Every grocer knows the best way to stack oranges into a bag in order to maximize the available space. But the mathematical explanation for this optimal packing of spheres into a confined space is surprisingly elusive, as mathematics professor John Conway explained in a lecture in Jadwin Hall last night.
Conway's lecture to a packed auditorium of about 100 students, faculty members and community members, titled "All the best ways to pack spheres," explained the major ideas behind a paper which Conway coauthored with AT&T engineer Neil Sloane, "What are all the best sphere packings in low dimensions?"
Conway said the best techniques for sphere packing in several different dimensions are suggested by string theory. He engaged his audience by demonstrating his arguments with tennis balls and diagrams on a chalkboard; interjecting humor with complex mathematical theorems helped keep the lecture lively. Conway specified low-dimensions as all dimensions below 10 and said that "optimal" scenarios of sphere packing can be loosely defined in terms of density.
He was able to develop his argument by making the diagrams progressively more complicated, moving from the first dimension's clear optimal packing to the highly complex ninth dimension's ideal sphere-packing scenario.
This visual approach made the lecture very clear, and the material was generally well received by the audience. "No matter how difficult the math is, Professor Conway finds a way to make it accessible to the man on the street," Joseph Shipman, a Princeton community member, said.
Conway noted that the possibilities for higher dimensions beyond the ninth are remarkably complex. "I think the optimal [sphere] packings could have no symmetry at all ... what happens in the ninth dimension could just be the beginning of all hell breaking loose," Conway said. "It is only up until nine dimensions that we can even claim to guess at the best packing."
The lecture was sponsored by the Princeton Center for Theoretical Physics as part of a new initiative to encourage interaction between physical science disciplines.
"Sphere-packing problems cover many disciplines including biology, computer science, mathematics and communication theory, among others," Salvatore Torquato, a chemistry professor and faculty fellow at the Center for Theoretical Physics who introduced the lecture, said. "Physical scientists are interested in sphere packing because it is related to the structure of matter at low temperatures and high densities, and [it] relates to protein packing and many other studies."
Conway explained that though his answers to sphere-packing dilemmas were hardly conclusive, the claims in his paper have gone unchallenged for 10 years. "These claims could be false, but our paper has stood for more than 10 years," Conway said.
"If you can see a better way to pack spheres, come tell us," he joked.
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