600-cell

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600-cell
Schlegel diagram (Vertex centered)
Type	Regular polychoron
Cells	600 (3.3.3)
Faces	1200 {3}
Edges	720
Vertices	120
Vertex figure	(3.3.3.3.3)
Schläfli symbol	{3,3,5}
Coxeter-Dynkin diagram
Symmetry group	H₄, [3,3,5]
Dual	120-cell
Properties	convex

Orthographic projection

Vertex figure: icosahedron formed by 20 tetrahedron cells meeting at a point.

In geometry, the 600-cell (or hexacosichoron) is the convex regular 4-polytope, or polychoron, with Schläfli symbol {3,3,5}.

It is sometimes thought of as the 4-dimensional analog of the icosahedron. It is also called a tetraplex and or polytetrahedron for being constructed of tetrahedron cells.

The boundary of the 600-cell is composed of 600 tetrahedral cells with 20 meeting at each vertex. Together they form 1200 triangular faces, 720 edges, and 120 vertices. The vertex figure is an icosahedron. The dual polytope of the 600-cell is the 120-cell.

The vertices of a 600-cell centered at the origin of 4-space, with edges of length 1/φ (where φ = (1+√5)/2 is the golden ratio), can be given as follows: 16 vertices of the form

(±½,±½,±½,±½),

and 8 vertices obtained from

(0,0,0,±1)

by permuting coordinates. The remaining 96 vertices are obtained by taking even permutations of

½(±1,±φ,±1/φ,0).

Note that the first 16 vertices are the vertices of a tesseract, the second eight are the vertices of a 16-cell, and that all 24 vertices together are vertices of a 24-cell. The final 96 vertices are the vertices of a snub 24-cell, which can be found by partitioning each of the 96 edges of another 24-cell (dual to the first) in the golden ratio in a consistent manner.

When interpreted as quaternions, the 120 vertices of the 600-cell form a group under quaternionic multiplication. This group is often called the binary icosahedral group as it is the double cover of the ordinary icosahedral group I. The binary icosahedral group is isomorphic to SL(2,5).

The symmetry group of the 600-cell is the Weyl group of H₄. This is a group of order 14400.

[edit] See also

Uniform polychora family with [5,3,3] symmetry

[edit] References

H. S. M. Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. ISBN 0-486-61480-8.
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
M. Möller: Definitions and computations to the Platonic and Archimedean polyhedrons, thesis (diploma), University of Hamburg, 2001