In four-dimensional geometry, a runcinated 120-cell (or runcinated 600-cell) is a convex uniform 4-polytope, being a runcination (a 3rd order truncation) of the regular 120-cell.

More information Orthogonal projections in H3 Coxeter plane ...

Four runcinations

120-cell	Runcinated 120-cell (Expanded 120-cell)	Runcitruncated 120-cell
600-cell	Runcitruncated 600-cell	Omnitruncated 120-cell
Orthogonal projections in H3 Coxeter plane

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There are 4 degrees of runcinations of the 120-cell including with permutations truncations and cantellations.

The runcinated 120-cell can be seen as an expansion applied to a regular 4-polytope, the 120-cell or 600-cell.

Runcinated 120-cell
Type	Uniform 4-polytope
Uniform index	38
Coxeter diagram
Cells	2640 total: 120 5.5.5 720 4.4.5 1200 4.4.3 600 3.3.3
Faces	7440: 2400{3}+3600{4}+ 1440{5}
Edges	7200
Vertices	2400
Vertex figure	Equilateral-triangular antipodium
Schläfli symbol	t0,3{5,3,3}
Symmetry group	H4, [3,3,5], order 14400
Properties	convex

The runcinated 120-cell or small disprismatohexacosihecatonicosachoron is a uniform 4-polytope. It has 2640 cells: 120 dodecahedra, 720 pentagonal prisms, 1200 triangular prisms, and 600 tetrahedra. Its vertex figure is a nonuniform triangular antiprism (equilateral-triangular antipodium): its bases represent a dodecahedron and a tetrahedron, and its flanks represent three triangular prisms and three pentagonal prisms.

Images

Schlegel diagram (Only tetrahedral cells shown)

Polyhedral rings

Cells on 5-fold axis

Cells on 3-fold axis

Cells on 2-fold axis

Orthogonal projections in Coxeter planes

H3

A2/B3

A3/B2

More information Runcitruncated 120-cell ...

Runcitruncated 120-cell
Type	Uniform 4-polytope
Uniform index	43
Coxeter diagram
Cells	2640 total: 120 (3.10.10) 720 (4.4.10) 1200 (3.4.4) 600 (3.4.3.4)
Faces	13440: 4800{3}+7200{4}+ 1440{10}
Edges	18000
Vertices	7200
Vertex figure	Irregular rectangular pyramid
Schläfli symbol	t0,1,3{5,3,3}
Symmetry group	H4, [3,3,5], order 14400
Properties	convex

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The runcitruncated 120-cell or prismatorhombated hexacosichoron is a uniform 4-polytope. It contains 2640 cells: 120 truncated dodecahedra, 720 decagonal prisms, 1200 triangular prisms, and 600 cuboctahedra. Its vertex figure is an irregular rectangular pyramid, with one truncated dodecahedron, two decagonal prisms, one triangular prism, and one cuboctahedron.

More information Runcitruncated 600-cell ...

Runcitruncated 600-cell
Type	Uniform 4-polytope
Uniform index	44
Coxeter diagram
Cells	2640 total: 120 3.4.5.4 720 4.4.5 1200 4.4.6 600 3.6.6
Faces	13440: 2400{3}+7200{4}+ 1440{5}+2400{6}
Edges	18000
Vertices	7200
Vertex figure	Trapezoidal pyramid
Schläfli symbol	t0,1,3{3,3,5}
Symmetry group	H4, [3,3,5], order 14400
Properties	convex

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The runcitruncated 600-cell or prismatorhombated hecatonicosachoron is a uniform 4-polytope. It is composed of 2640 cells: 120 rhombicosidodecahedron, 600 truncated tetrahedra, 720 pentagonal prisms, and 1200 hexagonal prisms. It has 7200 vertices, 18000 edges, and 13440 faces (2400 triangles, 7200 squares, and 2400 hexagons).

More information Omnitruncated 120-cell ...

Omnitruncated 120-cell
Type	Uniform 4-polytope
Uniform index	46
Coxeter diagram
Cells	2640 total: 120 4.6.10 720 4.4.10 1200 4.4.6 600 4.6.6
Faces	17040 total: 10800 {4}, 4800 {6} 1440 {10}
Edges	28800
Vertices	14400
Vertex figure	Chiral scalene tetrahedron
Schläfli symbol	t0,1,2,3{3,3,5}
Symmetry group	H4, [3,3,5], order 14400
Properties	convex

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The omnitruncated 120-cell or great disprismatohexacosihecatonicosachoron is a convex uniform 4-polytope, composed of 2640 cells: 120 truncated icosidodecahedra, 600 truncated octahedra, 720 decagonal prisms, and 1200 hexagonal prisms. It has 14400 vertices, 28800 edges, and 17040 faces (10800 squares, 4800 hexagons, and 1440 decagons). It is the largest nonprismatic convex uniform 4-polytope.

The vertices and edges form the Cayley graph of the Coxeter group H₄.

These polytopes are a part of a set of 15 uniform 4-polytopes with H₄ symmetry:

More information H4 family polytopes, 120-cell ...

H4 family polytopes
120-cell	rectified 120-cell	truncated 120-cell	cantellated 120-cell	runcinated 120-cell	cantitruncated 120-cell	runcitruncated 120-cell	omnitruncated 120-cell
{5,3,3}	r{5,3,3}	t{5,3,3}	rr{5,3,3}	t0,3{5,3,3}	tr{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	bitruncated 600-cell	cantitruncated 600-cell	runcitruncated 600-cell	omnitruncated 600-cell
{3,3,5}	r{3,3,5}	t{3,3,5}	rr{3,3,5}	2t{3,3,5}	tr{3,3,5}	t0,1,3{3,3,5}	t0,1,2,3{3,3,5}

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