Truncated 24-cells

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Schlegel wireframe 24-cell.png
24-cell
CDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png
Schlegel half-solid truncated 24-cell.png
Truncated 24-cell
CDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Bitruncated 24-cell Schlegel halfsolid.png
Bitruncated 24-cell
CDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png
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.

Truncated 24-cell[edit]

Schlegel half-solid truncated 24-cell.png
Schlegel diagram
Truncated 24-cell
Type Uniform polychoron
Schläfli symbol t{3,4,3}
tr{3,3,4}
Coxeter-Dynkin diagram CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
Cells 48 24 4.6.6 Truncated octahedron.png
24 4.4.4 Hexahedron.png
Faces 240 144 {4}
96 {6}
Edges 384
Vertices 192
Vertex figure Truncated 24-cell verf.png
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 or truncated icositetrachoron 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:

  1. F4 [3,4,3]: A truncation of the 24-cell.
  2. B4 [3,3,4]: A cantitruncation of the 16-cell, with two families of truncated octahedral cells.
  3. D4 [31,1,1]: An omnitruncation of the demitesseract, with three families of truncated octahedral cells.
Coxeter group {F}_3 = [3,4,3] {C}_3 = [4,3,3] {D}_3 = [3,31,1]
Schläfli symbol t{3,4,3} tr{3,3,4} t{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 CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel splitsplit1.pngCDel branch3 11.pngCDel node 1.png
Facets 3: CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png
1: CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
2: CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
1: CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.png
1: CDel node.pngCDel 4.pngCDel node 1.pngCDel 2.pngCDel node 1.png
1,1,1: CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
1: CDel node 1.pngCDel 2.pngCDel node 1.pngCDel 2.pngCDel node 1.png
Vertex figure Truncated 24-cell verf.png Cantitruncated 16-cell verf.png Omnitruncated demitesseract verf.png

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 truncated cuboctahedron.
  • 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]

orthographic projections
Coxeter plane F4
Graph 24-cell t01 F4.svg
Dihedral symmetry [12]
Coxeter plane B3 / A2 (a) B3 / A2 (b)
Graph 24-cell t01 B3.svg 24-cell t23 B3.svg
Dihedral symmetry [6] [6]
Coxeter plane B4 B2 / A2
Graph 24-cell t01 B4.svg 24-cell t01 B2.svg
Dihedral symmetry [8] [4]
Schlegel half-solid truncated 24-cell.png
Schlegel diagram
(cubic cells visible)
Schlegel half-solid cantitruncated 16-cell.png
Schlegel diagram
8 of 24 truncated octahedral cells visible
Truncated xylotetron stereographic oblique.png
Stereographic projection
Centered on truncated tetrahedron
Nets
Truncated 24-cell net.png
Truncated 24-cell
Dual tico net.png
Dual to truncated 24-cell

Bitruncated 24-cell[edit]

Bitruncated 24-cell
Bitruncated 24-cell Schlegel halfsolid.png
Schlegel diagram, centered on truncated cube, with alternate cells hidden
Type Uniform polychoron
Schläfli symbol 2t{3,4,3}
Coxeter-Dynkin diagram CDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png
Cells 48 (3.8.8) Truncated hexahedron.png
Faces 336 192 {3}
144 {8}
Edges 576
Vertices 288
Edge figure 3.8.8
Vertex figure Bitruncated 24-cell verf.png
tetragonal disphenoid
dual polytope Disphenoidal 288-cell
Symmetry group Aut(F4), [[3,4,3]], order 2304
Properties convex, isogonal, isotoxal, isochoric
Uniform index 26 27 28

The bitruncated 24-cell. 48-cell, or tetracontoctachoron 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]

orthographic projections
Coxeter plane F4 B4
Graph 24-cell t12 F4.svg 24-cell t12 B4.svg
Dihedral symmetry [[12]] [8]
Coxeter plane B3 / A2 B2 / A3
Graph 24-cell t12 B3.svg 24-cell t12 B2.svg
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.
Bitruncated-24cell-parallelproj-01.gif
The images of the 48 truncated cubes are laid out as follows:
  • The central truncated cube is the cell closest to the 4D viewpoint, highlighted to make it easier to see. To reduce visual clutter, the vertices and edges that lie on this central truncated cube have been omitted.
  • Surrounding this central truncated cube are 6 truncated cubes attached via the octagonal faces, and 8 truncated cubes attached via the triangular faces. These cells have been made transparent so that the central cell is visible.
  • The 6 outer square faces of the projection envelope are the images of another 6 truncated cubes, and the 12 oblong octagonal faces of the projection envelope are the images of yet another 12 truncated cubes.
  • The remaining cells have been culled because they lie on the far side the bitruncated 24-cell, and are obscured from the 4D viewpoint. These include the antipodal truncated cube, which would have projected to the same volume as the highlighted truncated cube, with 6 other truncated cubes surrounding it attached via octagonal faces, and 8 other truncated cubes surrounding it attached via triangular faces.
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.

Bitruncated 24cell perspective 04.gif

Stereographic projection
Bitruncated xylotetron stereographic close-up.png

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.

Disphenoidal 288-cell[edit]

Disphenoidal 288-cell
Type perfect[1] polychoron
Symbol f1,2F4[1]
(1,0,0,0)F4 ⊕ (0,0,0,1)F4[2]
Coxeter CDel node.pngCDel 3.pngCDel node f1.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node.png
Cells Disphenoid tetrahedron.png
288 congruent tetragonal disphenoids
Faces 576 congruent isosceles
  (2 short edges)
Edges 336 192 of length \scriptstyle 1
144 of length \scriptstyle \sqrt{2-\sqrt2}
Vertices 48
Vertex figure Triakisoctahedron.jpg
48 triakis octahedron, CDel node.pngCDel 3.pngCDel node f1.pngCDel 4.pngCDel node f1.png
Dual Bitruncated 24-cell
Coxeter group Aut(F4), [[3,4,3]], order 2304
Orbit vector (1, 2, 1, 1)
Properties convex, isochoric

The disphenoidal 288-cell, is the dual of the bitruncated 24-cell. It is a 4-dimensional polytope (or polychoron) derived from the 24-cell. It is constructed by doubling and rotating the 24-cell, then constructing the convex hull.

Being the dual of a uniform polychoron, it is cell-transitive, consisting of 288 congruent tetragonal disphenoids. In addition, it is vertex-transitive under the group Aut(F4).[2]

Images[edit]

Orthogonal projections
Coxeter planes B2 B3 F4
Disphenoidal
288-cell
Dual bitruncated 24-cell B2-3.png F4 roots by 24-cell duals.svg
Bitruncated
24-cell
24-cell t12 B2.svg 24-cell t12 B3.svg 24-cell t12 F4.svg

Geometry[edit]

The vertices of the 288-cell are precisely the 24 Hurwitz unit quaternions with norm squared 1, united with the 24 vertices of the dual 24-cell with norm squared 2, projected to the unit 3-sphere. These 48 vertices correspond to the binary octahedral group.

Thus, the 288-cell is the only non-regular 4-polytope which is the convex hull of a quaternionic group, disregarding the infinitely many dicyclic (same as binary dihedral) groups; the regular ones are the 24-cell (≘ 2T) and the 120-cell (≘ 2I). (The 16-cell corresponds to the binary dihedral group 2D2.)

The inscribed 3-sphere has radius 1/2+2/4 ≈ 0.853553 and touches the 288-cell at the centers of the 288 tetrahedra which are the vertices of the dual bitruncated 24-cell.

The vertices can be coloured in 2 colours, say red and yellow, with the 24 Hurwitz units in red and the 24 duals in yellow, the yellow 24-cell being congruent to the red one. Thus the product of 2 equally coloured quaternions is red and the product of 2 in mixed colours is yellow.

There are 192 long edges with length 1 connecting equal colours and 144 short edges with length 2–2 ≈ 0.765367 connecting mixed colours. 192*2/48 = 8 long and 144*2/48 = 6 short, that is together 14 edges meet at any vertex.

The 576 faces are isosceles with 1 long and 2 short edges, all congruent. The angles at the base are arccos(4+8/4) ≈ 49.210°. 576*3/48 = 36 faces meet at a vertex, 576*1/192 = 3 at a long edge, and 576*2/144 = 8 at a short one.

The 288 cells are tetrahedra with 4 short edges and 2 antipodal and perpendicular long edges, one of which connects 2 red and the other 2 yellow vertices. All the cells are congruent. 288*4/48 = 24 cells meet at a vertex. 288*2/192 = 3 cells meet at a long edge, 288*4/144 = 8 at a short one. 288*4/576 = 2 cells meet at a triangle.

Region Layer Latitude red yellow
Northern hemisphere 3 1 1 0
2 2/2 0 6
1 1/2 8 0
Equator 0 0 6 12
Southern hemisphere –1 –1/2 8 0
–2 2/2 0 6
–3 –1 1 0
Total 24 24

Placing a fixed red vertex at the north pole (1,0,0,0), there are 6 yellow vertices in the next deeper “latitude” at (2/2,x,y,z), followed by 8 red vertices in the latitude at (1/2,x,y,z). The next deeper latitude is the equator hyperplane intersecting the 3-sphere in a 2-sphere which is populated by 6 red and 12 yellow vertices.

Layer 2 is a 2-sphere circumscribing a regular octahedron whose edges have length 1. A tetrahedron with vertex north pole has 1 of these edges as long edge whose 2 vertices are connected by short edges to the north pole. Another long edge runs from the north pole into layer 1 and 2 short edges from there into layer 2.


Related polytopes[edit]

D4 uniform polychora
CDel node 1.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png
CDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes 10lu.png
CDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes 11.png
CDel node 1.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes 10lu.png
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel split1.pngCDel nodes.png
CDel node.pngCDel 3.pngCDel node 1.pngCDel split1.pngCDel nodes 10lu.png
CDel node.pngCDel 3.pngCDel node 1.pngCDel split1.pngCDel nodes 11.png
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel split1.pngCDel nodes 10lu.png
CDel node.pngCDel 3.pngCDel node 1.pngCDel split1.pngCDel nodes.png
CDel node 1.pngCDel splitsplit1.pngCDel branch3.pngCDel node.png
CDel node 1.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes 11.png
CDel node.pngCDel splitsplit1.pngCDel branch3 11.pngCDel node 1.png
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel split1.pngCDel nodes 11.png
CDel node 1.pngCDel splitsplit1.pngCDel branch3 11.pngCDel node 1.png
CDel node h.pngCDel 3.pngCDel node h.pngCDel split1.pngCDel nodes hh.png
CDel node h.pngCDel splitsplit1.pngCDel branch3 hh.pngCDel node h.png
4-demicube t0 D4.svg 4-demicube t02 D4.svg 4-demicube t01 D4.svg 4-demicube t012 D4.svg 4-demicube t1 D4.svg 4-demicube t023 D4.svg 4-demicube t0123 D4.svg 24-cell h01 B3.svg
{3,31,1}
h{4,3,3}
2r{3,31,1}
h3{4,3,3}
t{3,31,1}
h2{4,3,3}
2t{3,31,1}
h2,3{4,3,3}
r{3,31,1}
{31,1,1}={3,4,3}
rr{3,31,1}
r{31,1,1}=r{3,4,3}
tr{3,31,1}
t{31,1,1}=t{3,4,3}
sr{3,31,1}
s{31,1,1}=s{3,4,3}

BC4 family of unifom polytopes:

Name tesseract rectified
tesseract
truncated
tesseract
cantellated
tesseract
runcinated
tesseract
bitruncated
tesseract
cantitruncated
tesseract
runcitruncated
tesseract
omnitruncated
tesseract
Coxeter
diagram
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
= CDel nodes 11.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.png
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
= CDel nodes 11.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node.png
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Schläfli
symbol
{4,3,3} t1{4,3,3}
r{4,3,3}
t0,1{4,3,3}
t{4,3,3}
t0,2{4,3,3}
rr{4,3,3}
t0,3{4,3,3} t1,2{4,3,3}
2t{4,3,3}
t0,1,2{4,3,3}
tr{4,3,3}
t0,1,3{4,3,3} t0,1,2,3{4,3,3}
Schlegel
diagram
Schlegel wireframe 8-cell.png Schlegel half-solid rectified 8-cell.png Schlegel half-solid truncated tesseract.png Schlegel half-solid cantellated 8-cell.png Schlegel half-solid runcinated 8-cell.png Schlegel half-solid bitruncated 8-cell.png Schlegel half-solid cantitruncated 8-cell.png Schlegel half-solid runcitruncated 8-cell.png Schlegel half-solid omnitruncated 8-cell.png
B4 4-cube t0.svg 4-cube t1.svg 4-cube t01.svg 4-cube t02.svg 4-cube t03.svg 4-cube t12.svg 4-cube t012.svg 4-cube t013.svg 4-cube t0123.svg
 
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
diagram
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
= CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node 1.png
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
= CDel nodes.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node.png
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
= CDel nodes.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node 1.png
CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
= CDel nodes 11.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node 1.png
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
= CDel nodes 11.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node.png
CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
= CDel nodes 11.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node 1.png
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Schläfli
symbol
{3,3,4} t1{3,3,4}
r{3,3,4}
t0,1{3,3,4}
t{3,3,4}
t0,2{3,3,4}
rr{3,3,4}
t0,3{3,3,4} t1,2{3,3,4}
2t{3,3,4}
t0,1,2{3,3,4}
tr{3,3,4}
t0,1,3{3,3,4} t0,1,2,3{3,3,4}
Schlegel
diagram
Schlegel wireframe 16-cell.png Schlegel half-solid rectified 16-cell.png Schlegel half-solid truncated 16-cell.png Schlegel half-solid cantellated 16-cell.png Schlegel half-solid runcinated 16-cell.png Schlegel half-solid bitruncated 16-cell.png Schlegel half-solid cantitruncated 16-cell.png Schlegel half-solid runcitruncated 16-cell.png Schlegel half-solid omnitruncated 16-cell.png
B4 4-cube t3.svg 4-cube t2.svg 4-cube t23.svg 4-cube t13.svg 4-cube t03.svg 4-cube t12.svg 4-cube t123.svg 4-cube t023.svg 4-cube t0123.svg

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}
t{3,4,3}
s{3,4,3} t1{3,4,3}
r{3,4,3}
t0,2{3,4,3}
rr{3,4,3}
t1,2{3,4,3}
2t{3,4,3}
t0,1,2{3,4,3}
tr{3,4,3}
t0,3{3,4,3} t0,1,3{3,4,3} t0,1,2,3{3,4,3}
Coxeter
diagram
CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png CDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png CDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Schlegel
diagram
Schlegel wireframe 24-cell.png Schlegel half-solid truncated 24-cell.png Schlegel half-solid alternated cantitruncated 16-cell.png Schlegel half-solid cantellated 16-cell.png Cantel 24cell1.png Bitruncated 24-cell Schlegel halfsolid.png Cantitruncated 24-cell schlegel halfsolid.png Runcinated 24-cell Schlegel halfsolid.png Runcitruncated 24-cell.png Omnitruncated 24-cell.png
F4 24-cell t0 F4.svg 24-cell t01 F4.svg 24-cell h01 F4.svg 24-cell t1 F4.svg 24-cell t02 F4.svg 24-cell t12 F4.svg 24-cell t012 F4.svg 24-cell t03 F4.svg 24-cell t013 F4.svg 24-cell t0123 F4.svg
B4 24-cell t0 B4.svg 24-cell t01 B4.svg 24-cell h01 B4.svg 24-cell t1 B4.svg 24-cell t02 B4.svg 24-cell t12 B4.svg 24-cell t012 B4.svg 24-cell t03 B4.svg 24-cell t013 B4.svg 24-cell t0123 B4.svg
B3(a) 24-cell t0 B3.svg 24-cell t01 B3.svg 24-cell h01 B3.svg 24-cell t1 B3.svg 24-cell t02 B3.svg 24-cell t12 B3.svg 24-cell t012 B3.svg 24-cell t03 B3.svg 24-cell t013 B3.svg 24-cell t0123 B3.svg
B3(b) 24-cell t3 B3.svg 24-cell t23 B3.svg 24-cell t2 B3.svg 24-cell t13 B3.svg 24-cell t123 B3.svg 24-cell t023 B3.svg
B2 24-cell t0 B2.svg 24-cell t01 B2.svg 24-cell h01 B2.svg 24-cell t1 B2.svg 24-cell t02 B2.svg 24-cell t12 B2.svg 24-cell t012 B2.svg 24-cell t03 B2.svg 24-cell t013 B2.svg 24-cell t0123 B2.svg

References[edit]

  1. ^ a b On Perfect 4-Polytopes Gabor Gévay Contributions to Algebra and Geometry Volume 43 (2002), No. 1, 243-259 ] Table 2, page 252
  2. ^ a b Quaternionic Construction of the W(F4) Polytopes with Their Dual Polytopes and Branching under the Subgroups W(B4) and W(B3) × W(A1) Mehmet Koca 1, Mudhahir Al-Ajmi 2 and Nazife Ozdes Koca 3 Department of Physics, College of Science, Sultan Qaboos University P. O. Box 36, Al-Khoud 123, Muscat, Sultanate of Oman, p.18. 5.7 Dual polytope of the polytope (0, 1, 1, 0)F4 = W(F4)(ω23)

External links[edit]