Runcitruncated 16-cell
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| Runcitruncated 16-cell | |
|---|---|
 Schlegel diagram centered on rhombicuboctahedron, only truncated tetahedra cells shown |
|
| Type | Uniform polychoron |
| Cells | 8 3.4.4.4  16 3.6.6  24 4.4.4  32 4.4.6  |
| Faces | 64 {3} 240 {4} 64 {6} |
| Edges | 480 |
| Vertices | 192 |
| Vertex figure | Trapezoidal pyramid |
| Symmetry group | B4, [3,3,4] |
| Schläfli symbol | t0,1,3{3,3,4} |
| Properties | convex |
In geometry, the runcitruncated 16-cell is a convex uniform polychoron, or 4-dimensional polytope, bounded by 80 cells: 8 small rhombicuboctahedra, 16 truncated tetrahedra, 24 cubes, and 32 hexagonal prisms.
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[edit] Construction
The runcitruncated 16-cell may be constructed by contracting the small rhombicuboctahedral cells of the cantellated tesseract radially, and filling in the spaces between them with cubes. In the process, the octahedral cells expand into truncated tetrahedra (half of their triangular faces are expanded into hexagons by pulling apart the edges), and the triangular prisms expand into hexagonal prisms (each with its three original square faces joined, as before, to small rhombicuboctahedra, and its three new square faces joined to cubes).
[edit] Structure
The small rhombicuboctahedral cells are joined via their 6 axial square faces to the cubical cells, and joined via their 12 non-axial square faces to the hexagonal prisms. The cubical cells are joined to the rhombicuboctahedra via 2 opposite faces, and joined to the hexagonal prisms via the remaining 4 faces. The hexagonal prisms are connected to the truncated tetrahedra via their hexagonal faces, and to the rhombicuboctahedra via 3 of their square faces each, and to the cubes via the other 3 square faces. The truncated tetrahedra are joined to the rhombicuboctahedra via their triangular faces, and the hexagonal prisms via their hexagonal faces.
[edit] Projections
The following is the layout of the cells of the runcitruncated 16-cell under the parallel projection, small rhombicuboctahedron first, into 3-dimensional space:
- The projection envelope is a great rhombicuboctahedron.
- Six of the small rhombicuboctahedra project onto the 6 octagonal faces of this envelope, and the other two project to a small rhombicuboctahedron lying at the center of this envelope.
- The 6 cuboidal volumes connecting the axial square faces of the central small rhombicuboctahedron to the center of the octagons correspond with the image of 12 of the cubical cells (each pair of the twelve share the same image).
- The remaining 12 cubical cells project onto the 12 square faces of the great rhombicuboctahedral envelope.
- The 8 volumes connecting the hexagons of the envelope to the triangular faces of the central rhombicuboctahedron are the images of the 16 truncated tetrahedra.
- The remaining 12 spaces connecting the non-axial square faces of the central small rhombicuboctahedron to the square faces of the envelope are the images of 24 of the hexagonal prisms.
- Finally, the last 8 hexagonal prisms project onto the hexagonal faces of the envelope.
This layout of cells is similar to the layout of the faces of the great rhombicuboctahedron under the projection into 2-dimensional space. Hence, the runcitruncated 16-cell may be thought of as one of the 4-dimensional analogues of the great rhombicuboctahedron. The other analogue is the omnitruncated tesseract.
