Rectified 24-cell

Rectified 24-cell
Schlegel diagram 8 of 24 cuboctahedral cells shown
Type	Uniform 4-polytope
Schläfli symbols	r{3,4,3} = [math]\displaystyle{ \left\{\begin{array}{l}3\\4,3\end{array}\right\} }[/math] rr{3,3,4}=[math]\displaystyle{ r\left\{\begin{array}{l}3\\3,4\end{array}\right\} }[/math] r{31,1,1} = [math]\displaystyle{ r\left\{\begin{array}{l}3\\3\\3\end{array}\right\} }[/math]
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	50px50px 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

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.

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

Stereographic projection
Center of stereographic projection with 96 triangular faces blue

Symmetry constructions

There are three different symmetry constructions of this polytope. The lowest [math]\displaystyle{ {D}_4 }[/math] construction can be doubled into [math]\displaystyle{ {C}_4 }[/math] by adding a mirror that maps the bifurcating nodes onto each other. [math]\displaystyle{ {D}_4 }[/math] can be mapped up to [math]\displaystyle{ {F}_4 }[/math] 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 [math]\displaystyle{ {D}_4 }[/math] construction, and two colors (1:2 ratio) in [math]\displaystyle{ {C}_4 }[/math], and all identical cuboctahedra in [math]\displaystyle{ {F}_4 }[/math].

Coxeter group	[math]\displaystyle{ {F}_4 }[/math] = [3,4,3]	[math]\displaystyle{ {C}_4 }[/math] = [4,3,3]	[math]\displaystyle{ {D}_4 }[/math] = [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)

Related polytopes

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.

Related uniform polytopes

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

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). 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: 1_n1)
• 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.
  • 3. Convex uniform polychora based on the icositetrachoron (24-cell) - Model 23, George Olshevsky.
  • 7. Uniform polychora derived from glomeric tetrahedron B4 - Model 23, George Olshevsky.
• Klitzing, Richard. "4D uniform polytopes (polychora) o3x4o3o - rico". https://bendwavy.org/klitzing/dimensions/polychora.htm.