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600-Cell

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600-cell

The 600-cell is the finite regular four-dimensional polytope with Schläfli symbol {3,3,5}. It is also known as the hypericosahedron or hexacosichoron. It is composed of 600 tetrahedra, with 5 to an edge. The 600-cell has 120 vertices (Coxeter 1969) and 720 edges. It is one of the six regular polychora.

In the plate following p. 160, Coxeter (1973) gives two illustrations of the polytope.

The dual of the 600-cell is the 120-cell.

The vertices of the 600-cell with circumradius 2 and edge length 2/phi, where phi is the golden ratio, are given by the following sets (Coxeter 1969, pp. 403-404).

1. The 96 even permutations of (+/-phi,+/-1,+/-phi^(-1),0).

2. The 8 permutations of (+/-2,0,0,0).

3. The 16 permutations of (+/-1,+/-1,+/-1,+/-1).

There are 8 distinct nonzero distances between vertices of the 600-cell in 4-space.

The 600-cell graph is the skeleton of the 600-cell.

The 600-cell has

2^(188)·3^(102)·5^(20)·7^(36)·11^(48)·23^(48)·29^(30) approx 7.667×10^(308)

distinct nets (Buekenhout and Parker 1998).

See also

600-Cell Graph, 11-Cell, 16-Cell, 24-Cell, 57-Cell, 120-Cell, Cell, Hypercube, Pentatope, Polychoron, Polytope, Simplex

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References

Buekenhout, F. and Parker, M. "The Number of Nets of the Regular Convex Polytopes in Dimension <=4." Disc. Math. 186, 69-94, 1998.Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, 1969.Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York: Dover, pp. 136-137, 153-154, and 157, 1973.Swab, E. "600-Cell." https://web.archive.org/web/20021012140841/http://users.adelphia.net:80/~eswab/600cell.htm.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London, England: Penguin, p. 210, 1991.

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600-Cell

Cite this as:

Weisstein, Eric W. "600-Cell." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/600-Cell.html

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