# Cantellated 120-cell

(Redirected from Cantitruncated 120-cell)
 Orthogonal projections in H3 Coxeter plane 120-cell Cantellated 120-cell Cantellated 600-cell 600-cell Cantitruncated 120-cell Cantitruncated 600-cell

In four-dimensional geometry, a cantellated 120-cell is a convex uniform polychoron, being a cantellation (a 2nd order truncation) of the regular 120-cell.

There are four degrees of cantellations of the 120-cell including with permutations truncations. Two are expressed relative to the dual 600-cell.

## Cantellated 120-cell

Cantellated 120-cell
Type Uniform polychoron
Uniform index 37
Coxeter-Dynkin diagram
Cells 1920 total:
120 (3.4.5.4)
1200 (3.4.4)
600 (3.3.3.3)
Faces 4800{3}+3600{4}+720{5}
Edges 10800
Vertices 3600
Vertex figure
wedge
Schläfli symbol t0,2{5,3,3}
Symmetry group H4, [3,3,5], order 14400
Properties convex

The cantellated 120-cell is a uniform polychoron. It is named by its construction as a Cantellation operation applied to the regular 120-cell. It contains 1920 cells, including 120 rhombicosidodecahedra, 1200 triangular prisms, 600 octahedra. Its vertex figure is a wedge, with two rhombicosidodecahedra, two triangular prisms, and one octahedron meeting at each vertex.

### Alternative names

• Cantellated 120-cell Norman Johnson
• Cantellated hecatonicosachoron / Cantellated polydodecahedron
• Small rhombated hecatonicosachoron (Acronym srahi)(George Olshevsky and Jonathan Bowers)[1]
• Ambo-02 polydodecahedron (John Conway)

### Images

Orthographic projections by Coxeter planes
H3 A2 / B3 / D4

[10]

[6]
 Schlegel diagram. Pentagonal face are removed.

## Cantitruncated 120-cell

Cantitruncated 120-cell
Type Uniform polychoron
Uniform index 42
Schläfli symbol t0,1,2{5,3,3}
Coxeter-Dynkin diagram
Cells 1920 total:
120 (4.6.10)
1200 (3.4.4)
600 (3.6.6)
Faces 9120:
2400{3}+3600{4}+
2400{6}+720{10}
Edges 14400
Vertices 7200
Vertex figure
sphenoid
Symmetry group H4, [3,3,5], order 14400
Properties convex

The cantitruncated 120-cell is a uniform polychoron.

This polychoron is related to the regular 120-cell. The cantitruncation operation create new truncated tetrahedral cells at the vertices, and triangular prisms at the edges. The original dodecahedron cells are cantitruncated into great rhombicosidodecahedron cells.

The image shows the polychoron drawn as a Schlegel diagram which projects the 4 dimensional figure into 3-space, distorting the sizes of the cells. In addition, the decagonal faces are hidden, allowing us to see the elemented projected inside.

### Alternative names

• Cantitruncated 120-cell Norman Johnson
• Cantitruncated hecatonicosachoron / Cantitruncated polydodecahedron
• Great rhombated hecatonicosachoron (Acronym grahi) (George Olshevsky and Jonthan Bowers)[2]
• Ambo-012 polydodecahedron (John Conway)

### Images

Orthographic projections by Coxeter planes
H3 A2 / B3 / D4

[10]

[6]
 Centered on truncated icosidodecahedron cell with decagonal faces hidden.

## Cantellated 600-cell

Cantellated 600-cell
Type Uniform polychoron
Uniform index 40
Schläfli symbol t0,2{3,3,5}
Coxeter-Dynkin diagram
Cells 1440 total:
120 3.5.3.5
600 3.4.3.4
720 4.4.5
Faces 8640 total:
(1200+2400){3}
+3600{4}+1440{5}
Edges 10800
Vertices 3600
Vertex figure
isosceles triangular prism
Symmetry group H4, [3,3,5], order 14400
Properties convex

The cantellated 600-cell is a uniform polychoron. It has 1440 cells: 120 icosidodecahedra, 600 cuboctahedra, and 720 pentagonal prisms. Its vertex figure is an isosceles triangular prism, defined by one icosidodecahedron, two cuboctahedra, and two pentagonal prisms.

### Construction

This polychoron has cells at 3 of 4 positions in the fundamental domain, extracted from the Coxeter-Dynkin diagram by removing one node at a time:

Node Order Coxeter-Dynkin
Cell Picture
0 600 Cantellated tetrahedron
(Cuboctahedron)
1 1200 None
(Degenerate triangular prism)

2 720 Pentagonal prism
3 120 Rectified dodecahedron
(Icosidodecahedron)

There are 1440 pentagonal faces between the icosidodecahedra and pentagonal prisms. There are 3600 squares between the cuboctahedra and pentagonal prisms. There are 2400 triangular faces between the icosidodecahedra and cuboctahedra, and 1200 triangular faces between pairs of cuboctahedra.

There are two classes of edges: 3-4-4, 3-4-5: 3600 have two squares and a triangle around it, and 7200 have one triangle, one square, and one pentagon.

### Images

Orthographic projections by Coxeter planes
H4 -

[30]

[20]
F4 H3

[12]

[10]
A2 / B3 / D4 A3 / B2

[6]

[4]
 Stereographic projection with its 3600 green triangular faces and its 3600 blue square faces.

## Cantitruncated 600-cell

Cantitruncated 600-cell
Type Uniform polychoron
Uniform index 45
Coxeter-Dynkin diagram
Cells 1440 total:
120 (5.6.6)
720 (4.4.5)
600 (4.6.6)
Faces 8640:
3600{4}+1440{5}+
3600{6}
Edges 14400
Vertices 7200
Vertex figure
sphenoid
Schläfli symbol t0,1,2{3,3,5}
Symmetry group H4, [3,3,5], order 14400
Properties convex

The cantitruncated 600-cell is a uniform polychoron. It is composed of 1440 cells: 120 truncated icosahedra, 720 pentagonal prisms and 600 truncated octahedra. It has 7200 vertices, 14400 edges, and 8640 faces (3600 squares, 1440 pentagons, and 3600 hexagons). It has an irregular tetrahedral vertex figure, filled by one truncated icosahedron, one pentagonal prism and two truncated octahedra.

### Alternative names

• Cantitruncated 600-cell (Norman Johnson)
• Cantitruncated hexacosichoron / Cantitruncated polydodecahedron
• Great rhombated hexacosichoron (acronym grix) (George Olshevsky and Jonathan Bowers)[4]
• Ambo-012 polytetrahedron (John Conway)

### Images

Orthographic projections by Coxeter planes
H3 A2 / B3 / D4

[10]

[6]

## Related polytopes

H4 family polytopes by name, Coxeter diagram, and Schläfli symbol
120-cell rectified
120-cell
truncated
120-cell
cantellated
120-cell
runcinated
120-cell
bitruncated
120-cell
cantitruncated
120-cell
runcitruncated
120-cell
omnitruncated
120-cell
{5,3,3} t1{5,3,3} t0,1{5,3,3} t0,2{5,3,3} t0,3{5,3,3} t1,2{5,3,3} t0,1,2{5,3,3} t0,1,3{5,3,3} t0,1,2,3{5,3,3}
600-cell rectified
600-cell
truncated
600-cell
cantellated
600-cell
runcinated
600-cell
bitruncated
600-cell
cantitruncated
600-cell
runcitruncated
600-cell
omnitruncated
600-cell
{3,3,5} t1{3,3,5} t0,1{3,3,5} t0,2{3,3,5} t0,3{3,3,5} t1,2{3,3,5} t0,1,2{3,3,5} t0,1,3{3,3,5} t0,1,2,3{3,3,5}

## Notes

1. ^ Klitzing, (o3x3o5x - srahi)
2. ^ Klitzing, (o3x3x5x - grahi)
3. ^ Klitzing, (x3o3x5o - srix)
4. ^ Klitzing, (x3x3x5o - grix)