Rectified 24-cell

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]

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
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

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

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
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

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
  • 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)

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.

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
  1. Coxeter 1973, p. 154, §8.4.

References
  • 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-Strass, 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".
Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron Octahedron • Cube Demicube Dodecahedron • Icosahedron
Uniform polychoron Pentachoron 16-cell • Tesseract Demitesseract 24-cell 120-cell • 600-cell
Uniform 5-polytope 5-simplex 5-orthoplex • 5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex • 6-cube 6-demicube 122 • 221
Uniform 7-polytope 7-simplex 7-orthoplex • 7-cube 7-demicube 132 • 231 • 321
Uniform 8-polytope 8-simplex 8-orthoplex • 8-cube 8-demicube 142 • 241 • 421
Uniform 9-polytope 9-simplex 9-orthoplex • 9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex • 10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds
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