• Source: Truncated 24-cells

In geometry, a 24.3/info/truncated" target="_blank">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 truncations, including a bitruncation.




24.3/info/truncated" target="_blank">Truncated 24-cell



The 24.3/info/truncated" target="_blank">truncated 24-cell or 24.3/info/truncated" target="_blank">truncated icositetrachoron is a uniform 4-dimensional polytope (or uniform 4-polytope), which is bounded by 48 cells: 24 cubes, and 24 24.3/info/truncated" target="_blank">truncated octahedra. Each vertex joins three 24.3/info/truncated" target="_blank">truncated octahedra and one cube, in an equilateral triangular pyramid vertex figure.




= Construction

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The 24.3/info/truncated" target="_blank">truncated 24-cell can be constructed from polytopes 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 24.3/info/truncated" target="_blank">truncated octahedral cells.
D4 [31,1,1]: An omnitruncation of the demitesseract, with three families of 24.3/info/truncated" target="_blank">truncated octahedral cells.




= Zonotope

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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

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The Cartesian coordinates of the vertices of a 24.3/info/truncated" target="_blank">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

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The 24 cubical cells are joined via their square faces to the 24.3/info/truncated" target="_blank">truncated octahedra; and the 24 24.3/info/truncated" target="_blank">truncated octahedra are joined to each other via their hexagonal faces.




= Projections

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The parallel projection of the 24.3/info/truncated" target="_blank">truncated 24-cell into 3-dimensional space, 24.3/info/truncated" target="_blank">truncated octahedron first, has the following layout:

The projection envelope is a 24.3/info/truncated" target="_blank">truncated cuboctahedron.
Two of the 24.3/info/truncated" target="_blank">truncated octahedra project onto a 24.3/info/truncated" target="_blank">truncated octahedron lying in the center of the envelope.
Six cuboidal volumes join the square faces of this central 24.3/info/truncated" target="_blank">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 24.3/info/truncated" target="_blank">truncated octahedra.
The 8 (non-uniform) 24.3/info/truncated" target="_blank">truncated octahedral volumes lying between the hexagonal faces of the projection envelope and the central 24.3/info/truncated" target="_blank">truncated octahedron are the images of the remaining 16 24.3/info/truncated" target="_blank">truncated octahedra, a pair of cells to each image.




= Images

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= Related polytopes

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The convex hull of the 24.3/info/truncated" target="_blank">truncated 24-cell and its dual (assuming that they are congruent) is a nonuniform polychoron composed of 480 cells: 48 cubes, 144 square antiprisms, 288 tetrahedra (as tetragonal disphenoids), and 384 vertices. Its vertex figure is a hexakis triangular cupola.
Vertex figure




Bitruncated 24-cell



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 24.3/info/truncated" target="_blank">truncated cubes, and also edge-transitive, with 3 24.3/info/truncated" target="_blank">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

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Bitruncated 24-cell (Norman W. Johnson)
48-cell as a cell-transitive 4-polytope
Bitruncated icositetrachoron
Bitruncated polyoctahedron
Tetracontaoctachoron (Cont) (Jonathan Bowers)




= Structure

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The 24.3/info/truncated" target="_blank">truncated cubes are joined to each other via their octagonal faces in anti orientation; i. e., two adjoining 24.3/info/truncated" target="_blank">truncated cubes are rotated 45 degrees relative to each other so that no two triangular faces share an edge.
The sequence of 24.3/info/truncated" target="_blank">truncated cubes joined to each other via opposite octagonal faces form a cycle of 8. Each 24.3/info/truncated" target="_blank">truncated cube belongs to 3 such cycles. On the other hand, the sequence of 24.3/info/truncated" target="_blank">truncated cubes joined to each other via opposite triangular faces form a cycle of 6. Each 24.3/info/truncated" target="_blank">truncated cube belongs to 4 such cycles.
Seen in a configuration matrix, all incidence counts between elements are shown. The diagonal f-vector numbers are derived through the Wythoff construction, dividing the full group order of a subgroup order by removing one mirror at a time. Edges exist at 4 symmetry positions. Squares exist at 3 positions, hexagons 2 positions, and octagons one. Finally the 4 types of cells exist centered on the 4 corners of the fundamental simplex.




= Coordinates

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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

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Projection to 2 dimensions




Projection to 3 dimensions





= Related regular skew polyhedron

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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

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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).



Images




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 2O or <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 or <2,3,3>, order 24) and the 600-cell (≘ 2I or <2,3,5>, order 120). (The 16-cell corresponds to the binary dihedral group 2D2 or <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.

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 complete coordinates are given as linear combinations of the quaternionic units



1
,

i

,

j

,

k



{\displaystyle 1,\mathrm {i} ,\mathrm {j} ,\mathrm {k} }

, which at the same time can be taken as the elements of the group 2O. 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.
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.




Related polytopes



B4 family of uniform polytopes:

F4 family of uniform polytopes:




References



H.S.M. Coxeter:
Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [1]
(Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
(Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
(Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
Norman Johnson Uniform Polytopes, Manuscript (1991)
N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. (1966)
Klitzing, Richard. "4D uniform polytopes (polychora)". x3x4o3o=x3x3x4o - tico, o3x4x3o - cont
3. Convex uniform polychora based on the icositetrachoron (24-cell) - Model 24, 27, George Olshevsky.

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