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} = ${\displaystyle \left\{{\begin{array}{l}3\\4,3\end{array}}\right\}}$ rr{3,3,4}=${\displaystyle r\left\{{\begin{array}{l}3\\3,4\end{array}}\right\}}$ r{31,1,1} = ${\displaystyle r\left\{{\begin{array}{l}3\\3\\3\end{array}}\right\}}$
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 ${\displaystyle {D}_{4}}$ 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

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

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