by Ian Stewart
Double Bubble,Toil and Trouble
The dodecahedron has 20 vertices, 30 edges and 12 faces each with
five sides. But what solid has 22.9 vertices, 34.14 edges and 13.39 faces
each with 5.103 sides? Some kind of elaborate
fractal, perhaps? No, this solid is an ordinary,
familiar shape, one that you can probably find in your own home. Look out
for it when you drink a glass of cola or beer, take a shower or wash the
dishes.
I've cheated, of course. My bizarre solid can be found in the typical
home in much the same manner that, say, 2.3 children can be found in the
typical family. It exists only as an average. And it's not a solid; it's
a bubble. Foam contains thousands of bubbles, crowded together like tiny,
irregular polyhedraand the average number of vertices, edges and faces in
these polyhedra is 22.9, 34.14 and 13.39, respectively. If the average bubble
did exist, it would be like a
dodecahedron, only slightly more
so.

Coalescing Bubbles that enclose unequal volumes have shapes that remain a mathematical challenge. 
Bubbles have fascinated people ever since the invention of soap. But
the mathematics of bubbles and foam only really got going in the 1830s, when
Belgian physicist Joseph A. Plateau began dipping wire frames into soap solution
and was astounded by the results. Despite 170 years of research, we still
have not arrived at complete mathematical explanations  or even descriptions
of  several interesting phenomena that Plateau had observed.
A notorious case is the Double Bubble Conjecture, which states
that the shape formed when two bubbles coalesce consists of three spherical
surfaces. In 1995 Joel Hass of the University of California at Davis and
Roger Schiafly of Real Software in Soquel, Calif., announced a proof of this
conjecture in the special case when both bubbles enclose the same volume,
but the case of unequal volumes remains open. Many other phenomena found
by Plateau, however; are now well understood, and experiments with soap films
have repeatedly helped mathematicians develop rigorous proofs of important
geometric theorems.
In 1829 Plateau had carried out an optical experiment that involved
looking at the sun for 25 seconds: this damaged his eyes, and eventually
he became blind. Despite his loss of vision, he continued to make major
contributions to that most intensely visual area of mathematics,
threedimensional geometry.
Soap bubbles and films are examples of an immensely
important mathematical idea called a minimal
surface. This is a surface whose area is the smallest possible, subject
to certain additional constraints. Minimal surfaces relate to bubbles because
the energy caused by surface tension in a soap film is proportional to its
area. Nature likes to minimize energyso bubbles minimize area. For example,
the surface of smallest area that encloses a given volume is a sphere, and
that's why isolated soap bubbles are spherical.
A soap film is so thinabout a millionth of a meterthat it closely
resembles an infinitely thin mathematical surface. (Moving bubbles are another
matter; because dynamical forces can make them wobble into all kinds of fantastic
shapes.) Without some constraint, the area of a minimal surface would be
zero. The most common constraints are that the surface should enclose some
given volume or that its boundary should lie on some given surface or curve,
or both. A bubble that forms against a flat tabletop, for example, is usually
a hemisphere, and this is the smallest area surface that encloses a given
volume and has a boundary lying in a plane (the top of the table).
Surface of Least Area is always formed by a bubble. As a result,the soap film joining two parallel circles has the shape of a catenoid. A tetrahedron and a cube give rise to complicated arrangements of nearly flat surfaces that meet at characteristic angles. 
Plateau was especially interested in surfaces whose boundary was some
chosen curve. In his experiments the curve was represented by a length of
wire bent into shape or several wires joined together in a frame. What, for
instance, is the shape of a minimal surface whose boundary comprises two
identical parallel circles? A first guess might well be that it is a cylinder.
But we can do better. Leonhard
Euler proved that the true minimal surface with such a boundary is a
catenoid, formed by revolving a U shaped curve known as a catenary about
an axis running through the centers of the two circles.
The catenary is the shape formed by a heavy, uniform chain hanging
between two hooks of the same height: it looks rather like a parabola but
has a slightly fatter shape. (A hoary mathematical joke goes, "How do you
make a catenoid?" Answer: "By pulling its tail.") Euler's theorem can be
demonstrated by making two circular wire rings, with handlelike fishing
net frames. Hold them together, dip them into a bowl of soap solution or
detergent, and gently pull them apart to reveal the catenoid in all its
glistening beauty.
One of the most famous descriptions of soap films can be found in the
classic What is Mathematics? by Richard Courant and Herbert Robbins
(Oxford University Press, updated in 1996). They relate some of Plateau's
original experiments, in which he dipped wire frames shaped like regular
polyhedra. The simplest case, which they don't discuss, arises when the frame
is a tetrahedron, a shape with four triangular sides and six equal edges.
Here the minimal spanning surface consists of six triangles, all meeting
at the center of the tetrahedron.
A cubic frame leads to a more complicated arrangement of 13 nearly
flat surfaces. The tetrahedron case is fully understood, but a complete analysis
for the cube remains elusive.
The tetrahedral frame illustrates two general features of soap films, observed
by Plateau. Along the lines running from the vertices of the frame to its
central point, soap films meet in threes, at angles of 120 degrees; at the
central point, four edges meet at angles of 109 degrees 28 minutes. These
two angles are fundamental to any problem in which several soap films abut
one another. Angles of 120^{o }between faces and 109^{o }28'
between edges arise not just in the regular tetrahedron but in any arrangement
of soap films provided there is no trapped air or if there is, the pressures
on the two sides of each film are equal (hence canceling each other out).
The films in a foam are slightly curved but can be approximated by
plane faces: with this approximation, the two stated angles will be observed
in the interior of a foam, though not for films near the foam's external
surfaces. This fact is the basis of a curious calculation, which leads to
the strange numbers with which I began this column. By pretending that foam
is made from many identical polyhedra whose faces are regular polygons with
angles of 109^{o }28' (which is impossible, but who cares?), we can
estimate the average numbers of vertices, edges and faces in any foam.
Plateau's Rule for the angle between four bubble edges was proved by considering the possible ways in which six faces meet. The vertices are enclosed in a sphere, on which the faces meet at angles of 120 degrees. As shown, only 10 shapes meet this criterion; of these, only the first , three are physically plausible, because they correspond to minimal areas. 
Plateau's observation about the 120^{o }angle was quickly
established as a mathematical fact. The proof is often credited to the great
geometer Jacob Steiner in 1837, but Steiner was beaten to the punch by
Evangelista Torricelli and Francesco B. Cavalieri around 1640. All these
mathematicians actually studied an analogous problem for triangles. Given
a triangle and a point inside it, draw the three lines from that point to
the triangle's vertices and add up their lengths. Which point makes this
total distance smallest? Answer: the point that makes the three lines meet
at angles of 120^{o }. (Provided no angle of the triangle exceeds
120^{o }, that is  otherwise the desired point is the corresponding
vertex.) The problem for soap films can be reduced to that for triangles
by intersecting the films with a suitable plane.
In 1976 Frederick J. Almgren, Jr., then at Princeton University and
Jean E. Taylor; then at the Massachusetts Institute of Technology, proved
Plateau's second rule about 109^{o }28' angles. They started by
considering any vertex where six faces meet along four common edges. First,
they showed that the slight curvature that occurs in most soap films can
be ignored, so that the films can be taken as planar. They then considered
the system of circular arcs formed by these planes when they intersect a
small sphere centered on that vertex. Because the soap films are minimal
surfaces, these arcs are "minimal curves": their total length is as small
as possible. By the spherical analogue of the TorricelliCavalieri theorem,
these arcs must always meet in threes at angles of 120^{o }.
Almgren and Taylor proved that exactly 10 distinct configurations of
arcs [see illustration on this page] can satisfy this criterion. For each
case, they asked whether the total area of the films inside the sphere could
be made smaller by deforming the surfaces slightly, perhaps introducing new
bits of film. Any such cases could be discarded, because they could not
correspond to a true minimal surface. Exactly three cases survived this
treatment, the first three shown in the illustration above. The corresponding
arrangements of film are a single film, three meeting along an edge at
120^{o }, or six meeting at 109^{o }28'just as Plateau observed.
Radii of two coalescing bubbles (r and s) and their common surface (t) obey a simple relationship. 
The detailed techniques required for the proof went beyond geometry
into analysis  calculus and its more esoteric descendants. Almgren and Taylor
used abstract concepts known as measures to contemplate bubble shapes far
more complex than smooth surfaces.
The 120^{o }rule leads to a beautiful property of two coalescing
bubbles. It has long been assumed on empirical grounds that when two bubbles
stick together; they form three spherical surfaces, arranged as in the
illustration on the opposite page. This is the Double Bubble Conjecture.
If it is true, the radii of the spherical surfaces must satisfy a simple
relationship. Let the radii of the two bubbles be r and s and let the radius
of the surface along which they meet be t. Then the relationship is
1/r = 1/s + 1/t
This fact is proved in Cyril Isenberg's delightful hook The Science
of Soap Films and Soap Bubbles (Dover; 1992), using no more than elementary
geometry and the 120^{o }property. All that remains is to prove that
the surfaces are parts of spheres, and it is this that Hass and Schlafly
achieved in 1995but only by making the additional assumption that the bubbles
are of equal volume. Their proof required the assistance of a computer; which
had to work out 200,260 integrals associated with competing possibilitiesa
task that took the machine a mere 20 minutes!
One curious fact that is known about the unequal volume case is that
whatever the doublebubble minimal configuration is, it must he a surface
of revolution. The problem thus reduces to one about a system of curves in
the plane. Despite this simple feature, the answer remains as elusive as
it was when near blind Plateau dipped his first wire frame into a bowl of
sudsy water.
Further Reading "The Double Bubble Conjecture," by Frank Morgan in Focus (Mathematical Association of America), Vol.15, No.6, pages 67; December 1995. 
Chaos  Quantum  Logic  Cosmos  Conscious  Belief  Elect.  Art  Chem.  Maths 
SCIENTIFIC AMERICAN July 1994 File Info: Created // Updated 6/4/2018 Page Address: http://leebor2.100webspace.net/Zymic/bubble.html