Polyhedra:

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. | |

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, Using the value of 0.636 cc for the individual volume found
in the research papers, 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 Tea Room. | After 48 hours. |

Princeton lab web site

Ellipsoids in the 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