Download a high resolution photo of Doug Hardin and Ed Saff.
NASHVILLE, Tenn. ñ If you run into Ed Saff at a cocktail party and ask
him what he does for a living, the mathematician is likely to reply
that he is working on a "method for creating the perfect poppy-seed
bagel." Then he’ll pause and add, "Maybe that’s not the most accurate
description, but it’s the most digestible."
More accurately, Saff, who is a mathematics professor at Vanderbilt,
has been working with his colleague Associate Professor of Mathematics
Doug Hardin to come up with a new and improved way to distribute points
uniformly on various types of surfaces. Plotting a large set of
equidistant points on a flat surface doesn’t take a mathematician: Any
draftsman can do it. Throw in a curve or two, however, and the problem
gets much tougher. For complex surfaces like spheres and bagels (which
form a shape that mathematicians call a torus), it becomes so hard, in
fact, that mathematicians have not found a way to do it with absolute
precision.
Recently, Hardin and Saff analyzed a method for generating large
numbers of points that are spread with near uniformity over practically
any surface of any dimension. Their effort is described in the cover
article of the November issue of Notices of the American Mathematical
Society.
The procedure has a surprising number of applications. Among other
things, it comes in handy when trying to digitize curved surfaces for
computer graphics and animations with greater efficiency, in placing
the elements of a sonar net on the ocean bottom in the best locations
to detect the presence of submarines, and in testing radar systems in
aircraft to ensure uniform coverage.
Their theorems also help explain a variety of natural phenomena. They
describe some well known patterns such as that of spores on spherical
pollen grains and the way electrons distribute themselves on the
surface of a sphere. They also promise to provide new insights into the
nature of more complex patterns such as the surface structures of some
viruses and the locations of cracks in crystalline materials.
"It’s a nice mix of mathematical theory, computation and physics," says Hardin.
The new results are built on Saff’s previous work in collaboration with
A. B. J. Kuijlaars of Katholieke Universiteit Leuven, Belgium. It dealt
with optimal designs of soccer-ball shaped objects called fullerenes,
which are constructed from properly selected points on the surface of a
sphere. Saff and Hardin have extended that work so that it is no longer
limited to spherical surfaces, but now applies to elliptical, toroidal,
saddle and other two-dimensional surfaces, as well as three-dimensional
volumes. (They also handle surfaces with four or more dimensions, most
of which don’t have known physical manifestations but are of importance
in areas such as message coding.)
Hardin and Saff’s approach assumes that there is a repelling force
acting between the points, and then, depending on the strength of the
force, positions the points in the most energy-efficient manner (one
that minimizes the total amount of energy in the system). For example,
if the points strongly repel each other, then they tend to spread apart
until there is as much distance between them as possible.
In particular, the mathematicians investigated equilibrium cases in
which the repelling force between any pair of particles is inversely
proportional to the distance between them raised to a power. Their
formulation is a generalization of the inverse square law that is well
known in physics, where it describes the behavior of forces such as
electrical charge. In this case, however, the power depends on a
parameter labeled "s." Hardin and Saff have shown that when "s" is
small, the points act as if they are responding to a long-range force,
and when "s" is large, they act as if they are subject to a short-range
force. (In these terms, the familiar forces of gravity and
electromagnetism are long-range forces, while the strong force that
binds the atomic nucleus together is a short-range force.)
"Think of a room full of people," says Hardin. "In the short-range
case, they act like fighting dictators and move until they are spread
uniformly around the room. In the long-range case, however, they act as
if they are afraid of crowds and end up lining the edge of the room."
Such long-range/short-range behavior is illustrated by the distribution
of points on the surface of a torus (a.k.a. bagel). In the case of a
thousand points, when "s" is small, the points line up along the outer
rim of the bagel and look something like the pattern of treads on a
tire. As the value of "s" increases, however, the points spread out
over more and more of the surface until they reach the stage where they
are spread as uniformly as possible. As "s" increases beyond that
critical value, however, the points may move about but the overall
uniformity remains the same.
Hardin and Saff have also discovered ñ and rigorously justified ñ that
this critical value of "s" is precisely equal to the dimension of the
surface to which the points are restricted. In the case of the bagel,
for example, using a value of "s" greater than or equal to two produces
the most uniform distribution of seeds, resulting in the "nearly
perfect" poppy-seed bagel. But using a value of "s" less than two will
not spread the seeds evenly.
The insights that this model provides may be useful in creating new
materials on the micron scale with novel physical and electrical
properties. The mathematicians are collaborating with physicists Mark
Bowick of Syracuse University and Alex Travesset at Iowa State
University to explore this possibility. In particular, the researchers
believe that an improved understanding of the relationship between
certain chemical forces and surface shapes will allow them to create
new kinds of thin films and self-assembling membranes which could be
useful in certain medical applications.
"I think that we are really at the very beginning of something very big
and very exciting that we couldn’t see when we looked only at the
sphere," says Saff.
"Now, if we could only figure out how to design the perfect cheese danish!"
Media contact: David F. Salisbury, (615) 343-6803
david.salisbury@vanderbilt.edu