Joel Hass & Roger Schlafly. American Scientist. Volume 84, Issue 5. Sep/Oct 1996.
According to Roman mythology, Dido fled from her home in the ancient Phoenician capital, Tyre, after her brother murdered her husband. Her flight ended in North Africa, where she set out to purchase land for a new city. The local king let her buy all the land that could be enclosed with the skin of an ox. To obtain the most land, Dido had the skin cut into a long thong, and she laid it out in a circle that surrounded Byrsa Hill, which would become Carthage.
Like Dido, ancient farmers enclosing fields and lords building walls around castles must have wondered: Given a fixed length of material, what shape encloses the largest area? Equivalently, we can ask what shape uses the smallest length to enclose a given area. The Greeks dubbed that the isoperimetric (same-perimeter) problem. Although Dido made the best choice, because a circle encloses area more efficiently than any other two-dimensional shape, it took a long time for mathematicians to generate a proof. In the third or fourth century B.C., the ancient Greek mathematician Zenodorus apparently made the first mathematical stab at proving that-for a fixed perimeter-a circle surrounds the most area on a plane, but his proof contained some gaps. In the 19th century, the German mathematician Karl Weierstrass provided a complete proof.
The three-dimensional isoperimetric problem-showing that a sphere is the most efficient surface enclosing a given volume-turns out to be much harder. Archimedes, who wrote extensively on the areas and volumes of spheres and related surfaces, first investigated that problem. In 1882, the German mathematician Hermann Amandus Schwarz proved that the sphere has the smallest area among all surfaces enclosing a given volume. In other words, for a cube, sphere, torus or any other shape in three-dimensional shape that encloses the same volume, the sphere has the smallest surface area.
In nature, spheres and isoperimetric problems arise in many cases, including the shapes of cells and raindrops. Nevertheless, many factors-including air currents, gravity and the motion of molecules-reduce the accuracy of mathematical approximations of some aspects of natural shapes. Surprisingly, spheres formed from soap bubbles can be represented almost perfectly by mathematics. In fact, soap bubbles provide valuable tools for addressing isoperimetric problems. One issue in mathematics concerns two bubbles sticking together-a double bubble. That configuration triggered what can be called the double-bubble question: Can two volumes be enclosed most efficiently with a double bubble? In 1995, we showed that when the two enclosed regions are equal in volume, a double bubble enclosing a given pair of volumes has the least possible area.
Although anyone can blow soap bubbles, their ephemeral and fragile nature makes them hard to study systematically. During the 19th century, Belgian physicist Joseph Plateau carried out many ingenious experiments on the nature and properties of soap bubbles and soap films. In fact, he discovered four simple laws-now called Plateau’s laws-that summarize any soap bubble’s geometry. First, a soap film consists of a collection of smooth pieces. Second, the mean curvature-the average bending of the surface-of each smooth piece is constant. Third, where three soap-bubble surfaces meet, they form a smooth curve, and a 120-degree angle separates each surface. Fourth, where six curves converge they form a point at which the angles between any pair are all equal and approximately 109 degrees. These laws describe all soap bubbles-no matter how complicated.
Plateau carried out many experiments, looking for soap bubbles that broke his laws. He dipped cubes, octahedra, icosahedra and numerous other shapes into soap films, but all resulting configurations abided by his laws. It turns out that Plateau’s laws emerge as a consequence of a single principle, the minimizing principle, which states that a surface that minimizes area among all surfaces enclosing a given set of volumes looks like a soap bubble. In other words, surfaces that enclose given volumes with the least surface area look like soap bubbles.
Before applying any of Plateau’s work to the double-bubble question, we must deal with the existence question: How do we know that any surface is most efficient-enclosing the prescribed volumes with the least surface area? Could it be that no best surface exists? Although it might seem obvious that a best surface must exist, a rigorous proof of that escaped 19th-century mathematicians. In 1869, Weierstrass showed that a best surface may not exist when he described similar problems that lack a smallest solution.
More than a century later, in 1976, Fred Almgren of Princeton University and Jean Taylor of Rutgers proved that a smallest surface does exist for shapes enclosing two equal volumes. They based their argument on a powerful collection of recently developed ideas in a field called geometric measure theory. This method of geometrical analysis considers extremely complicated sets that possess one saving grace: Their areas can be measured. In this collection of fragmented sets, one can show that area-minimizing surfaces exist, and that they exhibit precisely the characteristics described by Plateau’s laws, so that they turn out to be nice bubble-type surfaces and not fragmented at all. Although Almgren and Taylor showed that a smallest surface exists, and that it satisfies Plateau’s laws, their methods did not reveal the surface’s shape.
In a mathematician’s mind, a tworegion bubble-a structure enclosing two volumes-could take on lots of shapes. It could be a double bubble. It could also be a torus bubble: a spherical bubble with a tube, or torus, around it-like a beach ball squeezed into the center of an inner tube. Perhaps surprisingly, a “region” can even consist of separate parts. Mathematically, one must consider the possibility of such disconnected regions. For instance, a torus bubble could consists of two or three tubes around a sphere, where one region includes the space inside all of the tubes and the other region consists of the inside of the sphere. In addition, a two-region bubble might look like beads on a necklace. Think of these as alternating in color-red, blue, red, blue-where all the red bubbles represent one region and the blue bubbles represent another. All of these two-region bubbles satisfy Plateau’s laws, and Almgren and Taylor’s results do not rule out these possibilities.
Making a Connection
After Almgren and Taylor’s work, progress on the double-bubble question stalled for some years. Recently, Frank Morgan of Williams College revived this question when he gave the two-dimensional, or planar, analogue of it to some talented undergraduates. They solved that problem, showing that a planar double bubble gives the shortest curve enclosing a given pair of areas. Then, Brian White of Stanford University showed that the doublebubble solution must be a surface of revolution, meaning that the most efficient surface can be generated by rotating an appropriate curve around an axis-just as rotating a straight line around an axis produces a cylinder, a semi-circle produces a sphere, and a circle produces an inner tube. White’s argument shows that one candidate for the best surface can be replaced by a better candidate that exhibits more symmetry, unless the surface already possesses all of the symmetries of a surface of revolution.
White’s conclusion vastly reduced the number of contenders for the most efficient two-region surface, but the field remained large. A further culling of competing surfaces came from Michael Hutchings, a graduate student at Harvard University who had studied with Morgan as an undergraduate. Hutchings showed that the smallest surface enclosing two equal volumes possesses two connected regions. As a result, only torus bubbles and double bubbles remained as contenders for the most efficient shape.
It turns out that there an infinite number of torus bubbles satisfy Plateau’s Laws. For a surface of revolution, the constant mean curvature condition gives rise to an ordinary differential equation, and the solutions of this equation can be classified, an undertaking first carried out by the French mathematician Charles Delaunay in 1841. The solutions can be pieced together to form an infinite collection of distinct torus bubbles. Any shape from this infinite collection is a candidate for a bubble that encloses two equal-volume regions more efficiently than a double bubble. We needed a method to rule out all these torus bubbles. Developing such a method depends on an accurate, but simple, way of describing torus bubbles.
Torus bubbles can be made by rotating two circular arcs-generating pieces of spheres-and two more complicated curves that generate pieces of so-called Delaunay surfaces. Despite the elaborate shape that results from rotating those four curves, it can be described completely with two numbers: the angle, theta, subtended by the smaller spherical piece, and the value of the mean curvature, h, of the outside film of the torus. These values, plus Plateau’s laws and geometrical arguments, completely determine the shape of a torus bubble.
Solving the double-bubble problem requires showing that no choice of values for theta and h gives a torus bubble that encloses two equal volumes more efficiently then a double bubble. Nevertheless, those parameters cover lots of ground: The value of theta ranges from zero to 180 degrees, and the value of h ranges from zero to infinity. For a fixed choice of theta and h, we can calculate the volume enclosed by each region of the resulting bubble. If those computations show that the two volumes are unequal, then we can discard that bubble as a potential rival to the double bubble. Fortunately, geometric calculations show that the volume contained in the torus piece is always smaller than half of the total volume if the value of h is greater than 10. In addition, a geometrical argument shows that if theta is greater than 90 degrees, the pieces of a torus bubble can be rearranged to form a new bubble that encloses the same volumes but with less area. By the Almgren-Taylor existence experiment, a most efficient bubble exists, and we need only show that this best bubble is not a torus bubble. Any torus bubble that can be improved on does not need to be further considered. Therefore, we need to consider only the domains of 0 <= h <= 10 and 0 <= theta <= 90.
Nevertheless, we needed to discard an infinite family of bubbles, one for every value of theta and h in the rectangle of possible values. It may seem that individual computations are futile-trying to fill a canvas with dots of zero area-but we shall show how to overcome that difficulty.
Computers in Mathematics
Our approach to the double-bubble question requires using a computer to perform a variety of calculations. In most fields, computers serve as indispensable laboratory tools-as essential as a cyclotron or an oscilloscope. For instance, an engineer studying fluid flows might consider a wind tunnel and a computer simulation as two experimental approaches to the same problem. Not all mathematicians, however, accept computers so readily
Many pure mathematicians regard computers as useful for electronic mail, typesetting and recording grades. These mathematicians will grudgingly grant that computers can find large prime numbers, but believe that computers contribute little to mathematics and can corrupt it if vigilance is not maintained. The beauty of mathematics, these investigators might say, lies in its abstract ideas and insights-concepts inherently inaccessible to computers. The value of mathematics, in this view, is intrinsic, disassociated from real-world applications. Although mathematics offers a myriad of useful applications, some mathematicians consider such uses to be secondary in nature. Moreover, using a computer generates an engineering flavor that some mathematicians find somewhat distasteful. In fact, some mathematicians would call using a computer for pure mathematics probably incorrect, certainly dangerous and best avoided.
These mathematicians raise several important issues about computers as mathematical tools. One objection to computer-assisted proofs arises from the foundation of mathematical philosophy: A proof should be comprehensible. In other words, a proof can be followed, step-by-step, from start to finish-all without hand waving about the results from one calculation or another. According to some mathematicians, a computer can never meet that criterion, because no one can show exactly-bit-by-bit and address-by-address-how a computer completed a “proof.” Another objection asks: How do we know if there is a bug in the microprocessor or in some key software? How can we claim to understand a proof that relies on the motion of electrons through a semiconductor for some of its steps? In this view, computers are inherently nonrigorous, and computerassisted proofs should be rejected.
We disagree with that opinion. In our view, a computer, like a pencil, serves as a tool for computation, and it can support a rigorous mathematical proof just as easily as it can support a sloppy pseudo-argument. In fact, several camps of mathematicians see value in computers. The theoreticians working in areas such as the complexity and efficiency of algorithms form one such group. Those mathematicians work with idealized models of computers that allow theories analogous to those of other mathematical areas. They also carry out traditional pencil-and-paper mathematical arguments, proving theorems about what computers can do. Applied mathematicians who are interested in solving scientific problems of practical interest-also use computers as an integral part of their work, but not usually for proofs.
The computational elements of our proof do rely on the computer components performing according to their specifications. To enhance the ability to check our work, however, we used only the operations of addition, subtraction, multiplication, division and square root, which are fully specified including the rounding of the last bit in a standard called IEEE 754. Moreover, we avoided commercial software packages in which reliability could not be verified because of proprietary algorithms.
In our search for the most efficient surface for enclosing two equal volumes, only two classes of shapes remained: double bubbles and torus bubbles (just ones in which 0 <= h <= 10 and 0 <= theta <= 90). We set out to test all of those torus bubbles, but even that restricted range of bubbles consisted of an infinite number of individual ones. To avoid infinite computer running time, we needed a mechanism for simultaneously handling a continuum of values, or a group of similar torus bubbles. We also needed to account for how our computer rounded off all numbers. Fortunately, interval arithmetic handles both of those issues.
As a brief introduction to interval arithmetic, consider the problem of minimizing the function e^x + 10/x (where x > 0). The solution cannot be found explicitly with the minimization methods of undergraduate calculus, because finding the zeros of the first derivative requires solving e^x = 10/x^2, which cannot be done algebraically. In fact, a straightforward computer attack gives a fairly accurate answer. One divides the domain into smaller intervals, makes a crude estimate of the function on each interval, and discards intervals where the function generates only large values. Then, one subdivides the remaining intervals-looking at them at higher resolution-until one finds an interval containing the function’s minimum.
Interval arithmetic proves particularly useful in computer calculations, because a computer cannot represent all real numbers. Instead, a computer operates on a finite collection of numbers, called representable numbers, which are usually integers divided by powers of 2. Other real numbers can be described by intervals with representable endpoints. For example, pi cannot be represented, but it lies in the interval [3.1415, 3.1416]. (These endpoints would be appropriate on a decimal computer-a more realistic example would use binary representations.) The usual operations on real numbers also apply to intervals. For example, [3.1415, 3.1416] squared equals [9.86902225, 9.86965056]. If using five-digit arithmetic, this interval can be rounded to [9.8690, 9.8697] with complete confidence that the square of any number in the original interval is included. Operating on intervals in this way, we can systematically account for all possible round-off errors. At the same time, we can evaluate the bounds of a function when the input consists of a continuum of numbers not a single number.
As an example, consider using these ideas to minimize the function e^x + 10/x on the interval [1, 2]. Over that interval, the range of e^x is [2.7182, 7.3891], and the range of 10/x is [5,10]. Adding these two intervals [2.7182 + 5, 7.3891 + 10] gives [7.7182, 17.3891], and that interval certainly contains the function’s minimum. Subdividing the domain [1, 2] into smaller intervals improves the accuracy of the result.
A very similar analysis reveals the minimum surface area of torus bubbles containing a fixed volume. We knew that we needed to evaluate only the torus bubbles where 0 <= h <= 10 and 0 <= theta <= 90, and those restrictions allowed us to concentrate on a specific rectangle of torus bubbles, or torus bubbles that we can represent by a graph in which the mean curvature h ranges from zero to 10 and the angle theta ranges from zero to 90 degrees. We subdivided that rectangle into smaller rectangles. Then we applied interval arithmetic to each interval, which involved 90,159 numerical integrals. From those calculations, a subrectangle could be rejected for a variety of reasons. For example, many of the subrectangles contained only torus bubbles in which their two regions surrounded unequal volumes, and other subrectangles included torus bubbles with impossible mean curvatures.
Some areas of the torus-bubble graph produced trouble spots. For instance, calculations on some of the subrectangles produced an insufficiently accurate answer. In such cases, we subdivided the subrectangle into four even smaller rectangles, and performed the calculations on each of them. We repeated that process until the rectangles were small enough to provide an accurate answer. Ultimately we examined 22,393 smaller rectangles. In addition, some extreme cases-such as where approaches zero-gave rise to singularities, or points where the equations for the curves generating the torus bubbles diverge to +/- (inf). Torus bubbles in those areas could not be evaluated with numerical calculations, and we had to make a geometrical analysis in those cases. For instance, if theta = 0, the equations do not make any sense, because no torus bubble exists for theta = 0. Fortunately, where the numerical scheme broke down, the configurations of torus bubbles made extremely inefficient volume containers, and direct geometric arguments sufficed to discard them.
The entire numerical computation took about six minutes, running on an ordinary 80486 personal computer. After the program completed its combination of mathematical approaches, it concluded with the printout: All torus bubbles rejected. With all competitors eliminated, the double bubble remains as the unique surface that solves the area minimization problem for two equal volumes.