Packing Ellipsoids

Oct. 27, 2004 talk for the PI Math Club

Filling space with solid shapes


Newton-Gregory problem - 1694-1953:

Maximum number of contacts for spheres = 12

Kepler problem - 1611-1998:

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)

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.