Truncated 24-cells


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 4-polytope (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


Schlegel diagram
Truncated 24-cell
Type Uniform 4-polytope
Schläfli symbols t{3,4,3}
tr{3,3,4}=
t{31,1,1} =
Coxeter 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 or truncated icositetrachoron is a uniform 4-dimensional polytope (or uniform 4-polytope), 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

The truncated 24-cell can be constructed from with three symmetry groups:

Coxeter group = [3,4,3] = [4,3,3] = [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 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

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 configuration 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

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

The parallel projection of the truncated 24-cell into 3-dimensional space, truncated octahedron first, has the following layout:

Images

orthographic projections
Coxeter plane F4
Graph
Dihedral symmetry [12]
Coxeter plane B3 / A2 (a) B3 / A2 (b)
Graph
Dihedral symmetry [6] [6]
Coxeter plane B4 B2 / A3
Graph
Dihedral symmetry [8] [4]

Schlegel diagram
(cubic cells visible)
Schlegel diagram
8 of 24 truncated octahedral cells visible

Stereographic projection
Centered on truncated tetrahedron
Nets

Truncated 24-cell
Dual to truncated 24-cell

Bitruncated 24-cell

Bitruncated 24-cell

Schlegel diagram, centered on truncated cube, with alternate cells hidden
Type Uniform 4-polytope
Schläfli symbol 2t{3,4,3}
Coxeter 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
dual polytope Disphenoidal 288-cell
Symmetry group Aut(F4), [[3,4,3]], order 2304
Properties convex, isogonal, isotoxal, isochoric
Uniform index 26 27 28
Net

The bitruncated 24-cell. 48-cell, or tetracontoctachoron is a 4-dimensional uniform polytope (or uniform 4-polytope) derived from the 24-cell.

E. L. Elte identified it in 1912 as a semiregular polytope.

It is constructed by bitruncating the 24-cell (truncating at halfway to the depth which would yield the dual 24-cell).

Being a uniform 4-polytope, 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

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.

Coordinates

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

Projection to 2 dimensions

orthographic projections
Coxeter plane F4 B4
Graph
Dihedral symmetry [[12]] = [24] [8]
Coxeter plane B3 / A2 B2 / A3
Graph
Dihedral symmetry [6] [[4]] = [8]

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:
  • 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.

Stereographic projection

Related regular skew polyhedron

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

Disphenoidal 288-cell
Type perfect[1] polychoron
Symbol f1,2F4[1]
(1,0,0,0)F4 ⊕ (0,0,0,1)F4[2]
Coxeter
Cells
288 congruent tetragonal disphenoids
Faces 576 congruent isosceles
  (2 short edges)
Edges 336 192 of length
144 of length
Vertices 48
Vertex figure
48 triakis octahedron,
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

Orthogonal projections
Coxeter planes B2 B3 F4
Disphenoidal
288-cell
Bitruncated
24-cell

Geometry

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, <2,3,4>, order 48.

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, <2,3,3>, order 24) and the 120-cell (≘ 2I, <2,3,5>, order 120). (The 16-cell corresponds to the binary dihedral group 2D2, <2,2,2>, order 16.)

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 LayerLatituderedyellow
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

B4 family of uniform polytopes:

F4 family of uniform polytopes:

References

  1. 1 2 On Perfect 4-Polytopes Gabor Gévay Contributions to Algebra and Geometry Volume 43 (2002), No. 1, 243-259 ] Table 2, page 252
  2. 1 2 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)(ω2+ω3)
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