Packing Ellipsoids
Oct. 27, 2004 talk for the PI Math Club
Filling space with solid shapes
Polyhedra:
NewtonGregory problem  16941953:
Maximum number of contacts for spheres = 12
Kepler problem  16111998:
Sphere packing  also with 12 contacts
The most efficient sphere packing (two views):
Experimental average # of contacts in a random sphere packing: 6.4.
Experimental packing efficiency in a random sphere packing: approx. 64%.

Demonstrating circular disk packing.
Most efficient packing: hexagonal array, approx. 90.7% area coverage.

Photos from the PI Math Club talk on random packing of ellipsoids 
Audience Participation 

The ellipsoids at the bottom of the container are not randomly packed 

Measuring the volume of 100
ellipsoids using the water displacement method
of Archimedes, with the assistance of PI Math
Club V.P. K. Boyd
The experimental conclusion: 68 cubic centimeters, or an average of
0.68 cc each. (however, this imprecise measurement may be a large
source of error in the next step of the experiment.)




4000 ellipsoids sorted into bags of 1000 

Pouring into a 5000 cubic centimeter container 
The first 1000 
The next 1000 


The next 1000 
The next 1000 


The conclusion: it takes about 5200 ellipsoids to fill the 5000 ml container. 

The calculations:
Using our estimate of the volume of an individual ellipsoid, 5200 ellipsoids, 0.68 cc each = 3536 cc, about 70.7% of the 5000 cc volume.
Using the value of 0.636 cc for the individual volume found in the research papers, 5200 ellipsoids is 3307.2 cc, about 66.14%of the 5000 cc volume.
Either way, we get a higher packing density than the experimentally measured 64% efficiency for randomly jammed spheres.
This confirms recent experiments and simulations with randomly packed ellipsoids.
(see news articles, below) 
After 24 hours in the Math Department TeaRoom. 
After 48 hours. 


Princeton lab web site
Ellipsoids in the news:
CNN article
Science News
Science Magazine summary (PDF)
Eurekalert.org
Another Container Shape.
Prof. Coffman's web site
on Linear systems of
ellipsoids  possibly applicable to the
"collision detection" problem for computer simulation of ellipsoid
packing
Thanks to the IPFW Department of Chemistry for a loan of equipment Backto Professor Adam Coffman's home page.
