spaces, corresponding to a cube's corners, are the images of the 16 truncated tetrahedra, a pair to each space.
This layout of cells in projection is similar to that of the cantellated tesseract.
