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# Cantellated 16-cell

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 Title: Cantellated 16-cell Author: World Heritage Encyclopedia Language: English Subject: Collection: Publisher: World Heritage Encyclopedia Publication Date:

### Cantellated 16-cell

 Rectified 24-cell Schlegel diagram8 of 24 cuboctahedral cells shown Type Uniform polychoron Schläfli symbol r{3,4,3}rr{3,3,4}r{31,1,1} Coxeter-Dynkin diagrams Cells 48 24 Faces 240 96 {3}144 {4} Edges 288 Vertices 96 Vertex figure Triangular prism Symmetry groups F4 [3,4,3], order 1152B4 [3,3,4], order 384D4 [31,1,1], order 192 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.

## 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 $\left\{D\right\}_3$ construction can be doubled into $\left\{C\right\}_3$ by adding a mirror that maps the bifurcating nodes onto each other. $\left\{D\right\}_3$ can be mapped up to $\left\{F\right\}_3$ 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 $\left\{D\right\}_3$ construction, and two colors (1:2 ratio) in $\left\{C\right\}_3$, and all identical cuboctahedra in $\left\{F\right\}_3$.

Coxeter group $\left\{F\right\}_3$ = [3,4,3] $\left\{C\right\}_3$ = [4,3,3] $\left\{D\right\}_3$ = [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-Dynkin diagram
Facets 3:
2:
2,2:
2:
1,1,1:
2:
Vertex figure

## Related uniform polytopes

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}
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 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}
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 Coxeter plane graph