[EAS] Ellipsoidal Packing

pjk pjk at design.eng.yale.edu
Wed May 12 17:22:27 EDT 2004

Subject:   Ellipsoidal Packing

Dear Colleagues -

Today feels like summer here (with apologies to my Australian
readers), the academic year is ending, _and_ I have a weakness for
M&Ms (despite growing up amidst more refined chocolate practices in
Austria). So I thought this item was worth sending out.

All best,  --PJK

The American Institute of Physics Bulletin of Physics News
Number 685 May 12, 2004  by Phillip F. Schewe, Ben Stein

THE BEST PACKING OF M&Ms, filling more than 77% of available volume,
has been achieved in a computer simulation performed at Princeton.
Actually the new results apply to any ellipsoid object, such as M&M
candy, fish eggs, or watermelons.  The modern understanding of dense
packing might be said to start in 1611, when Johannes Kepler
suggested that the most efficient packing of spheres in a container
occurred when the spheres were placed in a face-centered cubic
arrangement---the way a grocer stacks oranges.  "Kepler's
conjecture" was proved in 1998 and the filling factor was worked out
to be about 74%.  Unlike spheres, which still look the same after
you rotate them, ellipsoids' oblateness (they are squashed or
stretched in at least one direction) give them orientational degrees
of freedom that spheres don't have.  Consequentially, ellipsoids can
be packed more efficiently than spheres.  Depending on the aspect
ratio of the ellipsoid, the packing density can be anywhere between
74% and 77%.  The Princeton research (contact Salvatore Torquato,
609-258-3341, torquato at electron.princeton.edu) has a number of
practical implications: it shows that glassy states of matter, in
which molecules lie in a disordered arrangement, can have densities
almost as high as for crystals; it suggests that because of a high
contact number (in the high-density packings ellipsoids can touch 14
of their neighbors) stronger ceramics can be designed); and it
encourages researchers to investigate the effect of ellipsoidal
shape on evolutionary optimization in fish eggs.  (Donev et al.,
Physical Review Letters, upcoming article)

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