Rectified 600-cell

Rectified 600-cell
Schlegel diagram, shown as Birectified 120-cell, with 119 icosahedral cells colored
Type	Uniform 4-polytope
Uniform index	34
Schläfli symbol	t1{3,3,5} or r{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, [3,3,5], order 14400
Properties	convex, vertex-transitive, edge-transitive

In geometry, the rectified 600-cell or rectified hexacosichoron is a convex uniform 4-polytope 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.

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.

Semiregular polytope

It is one of three semiregular 4-polytopes 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.

E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as tC₆₀₀.

Alternate names

• octicosahedric (Thorold Gosset)
• 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	Net

Related polytopes

Diminished rectified 600-cell

120-diminished rectified 600-cell
Type	4-polytope
Cells	840 cells: 600 square pyramid 120 pentagonal prism 120 pentagonal antiprism
Faces	2640: 1800 {3} 600 {4} 240 {5}
Edges	2400
Vertices	600
Vertex figure	Bi-diminished pentagonal prism (1) 3.3.3.3 + (4) 3.3.4 (2) 4.4.5 (2) 3.3.3.5
Symmetry group	1/12[3,3,5], order 1200
Properties	convex

A related vertex-transitive polytope can be constructed with equal edge lengths removes 120 vertices from the rectified 600-cell, but isn't uniform because it contains square pyramid cells,^[1] discovered by George Olshevsky, calling it a swirlprismatodiminished rectified hexacosachoron, with 840 cells (600 square pyramids, 120 pentagonal prisms, and 120 pentagaonal antiprisms), 2640 faces (1800 triangles, 600 square, and 240 pentagons), 2400 edges, and 600 vertices. It has a chiral bi-diminished pentagonal prism vertex figure.

Each removed vertex creates a pentagonal prism cell, and diminishes two neighboring icosahedra into pentagonal antiprisms, and each octahedron into a square pyramid.^[2]

This polytope can be partitioned into 12 rings of alternating 10 pentagonal prisms and 10 antiprisms, and 30 rings of square pyramids.

Schlegel diagram	Orthogonal projection
Two orthogonal rings shown	2 rings of 30 red square pyramids, one ring along perimeter, and one centered.

Net

H4 family

H4 family 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	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}

Pentagonal prism vertex figures

r{p,3,5}

Space	S3	H3
Form	Finite	Compact		Paracompact	Noncompact
Name	r{3,3,5}	r{4,3,5}	r{5,3,5}	r{6,3,5}	r{7,3,5}	... r{∞,3,5}
Image
Cells {3,5}	r{3,3}	r{4,3}	r{5,3}	r{6,3}	r{7,3}	r{∞,3}

References

• 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

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)
• H4 uniform polytopes with coordinates: r{3,3,5}