# Cantellated 24-cells

| 24-cell | Cantellated 24-cell | Cantitruncated 24-cell |
| Orthogonal projections in F$_4$ Coxeter plane |||
In four-dimensional geometry, a cantellated 24-cell is a convex uniform 4-polytope, being a cantellation (a 2nd order truncation) of the regular 24-cell.
There are 2 unique degrees of cantellations of the 24-cell including permutations with truncations.

## Cantellated 24-cell

Cantellated 24-cell :
| Type | Uniform 4-polytope | Uniform 4-polytope |
| Schläfli symbol | rr{3,4,3} s$_2${3,4,3} | rr{3,4,3} s$_2${3,4,3} |
| Coxeter diagram | | |
| Cells | 144 | 24 (3.4.4.4) 24 (3.4.3.4) 96 (3.4.4) |
| Faces | 720 | 288 triangles 432 squares |
| Edges | 864 | 864 |
| Vertices | 288 | 288 |
| Vertex figure | Wedge | Wedge |
| Symmetry group | F$_4$, [3,4,3], order 1152 | F$_4$, [3,4,3], order 1152 |
| Properties | convex | convex |
| Uniform index | 24 25 26 | 24 25 26 |
The cantellated 24-cell or small rhombated icositetrachoron is a uniform 4-polytope.
The boundary of the cantellated 24-cell is composed of 24 truncated octahedral cells, 24 cuboctahedral cells and 96 triangular prisms. Together they have 288 triangular faces, 432 square faces, 864 edges, and 288 vertices.

### Construction

When the cantellation process is applied to 24-cell, each of the 24 octahedra becomes a small rhombicuboctahedron. In addition however, since each octahedra's edge was previously shared with two other octahedra, the separating edges form the three parallel edges of a triangular prism - 96 triangular prisms, since the 24-cell contains 96 edges. Further, since each vertex was previously shared with 12 faces, the vertex would split into 12 (24*12=288) new vertices. Each group of 12 new vertices forms a cuboctahedron.

### Coordinates

The Cartesian coordinates of the vertices of the cantellated 24-cell having edge length 2 are all permutations of coordinates and sign of:
    (0, √2, √2, 2+2√2)
    (1, 1+√2, 1+√2, 1+2√2)
The permutations of the second set of coordinates coincide with the vertices of an inscribed runcitruncated tesseract.
The dual configuration has all permutations and signs of:
    (0,2,2+√2,2+√2)
    (1,1,1+√2,3+√2)

### Structure

The 24 small rhombicuboctahedra are joined to each other via their triangular faces, to the cuboctahedra via their axial square faces, and to the triangular prisms via their off-axial square faces. The cuboctahedra are joined to the triangular prisms via their triangular faces. Each triangular prism is joined to two cuboctahedra at its two ends.

### Cantic snub 24-cell

A half-symmetry construction of the cantellated 24-cell, also called a cantic snub 24-cell, as , has an identical geometry, but its triangular faces are further subdivided. The cantellated 24-cell has 2 positions of triangular faces in ratio of 96 and 192, while the cantic snub 24-cell has 3 positions of 96 triangles.
The difference can be seen in the vertex figures, with edges representing faces in the 4-polytope:

### Images

orthographic projections :
| Coxeter plane | F$_4$ | F$_4$ |
| Graph | | |
| Dihedral symmetry | [12] | [12] |
| Coxeter plane | B$_3$ / A$_2$ (a) | B$_3$ / A$_2$ (b) |
| Graph | | |
| Dihedral symmetry | [6] | [6] |
| Coxeter plane | B$_4$ | B$_2$ / A$_3$ |
| Graph | | |
| Dihedral symmetry | [8] | [4] |
Schlegel diagrams :
| Schlegel diagram | Showing 24 cuboctahedra. | Showing 96 triangular prisms. |

### Related polytopes

The convex hull of two cantellated 24-cells in opposite positions is a nonuniform polychoron composed of 864 cells: 48 cuboctahedra, 144 square antiprisms, 384 octahedra (as triangular antipodiums), 288 tetrahedra (as tetragonal disphenoids), and 576 vertices. Its vertex figure is a shape topologically equivalent to a cube with a triangular prism attached to one of its square faces.

## Cantitruncated 24-cell

Cantitruncated 24-cell :
| Schlegel diagram, centered on truncated cuboctahedron |||
| Type | Uniform 4-polytope | Uniform 4-polytope |
| Schläfli symbol | tr{3,4,3} | tr{3,4,3} |
| Coxeter diagram | | |
| Cells | 144 | 24 4.6.8 96 4.4.3 24 3.8.8 |
| Faces | 720 | 192{3} 288{4} 96{6} 144{8} |
| Edges | 1152 | 1152 |
| Vertices | 576 | 576 |
| Vertex figure | sphenoid | sphenoid |
| Symmetry group | F$_4$, [3,4,3], order 1152 | F$_4$, [3,4,3], order 1152 |
| Properties | convex | convex |
| Uniform index | 27 28 29 | 27 28 29 |
The cantitruncated 24-cell or great rhombated icositetrachoron is a uniform 4-polytope derived from the 24-cell. It is bounded by 24 truncated cuboctahedra corresponding with the cells of a 24-cell, 24 truncated cubes corresponding with the cells of the dual 24-cell, and 96 triangular prisms corresponding with the edges of the first 24-cell.

### Coordinates

The Cartesian coordinates of a cantitruncated 24-cell having edge length 2 are all permutations of coordinates and sign of:
    (1,1+√2,1+2√2,3+3√2)
    (0,2+√2,2+2√2,2+3√2)
The dual configuration has coordinates as all permutations and signs of:
    (1,1+√2,1+√2,5+2√2)
    (1,3+√2,3+√2,3+2√2)
    (2,2+√2,2+√2,4+2√2)

### Projections

orthographic projections :
| Coxeter plane | F$_4$ | F$_4$ |
| Graph | | |
| Dihedral symmetry | [12] | [12] |
| Coxeter plane | B$_3$ / A$_2$ (a) | B$_3$ / A$_2$ (b) |
| Graph | | |
| Dihedral symmetry | [6] | [6] |
| Coxeter plane | B$_4$ | B$_2$ / A$_3$ |
| Graph | | |
| Dihedral symmetry | [8] | [4] |

## Related polytopes

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} | t$_{0,1}${3,4,3} t{3,4,3} | s{3,4,3} | t$_1${3,4,3} r{3,4,3} | t$_{0,2}${3,4,3} rr{3,4,3} | t$_{1,2}${3,4,3} 2t{3,4,3} | t$_{0,1,2}${3,4,3} tr{3,4,3} | t$_{0,3}${3,4,3} | t$_{0,1,3}${3,4,3} | t$_{0,1,2,3}${3,4,3} |
| Coxeter diagram | | | | | | | | | | |
| Schlegel diagram | | | | | | | | | | |
| F$_4$ | | | | | | | | | | |
| B$_4$ | | | | | | | | | | |
| B$_3$(a) | | | | | | | | | | |
| B$_3$(b) | | | | | | | | | | |
| B$_2$ | | | | | | | | | | |