Runcinated 24-cell

Runcinated 24-cell
24-cell t0 F4.svg
24-cell
CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
24-cell t03 F4.svg
Runcinated 24-cell
CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png
24-cell t013 F4.svg
Runcitruncated 24-cell
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png
24-cell t0123 F4.svg
Omnitruncated 24-cell
(Runcicantitruncated 24-cell)
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Orthogonal projections in F4 Coxeter plane

In four-dimensional geometry, a runcinated 24-cell is a convex uniform polychoron, being a runcination (a 3rd order truncation) of the regular 24-cell.

There are 3 unique degrees of runcinations of the 24-cell including with permutations truncations and cantellations.

Contents


Runcinated 24-cell

Runcinated 24-cell
Type Uniform polychoron
Schläfli symbol t0,3{3,4,3}
Coxeter-Dynkin diagram CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png
Cells 240 48 3.3.3.3Octahedron.png
192 3.4.4 Triangular prism.png
Faces 672 384{3}
288{4}
Edges 576
Vertices 144
Vertex figure Runcinated 24-cell verf.png
elongated square antiprism
Symmetry group F4, [[3,4,3]], order 2304
Properties convex, edge-transitive
Uniform index 25 26 27

In geometry, the runcinated 24-cell is a uniform polychoron bounded by 48 octahedra and 192 triangular prisms. The octahedral cells correspond with the cells of a 24-cell and its dual.

Coordinates

The Cartesian coordinates of the runcinated 24-cell having edge length 2 is given by all permutations of sign and coordinates of:

(0, 0, √2, 2+√2)
(1, 1, 1+√2, 1+√2)

The permutations of the second set of coordinates coincide with the vertices of an inscribed cantellated tesseract.

Projections

orthographic projections
Coxeter plane F4 B4
Graph 24-cell t03 F4.svg 24-cell t03 B4.svg
Dihedral symmetry [[12]] [8]
Coxeter plane B3 / A2 B2 / A3
Graph 24-cell t03 B3.svg 24-cell t03 B2.svg
Dihedral symmetry [6] [[4]]
3D perspective projections
Runcinated 24-cell Schlegel halfsolid.png
Schlegel diagram, centered on octahedron, with the octahedra shown.
Runcinated 24-cell-perspective-octahedron-first.gif
Perspective projection of the runcinated 24-cell into 3 dimensions, centered on an octahedral cell.

The rotation is only of the 3D image, in order to show its structure, not a rotation in 4-space. Fifteen of the octahedral cells facing the 4D viewpoint are shown here in red. The gaps between them are filled up by a framework of triangular prisms.

Runcinated 24cell1.png
Stereographic projection with 24 of its 48 octahedral cells

Related regular skew polyhedron

The regular skew polyhedron, {4,8|3}, exists in 4-space with 8 square around each vertex, in a zig-zagging nonplanar vertex figure. These square faces can be seen on the runcinated 24-cell, using all 576 edges and 288 vertices. The 384 triangular faces of the runcinated 24-cell can be seen as removed. The dual regular skew polyhedron, {8,3|3}, is similarly related to the octagonal faces of the bitruncated 24-cell.

Runcitruncated 24-cell

Runcitruncated 24-cell
Type Uniform polychoron
Schläfli symbol t0,1,3{3,4,3}
Coxeter-Dynkin diagram CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png
Cells 240 24 4.6.6 Truncated octahedron.png
96 4.4.6 Hexagonal prism.png
96 3.4.4 Triangular prism.png
24 3.4.4.4 Small rhombicuboctahedron.png
Faces 1104 192{3}
720{4}
192{6}
Edges 1440
Vertices 576
Vertex figure Runcitruncated 24-cell verf.png
Trapezoidal pyramid
Symmetry group F4, [3,4,3]
Properties convex
Uniform index 28 29 30

The runcitruncated 24-cell is a uniform polychoron derived from the 24-cell. It is bounded by 24 truncated octahedra, corresponding with the cells of a 24-cell, 24 rhombicuboctahedra, corresponding with the cells of the dual 24-cell, 96 triangular prisms, and 96 hexagonal prisms.

Coordinates

The Cartesian coordinates of an origin-centered runcitruncated 24-cell having edge length 2 are given by all permutations of coordinates and sign of:

(0, √2, 2√2, 2+3√2)
(1, 1+√2, 1+2√2, 1+3√2)

The permutations of the second set of coordinates give the vertices of an inscribed omnitruncated tesseract.

The dual configuration has coordinates generated from all permutations and signs of:

(1,1,1+√2,5+√2)
(1,3,3+√2,3+√2)
(2,2,2+√2,4+√2)

Projections

orthographic projections
Coxeter plane F4
Graph 24-cell t023 F4.svg
Dihedral symmetry [12]
Coxeter plane B3 / A2 (a) B3 / A2 (b)
Graph 24-cell t023 B3.svg 24-cell t013 B3.svg
Dihedral symmetry [6] [6]
Coxeter plane B4 B2 / A2
Graph 24-cell t023 B4.svg 24-cell t023 B2.svg
Dihedral symmetry [8] [4]
Runcitruncated 24-cell.png
Schlegel diagram
centered on rhombicuboctahedron
only triangular prisms shown

Omnitruncated 24-cell

Omnitruncated 24-cell
Type Uniform polychoron
Schläfli symbol t0,1,2,3{3,4,3}
Coxeter-Dynkin diagram CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Cells 240 48 (4.6.8) Great rhombicuboctahedron.png
192 (4.4.6) Hexagonal prism.png
Faces 1392 864{4}
384{6}
144{8}
Edges 2304
Vertices 1152
Vertex figure Omnitruncated 24-cell verf.png
Irreg. tetrahedron
Symmetry group F4, [[3,4,3]], order 2304
Properties convex
Uniform index 29 30 31

The omnitruncated 24-cell is a uniform polychoron derived from the 24-cell. It is composed of 1152 vertices, 2304 edges, and 1392 faces (864 squares, 384 hexagons, and 144 octagons). It has 240 cells: 48 great rhombicuboctahedra, 192 hexagonal prisms. Each vertex contains four cells in an irregular tetrahedral vertex figure: two hexagonal prisms, and two truncated cuboctahedra.

Structure

The 48 great rhombicuboctahedral cells are joined to each other via their octagonal faces. They can be grouped into two groups of 24 each, corresponding with the cells of a 24-cell and its dual. The gaps between them are filled in by a network of 192 hexagonal prisms, joined to each other via alternating square faces in alternating orientation, and to the great rhombicuboctahedra via their hexagonal faces and remaining square faces.

Coordinates

The Cartesian coordinates of an omnitruncated 24-cell having edge length 2 are all permutations of coordinates and sign of:

(1, 1+√2, 1+2√2, 5+3√2)
(1, 3+√2, 3+2√2, 3+3√2)
(2, 2+√2, 2+2√2, 4+3√2)

Projections

orthographic projections
Coxeter plane F4 B4
Graph 24-cell t0123 F4.svg 24-cell t0123 B4.svg
Dihedral symmetry [[12]] [8]
Coxeter plane B3 / A2 B2 / A3
Graph 24-cell t0123 B3.svg 24-cell t0123 B2.svg
Dihedral symmetry [6] [[4]]
3D perspective projections
Omnitruncated 24-cell.png
Schlegel diagram
Omnitruncated 24-cell perspective-great rhombicuboctahedron-first-01.png
Perspective projection into 3D centered on a great rhombicuboctahedron. The nearest great rhombicuboctahedral cell to the 4D viewpoint is shown in red, with the six surrounding great rhombicuboctahedra in yellow. Twelve of the hexagonal prisms sharing a square face with the nearest cell and hexagonal faces with the yellow cells are shown in blue. The remaining cells are shown in green. Cells lying on the far side of the polytope from the 4D viewpoint have been culled for clarity.

Related polytopes

Name 24-cell truncated 24-cell rectified 24-cell cantellated 24-cell bitruncated 24-cell cantitruncated 24-cell runcinated 24-cell runcitruncated 24-cell omnitruncated 24-cell snub 24-cell
Schläfli
symbol
{3,4,3} t0,1{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} h0,1{3,4,3}
Coxeter-Dynkin
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.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 CDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
Schlegel
diagram
Schlegel wireframe 24-cell.png Schlegel half-solid truncated 24-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 Schlegel half-solid alternated cantitruncated 16-cell.png
F4 24-cell t0 F4.svg 24-cell t01 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 24-cell h01 F4.svg
B4 24-cell t0 B4.svg 24-cell t01 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 24-cell h01 B4.svg
B3(a) 24-cell t0 B3.svg 24-cell t01 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 24-cell h01 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 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 24-cell h01 B2.svg

References


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