Four cantellations

120-cell	Cantellated 120-cell	Cantellated 600-cell
600-cell	Cantitruncated 120-cell	Cantitruncated 600-cell
Orthogonal projections in H3 Coxeter plane

In four-dimensional geometry, a cantellated 120-cell is a convex uniform 4-polytope, 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 4-polytope
Uniform index	37
Coxeter 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

Net

The cantellated 120-cell is a uniform 4-polytope. 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.

Images

Orthographic projections by Coxeter planes

H3	A2 / B3 / D4	A3 / B2
[10]	[6]	[4]

Schlegel diagram. Pentagonal face are removed.

Cantitruncated 120-cell

Cantitruncated 120-cell
Type	Uniform 4-polytope
Uniform index	42
Schläfli symbol	t0,1,2{5,3,3}
Coxeter 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

Net

The cantitruncated 120-cell is a uniform polychoron.

This 4-polytope 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 4-polytope 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.

Images

Orthographic projections by Coxeter planes

H3	A2 / B3 / D4	A3 / B2
[10]	[6]	[4]

Schlegel diagram

Centered on truncated icosidodecahedron cell with decagonal faces hidden.

Cantellated 600-cell

Cantellated 600-cell
Type	Uniform 4-polytope
Uniform index	40
Schläfli symbol	t0,2{3,3,5}
Coxeter 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

Net

The cantellated 600-cell is a uniform 4-polytope. 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 4-polytope has cells at 3 of 4 positions in the fundamental domain, extracted from the Coxeter diagram by removing one node at a time:

Node	Order	Coxeter diagram	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]

Schlegel diagrams

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

Cantitruncated 600-cell

Cantitruncated 600-cell
Type	Uniform 4-polytope
Uniform index	45
Coxeter 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

Net

The cantitruncated 600-cell is a uniform 4-polytope. 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.

Images

Schlegel diagram

Orthographic projections by Coxeter planes

H3	A2 / B3 / D4	A3 / B2
[10]	[6]	[4]

Related polytopes

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}

Notes

References

• Convex uniform polychora based on the hecatonicosachoron (120-cell) and hexacosichoron (600-cell) - Model 37, George Olshevsky.
• Archimedisches Polychor Nr. 57 (cantellated 120-cell) Marco Möller's Archimedean polytopes in R⁴ (German)
• Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6
  • (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
  • (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
  • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
• J.H. Conway and M.J.T. Guy: Four-Dimensional Archimedean Polytopes, Proceedings of the Colloquium on Convexity at Copenhagen, page 38 und 39, 1965
• N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
• Four-dimensional Archimedean Polytopes (German), Marco Möller, 2004 PhD dissertation Archived 2005-03-22 at the Wayback Machine m63 m61 m56
• Convex uniform polychora based on the hecatonicosachoron (120-cell) and hexacosichoron (600-cell) - Model 40, 42, 45, George Olshevsky.
• Klitzing, Richard. "4D uniform polytopes (polychora)". o3x3o5x - srahi, o3x3x5x - grahi, x3o3x5o - srix, x3x3x5o - grix

External links

• Four-Dimensional Polytope Projection Barn Raisings (A Zometool construction of the cantitruncated 120-cell), George W. Hart
• Renaissance Banff 2005 Zome Project: a Zome model of a 3D orthogonal projection of the cantellated 600-cell.
• H4 uniform polytopes with coordinates: rr{3,3,5} rr{5,3,3} tr{3,3,5} tr{5,3,3}