Truncated 24-cell
24-cell |
Truncated 24-cell |
Bitruncated 24-cell |
|
| Schlegel diagrams centered on one [3,4] (cells at opposite at [4,3]) | |||
In geometry, a truncated 24-cell is a uniform polychoron (4-dimensional uniform polytope) formed as the truncation of the regular 24-cell.
There are two degrees of trunctions, including a bitruncation.
Contents |
Truncated 24-cell [edit]
Schlegel diagram |
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|---|---|---|
| Truncated 24-cell | ||
| Type | Uniform polychoron | |
| Schläfli symbol | t0,1{3,4,3} t0,1,2{3,3,4} |
|
| Coxeter-Dynkin diagram | ||
| Cells | 48 | 24 4.6.6 24 4.4.4 |
| Faces | 240 | 144 {4} 96 {6} |
| Edges | 384 | |
| Vertices | 192 | |
| Vertex figure | equilateral triangular pyramid |
|
| Symmetry group | F4 [3,4,3], order 1152 | |
| Rotation subgroup | [3,4,3]+, order 576 | |
| Commutator subgroup | [3+,4,3+], order 288 | |
| Properties | convex zonohedron | |
| Uniform index | 23 24 25 | |
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. Each vertex joins three truncated octahedra and one cube, in an equilateral triangular pyramid vertex figure.
Construction [edit]
The truncated 24-cell can be constructed from with three symmetry groups:
- F4 [3,4,3]: A truncation of the 24-cell.
- B4 [3,3,4]: A cantitruncation of the 16-cell, with two families of truncated octahedral cells.
- D4 [31,1,1]: An omnitruncation of the demitesseract, with three families of truncated octahedral cells.
| Coxeter group | = [3,4,3] |
= [4,3,3] |
= [3,31,1] |
|---|---|---|---|
| Schläfli symbol | t0,1{3,4,3} | t0,1,2{3,3,4} | t0,1,2,3{31,1,1} |
| Order | 1152 | 384 | 192 |
| Full symmetry group |
[3,4,3] | [4,3,3] | <[3,31,1]> = [4,3,3] [3[31,1,1]] = [3,4,3] |
| Coxeter-Dynkin diagram | |||
| Facets | 3: 1: |
2: 1: 1: |
1,1,1: 1: |
| Vertex figure |
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).
Cartesian coordinates [edit]
The Cartesian coordinates of the vertices of a truncated 24-cell having edge length sqrt(2) are all coordinate permutations and sign combinations of:
- (0,1,2,3) [4!×23 = 192 vertices]
The dual configuation has coordinates at all coordinate permutation and signs of
- (1,1,1,5) [4×24 = 64 vertices]
- (1,3,3,3) [4×24 = 64 vertices]
- (2,2,2,4) [4×24 = 64 vertices]
Structure [edit]
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.
Projections [edit]
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.
Images [edit]
| Coxeter plane | F4 | |
|---|---|---|
| Graph | ||
| Dihedral symmetry | [12] | |
| Coxeter plane | B3 / A2 (a) | B3 / A2 (b) |
| Graph | ||
| Dihedral symmetry | [6] | [6] |
| Coxeter plane | B4 | B2 / A2 |
| Graph | ||
| Dihedral symmetry | [8] | [4] |
Schlegel diagram (cubic cells visible) |
Schlegel diagram 8 of 24 truncated octahedral cells visible |
net |
Stereographic projection Centered on truncated tetrahedron |
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Bitruncated 24-cell [edit]
| Bitruncated 24-cell | ||
|---|---|---|
Schlegel diagram, centered on truncated cube, with alternate cells hidden |
||
| Type | Uniform polychoron | |
| Schläfli symbol | t1,2{3,4,3} | |
| Coxeter-Dynkin diagram | ||
| Cells | 48 (3.8.8) |
|
| Faces | 336 | 192 {3} 144 {8} |
| Edges | 576 | |
| Vertices | 288 | |
| Edge figure | 3.8.8 | |
| Vertex figure | tetragonal disphenoid |
|
| Symmetry group | 2×F4 [[3,4,3]], order 2304 | |
| Properties | convex, isogonal, isotoxal, isochoric | |
| Uniform index | 26 27 28 | |
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 √(2+√2).
Alternative names [edit]
- Bitruncated 24-cell (Norman W. Johnson)
- 48-cell as a cell-transitive 4-polytope
- Bitruncated icositetrachoron
- Bitruncated polyoctahedron
- Tetracontaoctachoron (Cont) (Jonathan Bowers)
Structure [edit]
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.
Coordinates [edit]
The Cartesian coordinates of a bitruncated 24-cell having edge length 2 are all permutations of coordinates and sign of:
- (0, 2+√2, 2+√2, 2+2√2)
- (1, 1+√2, 1+√2, 3+2√2)
Projections [edit]
Projection to 2 dimensions [edit]
| Coxeter plane | F4 | B4 |
|---|---|---|
| Graph | ||
| Dihedral symmetry | [[12]] | [8] |
| Coxeter plane | B3 / A2 | B2 / A3 |
| Graph | ||
| Dihedral symmetry | [6] | [[4]] |
Projection to 3 dimensions [edit]
| 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:
|
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. |
Related regular skew polyhedron [edit]
The regular skew polyhedron, {8,4|3}, exists in 4-space with 4 octagonal around each vertex, in a zig-zagging nonplanar vertex figure. These octagonal faces can be seen on the bitruncated 24-cell, using all 576 edges and 288 vertices. The 192 triangular faces of the bitruncated 24-cell can be seen as removed. The dual regular skew polyhedron, {4,8|3}, is similarly related to the square faces of the runcinated 24-cell.
Related polytopes [edit]
BC4 family of unifom polytopes:
| Name | tesseract | rectified tesseract |
truncated tesseract |
cantellated tesseract |
runcinated tesseract |
bitruncated tesseract |
cantitruncated tesseract |
runcitruncated tesseract |
omnitruncated tesseract |
|---|---|---|---|---|---|---|---|---|---|
| Coxeter-Dynkin diagram |
|||||||||
| Schläfli symbol |
{4,3,3} | t1{4,3,3} | t0,1{4,3,3} | t0,2{4,3,3} | t0,3{4,3,3} | t1,2{4,3,3} | t0,1,2{4,3,3} | t0,1,3{4,3,3} | t0,1,2,3{4,3,3} |
| Schlegel diagram |
|||||||||
| B4 Coxeter plane graph | |||||||||
| Name | 16-cell | rectified 16-cell |
truncated 16-cell |
cantellated 16-cell |
runcinated 16-cell |
bitruncated 16-cell |
cantitruncated 16-cell |
runcitruncated 16-cell |
omnitruncated 16-cell |
| Coxeter-Dynkin diagram |
|||||||||
| Schläfli symbol |
{3,3,4} | t1{3,3,4} | t0,1{3,3,4} | t0,2{3,3,4} | t0,3{3,3,4} | t1,2{3,3,4} | t0,1,2{3,3,4} | t0,1,3{3,3,4} | t0,1,2,3{3,3,4} |
| Schlegel diagram |
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| B4 Coxeter plane graph | |||||||||
F4 family of unifom polytopes:
| Name | 24-cell | truncated 24-cell | snub 24-cell | rectified 24-cell | cantellated 24-cell | bitruncated 24-cell | cantitruncated 24-cell | runcinated 24-cell | runcitruncated 24-cell | omnitruncated 24-cell |
|---|---|---|---|---|---|---|---|---|---|---|
| Schläfli symbol |
{3,4,3} | t0,1{3,4,3} | h0,1{3,4,3} | t1{3,4,3} | t0,2{3,4,3} | t1,2{3,4,3} | t0,1,2{3,4,3} | t0,3{3,4,3} | t0,1,3{3,4,3} | t0,1,2,3{3,4,3} |
| Coxeter-Dynkin diagram |
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| Schlegel diagram |
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| F4 | ||||||||||
| B4 | ||||||||||
| B3(a) | ||||||||||
| B3(b) | ||||||||||
| B2 |
External links [edit]
- 3. Convex uniform polychora based on the icositetrachoron (24-cell) - Model 24, 27, George Olshevsky.
- Richard Klitzing, 4D, uniform polytopes (polychora) x3x4o3o - tico, o3x4x3o - cont
= [3,4,3]
= [4,3,3]
= [3,31,1]
