Rectified 24-cell

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Rectified 24-cell
Schlegel half-solid cantellated 16-cell.png
Schlegel diagram
8 of 24 cuboctahedral cells shown
Type Uniform polychoron
Schläfli symbol r{3,4,3}
rr{3,3,4}
r{31,1,1}
Coxeter-Dynkin diagrams CDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes 11.png or CDel node.pngCDel splitsplit1.pngCDel branch3 11.pngCDel node 1.png
Cells 48 24 3.4.3.4 Cuboctahedron.png
24 4.4.4 Hexahedron.png
Faces 240 96 {3}
144 {4}
Edges 288
Vertices 96
Vertex figure Rectified 24-cell verf.pngCantellated 16-cell verf.pngRuncicantellated demitesseract verf.png
Triangular prism
Symmetry groups F4 [3,4,3], order 1152
B4 [3,3,4], order 384
D4 [31,1,1], order 192
Properties convex, edge-transitive
Uniform index 22 23 24

In geometry, the rectified 24-cell or rectified icositetrachoron is a uniform 4-dimensional polytope (or uniform polychoron), which is bounded by 48 cells: 24 cubes, and 24 cuboctahedra. It can be obtained by reducing the icositetrachoron's cells to cubes or cuboctahedra.

It can also be considered a cantellated 16-cell with the lower symmetries B4 = [3,3,4]. B4 would lead to a bicoloring of the cuboctahedral cells into 8 and 16 each. It is also called a runcicantellated demitesseract in a D4 symmetry, giving 3 colors of cells, 8 for each.

Cartesian coordinates[edit]

A rectified 24-cell having an edge length of √2 has vertices given by all permutations and sign permutations of the following Cartesian coordinates:

(0,1,1,2) [4!/2!×23 = 96 vertices]

The dual configuration with edge length 2 has all coordinate and sign permutations of:

(0,2,2,2) [4×23 = 32 vertices]
(1,1,1,3) [4×24 = 64 vertices]

Images[edit]

orthographic projections
Coxeter plane F4
Graph 24-cell t1 F4.svg
Dihedral symmetry [12]
Coxeter plane B3 / A2 (a) B3 / A2 (b)
Graph 24-cell t1 B3.svg 24-cell t2 B3.svg
Dihedral symmetry [6] [6]
Coxeter plane B4 B2 / A2
Graph 24-cell t1 B4.svg 24-cell t1 B2.svg
Dihedral symmetry [8] [4]
Stereographic projection
Rectified 24cell.png
Center of stereographic projection
with 96 triangular faces blue

Symmetry constructions[edit]

There are three different symmetry constructions of this polytope. The lowest {D}_3 construction can be doubled into {C}_3 by adding a mirror that maps the bifurcating nodes onto each other. {D}_3 can be mapped up to {F}_3 symmetry by adding two mirror that map all three end nodes together.

The vertex figure is a triangular prism, containing two cubes and three cuboctahedra. The three symmetries can be seen with 3 colored cuboctahedra in the lowest {D}_3 construction, and two colors (1:2 ratio) in {C}_3, and all identical cuboctahedra in {F}_3.

Coxeter group {F}_3 = [3,4,3] {C}_3 = [4,3,3] {D}_3 = [3,31,1]
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 CDel node.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.pngCDel 3.pngCDel node 1.png CDel nodes 11.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node 1.png
Facets 3: CDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png
2: CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
2,2: CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
2: CDel node.pngCDel 4.pngCDel node 1.pngCDel 2.pngCDel node 1.png
1,1,1: CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
2: CDel node 1.pngCDel 2.pngCDel node 1.pngCDel 2.pngCDel node 1.png
Vertex figure Rectified 24-cell verf.png Cantellated 16-cell verf.png Runcicantellated demitesseract verf.png

Alternate names[edit]

Related uniform 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}
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

The rectified 24-cell can also be derived as a cantellated 16-cell:

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

References[edit]