Rectified 24-cell

(Redirected from Cantellated 16-cell)
Rectified 24-cell

Schlegel diagram
8 of 24 cuboctahedral cells shown
Type Uniform 4-polytope
Schläfli symbols r{3,4,3} =
rr{3,3,4}=
r{31,1,1} =
Coxeter diagrams

or
Cells 48 24 3.4.3.4
24 4.4.4
Faces 240 96 {3}
144 {4}
Edges 288
Vertices 96
Vertex figure
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 4-polytope), which is bounded by 48 cells: 24 cubes, and 24 cuboctahedra. It can be obtained by rectification of the 24-cell, reducing its octahedral cells to cubes and cuboctahedra.[1]

Net

E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as tC24.

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.

Construction

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The rectified 24-cell can be derived from the 24-cell by the process of rectification: the 24-cell is truncated at the midpoints. The vertices become cubes, while the octahedra become cuboctahedra.

Cartesian coordinates

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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

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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]
Stereographic projection
 
Center of stereographic projection
with 96 triangular faces blue

Symmetry constructions

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There are three different symmetry constructions of this polytope. The lowest   construction can be doubled into   by adding a mirror that maps the bifurcating nodes onto each other.   can be mapped up to   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   construction, and two colors (1:2 ratio) in  , and all identical cuboctahedra in  .

Coxeter group   = [3,4,3]   = [4,3,3]   = [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                      
Facets 3:      
2:      
2,2:      
2:      
1,1,1:      
2:      
Vertex figure      

Alternate names

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  • Rectified 24-cell, Cantellated 16-cell (Norman Johnson)
  • Rectified icositetrachoron (Acronym rico) (George Olshevsky, Jonathan Bowers)
    • Cantellated hexadecachoron
  • Disicositetrachoron
  • Amboicositetrachoron (Neil Sloane & John Horton Conway)
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The convex hull of the rectified 24-cell and its dual (assuming that they are congruent) is a nonuniform polychoron composed of 192 cells: 48 cubes, 144 square antiprisms, and 192 vertices. Its vertex figure is a triangular bifrustum.

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D4 uniform polychora
     
     
     
     
     
     
     
     
     
    
     
    
     
    
     
    
               
{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}
24-cell family polytopes
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
                                                                               
Schlegel
diagram
                   
F4                    
B4                    
B3(a)                    
B3(b)            
B2                    

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

B4 symmetry polytopes
Name tesseract rectified
tesseract
truncated
tesseract
cantellated
tesseract
runcinated
tesseract
bitruncated
tesseract
cantitruncated
tesseract
runcitruncated
tesseract
omnitruncated
tesseract
Coxeter
diagram
               
=      
                               
=      
                       
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
                 
B4                  
 
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
       
=      
       
=      
       
=      
       
=      
               
=      
       
=      
               
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
                 
B4                  

Citations

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  1. ^ Coxeter 1973, p. 154, §8.4.

References

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  • T. Gosset: On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900
  • Coxeter, H.S.M. (1973) [1948]. Regular Polytopes (3rd ed.). New York: Dover.
  • John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 26. pp. 409: Hemicubes: 1n1)
  • Norman Johnson Uniform Polytopes, Manuscript (1991)
    • N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. (1966)
  • 2. Convex uniform polychora based on the tesseract (8-cell) and hexadecachoron (16-cell) - Model 23, George Olshevsky.
  • Klitzing, Richard. "4D uniform polytopes (polychora) o3x4o3o - rico".
Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
Uniform polychoron Pentachoron 16-cellTesseract Demitesseract 24-cell 120-cell600-cell
Uniform 5-polytope 5-simplex 5-orthoplex5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex6-cube 6-demicube 122221
Uniform 7-polytope 7-simplex 7-orthoplex7-cube 7-demicube 132231321
Uniform 8-polytope 8-simplex 8-orthoplex8-cube 8-demicube 142241421
Uniform 9-polytope 9-simplex 9-orthoplex9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplexn-cube n-demicube 1k22k1k21 n-pentagonal polytope
Topics: Polytope familiesRegular polytopeList of regular polytopes and compounds