An ellipsoid packing is an arrangement of nonoverlapping ellipsoids in space. Bezdek and Kuperberg (1991) constructed packings of identical ellipsoids
with density . G. Kuperberg refined their construction
to obtain densities arbitrarily close to
(OEIS A093824; Wills 1991), greater than the maximum density of
(OEIS A093825) that is possible for sphere
packing (Sloane 1998), as established by proof of the Kepler
conjecture. Wills (1991) combined his refinement with G. Kuperberg's to
obtain a density of .
Donev et al. (2004) showed that a maximally random jammed state of M&Ms chocolate candies has a packing density of about 68%, about 4 percentage points above
the roughly 64% density of random jammed sphere packings. Furthermore, Donev et
al. (2004) also showed by computer simulations other ellipsoid packings resulted
in random packing densities approaching that of the densest sphere packings, i.e.,
filling nearly 74% of space.
Bezdek, A. and Kuperberg, W. In Applied Geometry and Discrete Mathematics: The Victor Klee Festschrift (Ed. P. Gritzmann
and B. Sturmfels). Providence, RI: Amer. Math. Soc., pp. 71-80, 1991.Donev,
A.; Cisse, I.; Sachs, D.; Variano, E. A.; Stillinger, F. H.; Connelly,
R.; Torquato, S.; and Chaikin, P. M. "Improving the Density of Jammed Disordered
Packings Using Ellipsoids." Science303, 990-993, 2004. https://doi.org/10.1126/science.1093010.Sloane,
N. J. A. "Kepler's Conjecture Confirmed." Nature395,
435-436, 1998.Wills, J. M. "An Ellipsoid Packing in of Unexpected High Density." Mathematika38,
318-320, 1991. https://doi.org/10.1112/S0025579300006653.Sloane,
N. J. A. Sequences A093824 and A093825 in "The On-Line Encyclopedia of Integer
Sequences."
. G. Kuperberg refined their construction
to obtain densities arbitrarily close to
(OEIS A093825) that is possible for sphere
packing (Sloane 1998), as established by proof of the Kepler
conjecture. Wills (1991) combined his refinement with G. Kuperberg's to
obtain a density of 
.
of Unexpected High Density." Mathematika 38,
318-320, 1991.