SOME RESULTS ON BUBBLES
Bubbles are nature's way of finding optimal shapes to enclose
certain volumes. Bubbles are studied in the fields of mathematics
called Differential Geometry and Calculus of Variations.
While it is possible to produce many bubbles through physical experiments,
many of the mathematical properties of bubbles remain elusive.
One question that has been asked by physicists and mathematicians is whether
bubbles form the optimal (meaning smallest surface area) surfaces
for enclosing given volumes. In work with Roger Schlafly, we made progress
on this problem, proving that the Double Bubble gives the best way
of enclosing two equal volumes.
A double bubble and a competitor, called a torus bubble.
Thanks to
John Sullivan
of the University of Minnesota,
for generating these images.
Here are
more images of bubbles, at the Scientific Graphics Project
SGP
at MSRI.
A symmetric torus bubble.
A nonsymmetric torus bubble.
In this one the inner Delaunay surface is generated by a curve
that includes a full period, though only
one local maximum and one local minimum.
In tiff format 
Another nonsymmetric torus bubble.
In this one the inner Delaunay surface is generated by a curve
that includes several full periods.
For additional pictures of these and other surfaces made by
Jim Hoffman, see
More Images .
You can get a preprint with this result,
written jointly with Roger Schlafly, by sending
me an email request, which you can do by clicking here:
hass@math.ucdavis.edu
An announcement of the result, titled "The Double Bubble Conjecture",
joint with Michael Hutchings and Roger Schlafly,
appeared in Issue 3 of
Electronic Research Announcements of the AMS.
Click below to get a copy.
 Get Announcement:

DVI
PostScript
Tex
You can download a preprint of the paper "Double Bubbles Minimize",
joint with Roger Schlafly, by
clicking below.
 Get Article:

DVI 
PostScript
 Get text file containing C++ source code:

C++ source code 
A copy of the C++ source file for this paper is available by clicking above.
Instructions for running the code are contained in
its introduction. It has been tested on Pentium and UNIX/LINUX machines,
and on other machines with a C++ compiler and IEEE 754 floating
point arithmetic.
It takes about 10 seconds to run on a fast 1999 PC.
You will need a C++ compiler to run it. These are found on just
about any UNIX or LINUX system.
 Download Pictures:

John Sullivan's picture of a standard double bubble 1,
tiff format 
John Sullivan's picture of a standard double bubble 2
tiff format 
John Sullivan's picture of a torus bubble tiff format 
Here are some links to articles about these results in
American Scientist.
GEO (German).
La Recherche (French).