Rectified 24-cell

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.

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]

There are three different symmetry constructions of this polytope. The lowest ${\displaystyle {D}_{4}}$ construction can be doubled into ${\displaystyle {C}_{4}}$ by adding a mirror that maps the bifurcating nodes onto each other. ${\displaystyle {D}_{4}}$ can be mapped up to ${\displaystyle {F}_{4}}$ 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 ${\displaystyle {D}_{4}}$ construction, and two colors (1:2 ratio) in ${\displaystyle {C}_{4}}$, and all identical cuboctahedra in ${\displaystyle {F}_{4}}$.

Coxeter group	${\displaystyle {F}_{4}}$ = [3,4,3]	${\displaystyle {C}_{4}}$ = [4,3,3]	${\displaystyle {D}_{4}}$ = [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

• 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

• 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: 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".