Truncated 24-cell
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| Truncated 24-cell | |
|---|---|
 Schlegel diagram (cubic cells visible) |
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| Type | Uniform polychoron |
| Schläfli symbol | t0,1{3,4,3} t0,1,2{3,3,4} t0,1{31,1,1} |
| Coxeter-Dynkin diagrams |         |
| Cells | 24 4.6.6  24 4.4.4  |
| Faces | 144 {4} 96 {6} |
| Edges | 384 |
| Vertices | 192 |
| Symmetry groups | F4 [3,4,3] B4 [3,3,4] D4 [31,1,1] |
| Properties | convex |
Vertex figure Three truncated octahedrons and one cube meet at each vertex in an equilateral triangular pyramid arrangement. |
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In geometry, the truncated 24-cell is a uniform 4-dimensional polytope (or uniform polychoron), which is bounded by 48 cells: 24 cubes, and 24 truncated octahedra.
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[edit] Construction
The truncated 24-cell, as its name implies, can be constructed from the 24-cell by truncating its vertices at 1/3 of the edge length. It is also a zonotope: it can be formed as the Minkowski sum of the six line segments connecting opposite pairs among the twelve permutations of the vector (+1,−1,0,0).
[edit] Images
 Schlegel diagram 8 of 24 of truncated octahedral cells visible |
[edit] Structure
The 24 cubical cells are joined via their square faces to the truncated octahedra; and the 24 truncated octahedra are joined to each other via their hexagonal faces.
[edit] Projections
The parallel projection of the truncated 24-cell into 3-dimensional space, truncated octahedron first, has the following layout:
- The projection envelope is a great rhombicuboctahedron.
- Two of the truncated octahedra project onto a truncated octahedron lying in the center of the envelope.
- Six cuboidal volumes join the square faces of this central truncated octahedron to the center of the octagonal faces of the great rhombicuboctahedron. These are the images of 12 of the cubical cells, a pair of cells to each image.
- The 12 square faces of the great rhombicuboctahedron are the images of the remaining 12 cubes.
- The 6 octagonal faces of the great rhombicuboctahedron are the images of 6 of the truncated octahedra.
- The 8 (non-uniform) truncated octahedral volumes lying between the hexagonal faces of the projection envelope and the central truncated octahedron are the images of the remaining 16 truncated octahedra, a pair of cells to each image.
[edit] Images
 net |
 stereographic projection (centered on truncated tetrahedron) |
