Runcinated 24-cells

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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.


Runcinated 24-cell[edit]

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 Aut(F4), [[3,4,3]], order 2304
Properties convex, edge-transitive
Uniform index 25 26 27

In geometry, the runcinated 24-cell or small prismatotetracontoctachoron 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[edit]

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[edit]

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[edit]

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, {8,4|3}, is similarly related to the octagonal faces of the bitruncated 24-cell.

Runcitruncated 24-cell[edit]

Runcitruncated 24-cell
Type Uniform polychoron
Schläfli symbol t0,1,3{3,4,3}
s2,3{3,4,3}
Coxeter-Dynkin diagram CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.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], order 1152
Properties convex
Uniform index 28 29 30

The runcitruncated 24-cell or prismatorhombated icositetrachoron 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[edit]

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[edit]

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

Runcicantic snub 24-cell[edit]

A half-symmetry construction of the runcitruncated 24-cell (or runcicantellated 24-cell), as CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.png, also called a runcicantic snub 24-cell, as CDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.png, has an identical geometry, but its triangular faces are further subdivided. Like the snub 24-cell, it has symmetry [3+,4,3], order 576. The runcitruncated 24-cell has 192 identical hexagonal faces, while the runcicantic snub 24-cell has 2 constructive sets of 96 hexagons. The difference can be seen in the vertex figures:

Runcitruncated 24-cell verf.png
CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Runcicantic snub 24-cell verf.png
CDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.png

Runcic snub 24-cell[edit]

Orthographic projections
Runcic snub 24-cell.png
CDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png
Runcic snub 24-cell
Icosahedron centered
Ortho solid 969-uniform polychoron 343-snub.png
CDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
Snub 24-cell
Icosahedron centered
Runcitruncated 24-cell-ortho.png
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png
Runcitruncated 24-cell
Rhombicuboctahedron centered

A related polychoron is the runcic snub 24-cell or prismatorhombisnub icositetrachoron, s3{3,4,3}, CDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png. It is not uniform, but it is vertex-uniform and has all regular polygon faces. It is constructed with 24 icosahedra, 24 truncated tetrahedra, 96 triangular prisms, and 96 triangular cupolae in the gaps, for a total of 240 cells, 960 faces, 1008 edges, and 288 vertices. Like the snub 24-cell, it has symmetry [3+,4,3], order 576.[1]

The vertex figure contains one icosahedron, two triangular prisms, one truncated tetrahedron, and 3 triangular cupolae:

Runcic snub 24-cell verf.png


Omnitruncated 24-cell[edit]

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
Phyllic disphenoid
Symmetry group Aut(F4), [[3,4,3]], order 2304
Properties convex
Uniform index 29 30 31

The omnitruncated 24-cell or great prismatotetracontoctachoron 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 truncated cuboctahedra, 192 hexagonal prisms. Each vertex contains four cells in an irregular tetrahedral vertex figure: two hexagonal prisms, and two truncated cuboctahedra.

Structure[edit]

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[edit]

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)

Images[edit]

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 truncated cuboctahedron. 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.
Net
Great prismatotetracontoctachoron net.png
Omnitruncated 24-cell
Dual gippic net.png
Dual to omnitruncated 24-cell

Full snub 24-cell[edit]

Vertex figure for full snub 24-cell

The uniform snub 24-cell is more accurately called a semi-snub 24-cell with Coxeter diagram CDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png within the F4 family, although it is a full snub within the D4 family, as CDel node h.pngCDel 3.pngCDel node h.pngCDel split1.pngCDel nodes hh.png.

In contrast a full snub 24-cell or omnisnub 24-cell, defined as an alternation of the omnitruncated 24-cell, can not be made uniform, but it can be given Coxeter diagram CDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.png, and symmetry [[3,4,3]]+, order 1152, and constructed from 48 snub cubes, 192 octahedrons, and 576 tetrahedrons filling the gaps at the deleted vertices. Its vertex figure contains 4 tetrahedra, 2 octahedra, and 2 snub cubes. It has 816 cells, 2832 faces, 2592 edges, and 576 vertices.[2]

Related polytopes[edit]

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

Notes[edit]

  1. ^ Richard Klitzing, 4D, s3s4o3x
  2. ^ Richard Klitzing, 4D, s3s4s3s

References[edit]