Rectified 24-cell

Rectified 24-cell

Schlegel diagram
8 of 24 cuboctahedral cells shown
Type Uniform polychoron
Schläfli symbol t1{3,4,3}
t0,2{3,3,4}
t0,2,3{31,1,1}
Coxeter-Dynkin diagrams

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]
B4 [3,3,4]
D4 [31,1,1]
Properties convex, edge-transitive
Uniform index 22 23 24

In geometry, the rectified 24-cell is a uniform 4-dimensional polytope (or uniform polychoron), which is bounded by 48 cells: 24 cubes, and 24 cuboctahedra. It can be obtained by reducing the icositetrachoron's cells to cubes or cuboctahedra.

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.

Contents

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!x23 = 96 vertices]

The dual configuration has all coordinate and sign permutations of:

(0,2,2,2) [4x23 = 32 vertices]
(1,1,1,3) [4x24 = 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 / A2
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 {D}_3 construction can be doubled into {C}_3 by adding a mirror that maps the bifurcating nodes onto each other. {D}_3 can be mapped up to {F}_3 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 {D}_3 construction, and two colors (1:2 ratio) in {C}_3, and all identical cuboctahedra in {F}_3.

In {F}_3 symmetry one further symmetry exists that maps the two cubes in the vertex figure onto each other, represented by Coxeter symmetry notation [[3,4,3]], and having a doubled order of 2304.

Coxeter group Order Full
symmetry
group		 Coxeter-Dynkin diagram Facets Vertex figure
{F}_3 = [3,4,3] 1152
(2304)		 [[3,4,3]] 3:
2:
{C}_3 = [4,3,3] 384 [4,3,3] 2,2:
2:
{D}_3 = [3,31,1] 192 <[3,31,1]> = [4,3,3]
[3[31,1,1]] = [3,4,3]		 1,1,1:
2:

Alternate names

Related uniform polytopes

Name 24-cell truncated 24-cell rectified 24-cell cantellated 24-cell bitruncated 24-cell cantitruncated 24-cell runcinated 24-cell runcitruncated 24-cell omnitruncated 24-cell snub 24-cell
Schläfli
symbol		 {3,4,3} t0,1{3,4,3} t1{3,4,3} t0,2{3,4,3} t1,2{3,4,3} t0,1,2{3,4,3} t0,3{3,4,3} t0,1,3{3,4,3} t0,1,2,3{3,4,3} h0,1{3,4,3}
Coxeter-Dynkin
diagram
Schlegel
diagram
F4
B4
B3(a)
B3(b)
B2

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

Name tesseract rectified
tesseract		 truncated
tesseract		 cantellated
tesseract		 runcinated
tesseract		 bitruncated
tesseract		 cantitruncated
tesseract		 runcitruncated
tesseract		 omnitruncated
tesseract
Coxeter-Dynkin
diagram
Schläfli
symbol		 {4,3,3} t1{4,3,3} t0,1{4,3,3} t0,2{4,3,3} t0,3{4,3,3} t1,2{4,3,3} t0,1,2{4,3,3} t0,1,3{4,3,3} t0,1,2,3{4,3,3}
Schlegel
diagram
B4 Coxeter plane graph
 
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-Dynkin
diagram
Schläfli
symbol		 {3,3,4} t1{3,3,4} t0,1{3,3,4} t0,2{3,3,4} t0,3{3,3,4} t1,2{3,3,4} t0,1,2{3,3,4} t0,1,3{3,3,4} t0,1,2,3{3,3,4}
Schlegel
diagram
B4 Coxeter plane graph

References