Rectified 600-cell

Four rectifications

600-cell	Rectified 600-cell	Rectified 120-cell	120-cell
Orthogonal projections in H3 Coxeter plane

In geometry, a rectified 600-cell is a uniform polychoron (4-dimensional uniform polytope) formed as the rectification of the regular 600-cell.

There are four rectifications of the 600-cell, including the zeroth, the 600-cell itself. Tbe birectified 600-cell is more easily seen as a rectified 120-cell, and the trirectified 600-cell is the same as the dual 120-cell.

Rectified 600-cell

Rectified 600-cell
Schlegel diagram, shown as Birectified 120-cell, with 119 icosahedral cells colored
Type	Uniform polychoron
Uniform index	34
Schläfli symbol	t1{3,3,5}
Coxeter-Dynkin diagram
Cells	600 (3.3.3.3) 120 {3,5}
Faces	1200+2400 {3}
Edges	3600
Vertices	720
Vertex figure	pentagonal prism
Symmetry group	H4 or [3,3,5]
Properties	convex, edge-transitive

Vertex figure: pentagonal prism
7 faces:

5 (3.3.3.3) and 2 (3.3.3.3.3)

In geometry, the rectified 600-cell is a convex uniform polychoron composed of 600 regular octahedra and 120 icosahedra cells. Each edge has two octahedra and one icosahedron. Each vertex has five octahedra and two icosahedra. In total it has 3600 triangle faces, 3600 edges, and 720 vertices.

It is one of three semiregular polychora made of two or more cells which are platonic solids, discovered by Thorold Gosset in his 1900 paper. He called it a octicosahedric for being made of octahedron and icosahedron cells.

Containing the cell realms of both the regular 120-cell and the regular 600-cell, it can be considered analogous to the polyhedron icosidodecahedron, which is a rectified icosahedron and rectified dodecahedron.

The vertex figure of the rectified 600-cell is a uniform pentagonal prism.

Alternate names

• Icosahedral hexacosihecatonicosachoron
• Rectified 600-cell (Norman W. Johnson)
• Rectified hexacosichoron
• Rectified polytetrahedron
• Rox (Jonathan Bowers)

Images

Orthographic projections by Coxeter planes

H4	-	F4
[30]	[20]	[12]
H3	A2 / B3 / D4	A3 / B2
[10]	[6]	[4]

Stereographic projection

Related polytopes

H₄ family polytopes by name, Coxeter-Dynkin 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}

References

• Kaleidoscopes: Selected Writings of H.S.M. Coxeter, editied by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [1]
  • (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 [2]

External links

• Convex uniform polychora based on the hecatonicosachoron (120-cell) and hexacosichoron (600-cell) - Model 34, George Olshevsky.
• Richard Klitzing, 4D uniform polytopes (polychora), o3x3o5o - rox
• Archimedisches Polychor Nr. 45 (rectified 600-cell) Marco Möller's Archimedean polytopes in R⁴ (German)