120-Cell
The 120-cell is a finite regular four-dimensional polytope with Schläfli symbol 
. It is also
known as the hyperdodecahedron or hecatonicosachoron, and is composed of 120 dodecahedra,
with 3 to an edge, and 720 pentagons (Coxeter 1973,
p. 264). The 120-cell has 600 vertices (Coxeter 1969) and 1200 edges. It is
one of the six regular polychora.
In the plate following p. 176, Coxeter (1973) illustrates the polytope.
The dual of the 120-cell is the 600-cell.
The vertices of the 120-cell with circumradius 
and edge
length 
are given by the following sets, where
is the golden ratio
(Coxeter 1969, p. 404).
1. A set of 24 vectors given by 
and all its permutations.
2. A set of 64 given by 
and all its
permutations.
3. A set of 64 given by 
and
all its permutations.
4. A set of 64 given by 
and all its permutations.
5. A set of 96 given by 
and all
its even permutations.
6. A set of 96 given by 
and
all its even permutations.
7. A set of 192 given by 
and all
its even permutations.
There are 30 distinct nonzero distances between vertices of the 120-cell in 4-space.
The top image above shows a projection of the 120-cell laser-etched into glass, and the two bottom images show a projection visualized as a metal sculpture. Both works were created by digital sculptor Bathsheba Grossman (http://www.bathsheba.com/).
The skeleton of the 120-cell, shown above in several projections, is a 4-regular graph (i.e., a quartic graph of girth 5 (pentagonal
cycles) and diameter 15. The numbers of vertices at graph distance 
, 1, 2, ... from
a given vertex on the skeleton of the 120-cell are 1, 4, 12, 24, 36, 52, 68, 76,
78, 72, 64, 56, 40, 12, 4, and 1 (OEIS A108997).
The 120-cell has graph spectrum
![4^1(alpha,beta,gamma)^(25)(mu,eta,sigma)^(36)[1/2(3+/-sqrt(13))]^(16)[1/2(-1+/-sqrt(21))]^(16)(-1+/-sqrt(2))^(48)[1/2(5+/-sqrt(5))]^91^(40)0^(18)(-1)^8(-2)^8(-3phi+2)^4(3phi-1)^4(+/-sqrt(5))^(24)phi^(24)(1-phi)^(24)(phi-2)^(24)(-1-phi)^(30),](/National_Library/20160521004321im_/http://mathworld.wolfram.com/images/equations/120-Cell/NumberedEquation1.gif) |
where 
, 
, and 
are the real
roots of 
, 
, 
, and 
are the roots
of 
, and 
is the golden ratio. The skeleton of the 120-cell
is implemented in the Wolfram Language
as GraphData["HundredTwentyCellGraph"].
The independence number of the 120-cell skeleton is 220 (Debroni et al. 2010) and its chromatic number is 3 (S. Wagon and R. Pratt, pers. comm., Dec. 2, 2011). R. Pratt has also found a balanced 3-coloring having 200 vertices of each color.
The 120-cell has
 |
distinct nets (Buekenhout and Parker 1998). The order of its automorphism group is 
(Buekenhout and Parker 1998).
Buekenhout, F. and Parker, M. "The Number of Nets of the Regular Convex Polytopes in Dimension 
." Disc. Math. 186,
69-94, 1998.
Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, p. 404, 1969.
Coxeter, H. S. M. "Stellating 
." §14.2
in Regular
Polytopes, 3rd ed. New York: Dover, pp. 136-137, 157, 264-267, and 292,
1973.
Debroni, S.; Delisle, E.; Myrvold, W.; Sethi, A.; Whitney, J.; Woodcock, J.; Fowler, P. W.; de La La Vaissière, B.; and Deza, M. "Maximum Independent Sets of the 120-Cell and Other Regular Polyhedra." To appear in Ars Mathematica Contemporanea. 2010. http://www.liga.ens.fr/~deza/withFowler/120-cell_2010.pdf.
Grossman, B. "The 120-Cell." http://www.bathsheba.com/math/120cell/.
Grossman, B. "120-Cell Crystal." http://www.bathsheba.com/crystalsci/120cell/.
Sloane, N. J. A. Sequence A108997 in "The On-Line Encyclopedia of Integer Sequences."
Stillwell, J. "The Story of the 120-Cell." Not. Amer. Math. Soc. 48, 17-24, 2001.
Swab, E. "120-Cell." http://users.adelphia.net/~eswab/120cell.htm.
Weimholt, A. "120-Cell Foldout." http://www.weimholt.com/andrew/120.html.
Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 210, 1991.
Weisstein, Eric W. "120-Cell." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/120-Cell.html
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120-cell
