Bitruncated 24-cell
From Wikipedia, the free encyclopedia
| Bitruncated 24-cell | ||
Stereographic projection close-up |
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| Type | Uniform polychoron | |
| Vertices | 288 | |
| Edges | 576 | |
| Faces | 336 | 192 {3} 144 {8} |
| Cells | 48 (3.8.8) |
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| Edge figure | 3.8.8 | |
| Vertex figure | tetragonal disphenoid |
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| Schläfli symbol | t1,2{3,4,3} | |
| Coxeter-Dynkin diagram | ||
| Symmetry group | [3,4,3], order 2304 | |
| Properties | convex, isogonal, isotoxal, isochoric | |
| Uniform index | 26 27 28 | |
In geometry, the bitruncated 24-cell is a 4-dimensional uniform polytope (or uniform polychoron) derived from the 24-cell. It is constructed by bitruncating the 24-cell (truncating at halfway to the depth which would yield the dual 24-cell).
Being a uniform polychoron, it is vertex-transitive. In addition, it is cell-transitive, consisting of 48 truncated cubes, and also edge-transitive, with 3 truncated cubes cells per edge and with one triangle and two octagons around each edge.
The 48 cells of the bitruncated 24-cell correspond with the 24 cells and 24 vertices of the 24-cell. As such, the centers of the 48 cells form the root system of type F4.
Its vertex figure is a tetragonal disphenoid, a tetrahedron with 2 opposite edges length 1 and all 4 lateral edges length sqrt(2+sqrt(2)).
Contents |
[edit] Structure
The truncated cubes are joined to each other via their octagonal faces in anti orientation; i. e., two adjoining truncated cubes are rotated 45 degrees relative to each other so that no two triangular faces share an edge.
The sequence of truncated cubes joined to each other via opposite octagonal faces form a cycle of 8. Each truncated cube belongs to 3 such cycles. On the other hand, the sequence of truncated cubes joined to each other via opposite triangular faces form a cycle of 6. Each truncated cube belongs to 4 such cycles.
[edit] Coordinates
The Cartesian coordinates of a bitruncated 24-cell having edge length 2 are all permutations of coordinates and sign of:
[edit] Projections
[edit] Projection to 2 dimensions
| Centered on octagon | Centered on triangle |
[edit] Projection to 3 dimensions
| Orthographic | Perspective |
|---|---|
The following animation shows the orthographic projection of the bitruncated 24-cell into 3 dimensions. The animation itself is a perspective projection from the static 3D image into 2D, with rotation added to make its structure more apparent.![]() The images of the 48 truncated cubes are laid out as follows:
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The following animation shows the cell-first perspective projection of the bitruncated 24-cell into 3 dimensions. Its structure is the same as the previous animation, except that there is some foreshortening due to the perspective projection. |
[edit] See also
- 24-cell
- Bitruncated 5-cell, the other cell-transitive, non-regular uniform polychoron
- Uniform polychoron
- Polychoron



