From Violins to Videos : Nature's Numbers Chapter 5

It has become conventional, as I have noted, to separate mathematics into two distinct subdisciplines labeled pure mathematics and applied mathematics. This is a separation that would have baffled the great mathematicians of classical times. Carl Friedrich Gauss, for example, was happiest in the ivory tower of number theory, where he delighted in abstract numerical patterns simply because they were beautiful and challenging. He called number theory "the queen of mathematics," and the poetic idea that queens are delicate beauties who do not sully their hands with anything useful was not far from his mind. However, he also calculated the orbit of Ceres, the first asteroid to be discovered. Soon after its discovery, Ceres passed behind the Sun, as seen from Earth, and could no longer be observed. Unless its orbit could be calculated accurately, astronomers would not be able to find it when it again became visible, months later. But the number of observations of the asteroid was so small that the standard methods for calculating orbits could not provide the required level of accuracy. So Gauss made several major innovations, some of which remain in use to this day. It was a virtuoso performance, and it made his public reputation. Nor was that his only practical application of his subject: among other things, he was also responsible for major developments in surveying, telegraphy, and the understanding of magnetism.

In Gauss's time, it was possible for one person to have a fairly good grasp of the whole of mathematics. But because all of the classical branches of science have grown so vast that no single mind can likely encompass even one of them, we now live in an age of specialists. The organizational aspects of mathematics function more tidily if people specialize either in the theoretical areas of the subject or its practical ones. Because most people feel happier working in one or the other of these two styles, individual preferences tend to reinforce this distinction. Unfortunately, it is then very tempting for the outside world to assume that the only useful part of mathematics is applied mathematics; after all, that is what the name seems to imply. This assumption is correct when it comes to established mathematical techniques: anything really useful inevitably ends up being considered "applied," no matter what its origins may have been. But it gives a very distorted view of the origins of new mathematics of practical importance. Good ideas are rare, but they come at least as often from imaginative dreams about the internal structure of mathematics as they do from attempts to solve a specific, practical problem. This chapter deals with a case history of just such a development, whose most powerful application is television- an invention that arguably has changed our world more than any other. It is a story in which the pure and applied aspects of mathematics combine to yield something far more powerful and compelling than either could have produced alone. And it begins at the start of the sixteenth century, with the problem of the vibrating violin string. Although this may sound like a practical question, it was studied mainly as an exercise in the solution of differential equations; the work was not aimed at improving the quality of musical instruments. Imagine an idealized violin string, stretched in a straight line between two fixed supports. If you pluck the string, pulling it away from the straight-line position and then letting go, what happens? As you pull it sideways, its elastic tension increases, which produces a force that pulls the string back toward its original position. When you let go, it begins to accelerate under the action of this force, obeying Newton's law of motion. However, when it returns to its initial position it is moving rapidly, because it has been accelerating the whole time-so it overshoots the straight line and keeps moving. Now the tension pulls in the opposite direction, slowing it down until it comes to a halt. Then the whole story starts over. If there is no friction, the string will vibrate from side to side forever.

That's a plausible verbal description; one of the tasks for a mathematical theory is to see whether this scenario really holds good, and if so, to work out the details, such as the shape that the string describes at any instant. It's a complex problem, because the same string can vibrate in many different ways, depending upon how it is plucked. The ancient Greeks knew this, because their experiments showed that a vibrating string can produce many different musical tones. Later generations realized that the pitch of the tone is determined by the frequency of vibration-the rate at which the string moves to and fro-so the Greek discovery tells us that the same string can vibrate at many different frequencies. Each frequency corresponds to a different configuration of the moving string, and the same string can take up many different shapes.

Strings vibrate much too fast for the naked eye to see any one instantaneous shape, but the Greeks found important evidence for the idea that a string can vibrate at many different frequencies. They showed that the pitch depends on the positions of the nodes-places along the length of the string which remain stationary. You can test this on a violin, banjo, or guitar. When the string is vibrating in its "fundamental" frequency-that is, with the lowest possible pitch-only the end points are at rest. If you place a finger against the center of the string, creating a node, and then pluck the string, it produces a note one octave higher. If you place your finger onethird of the way along the string, you actually create two nodes (the other being two-thirds of the way along, and this produces a yet higher note. The more nodes, the higher the frequency. In general, the number of nodes is an integer, and the nodes are equally spaced. The corresponding vibrations are standing waves, meaning waves that move up and down but do not travel along the string. The size of the up-and-down movement is known as the amplitude of the wave, and this determines the tone's loudness. The waves are sinusoidal-shaped like a sine curve, a repetive wavy line of rather elegant shape that arises in trigonometry.

In 1714, the English mathematician Brook Taylor published the fundamental vibrational frequency of a violin string in terms of its length, tension, and density. In 1746, the Frenchman Jean Le Rond d'Alembert showed that many vibrations of a violin string are not sinusoidal standing waves. In fact, he proved that the instantaneous shape of the wave can be anything you like. In 1748, in response to d'Alembert's work, the prolific Swiss mathematician Leonhard Euler worked out the "wave equation" for a string. In the spirit of Isaac Newton, this is a differential equation that governs the rate of change of the shape of the string. In fact it is a "partial differential equation," meaning that it involves not only rates of change relative to time but also rates of change relative to space-the direction along the string. It expresses in mathematical language the idea that the acceleration of each tiny segment of the string is proportional to the tensile forces acting upon that segment; so it is a consequence of Newton's law of motion.

Not only did Euler formulate the wave equation: he solved it. His solution can be described in words. First, deform the string into any shape you care to choose-a parabola, say, or a triangle, or a wiggly and irregular curve of your own devising. Then imagine that shape propagating along the string toward the right. Call this a rightward-traveling wave. Then turn the chosen shape upside down, and imagine it propagating the other way, to create a leftward-traveling wave. Finally, superpose these two waveforms. This process leads to all possible solutions of the wave equation in which the ends of the string remain fixed.

Almost immediately, Euler got into an argument with Daniel Bernoulli, whose family originally hailed from Antwerp but had moved to Germany and then Switzerland to escape religious persecution. Bernoulli also solved the wave equation, but by a totally different method. According to Bernoulli, the most general solution can be represented as a superposition of infinitely many sinusoidal standing waves. This apparent disagreement began a century-long controversy, eventually resolved by declaring both Euler and Bernoulli right. The reason that they are both right is that every periodically varying shape can be represented as a superposition of an infinite number of sine curves. Euler thought that his approach led to a greater variety of shapes, because he didn't recognize their periodicity. However, the mathematical analysis works with an infinitely long curve. Because the only part of the curve that matters is the part between the two endpoints, it can be repeated periodically along a very long string without any essential change. So Euler's worries were unfounded. The upshot of all this work, then, is that the sinusoidal waves are the basic vibrational components. The totality of vibrations that can occur is given by forming all possible sums of finitely or infinitely many sinusoidal waves of all possible amplitudes. As Daniel Bernoulli had maintained all along, "all new curves given by d' Alembert and Euler are only combinations of the Taylor vibrations."

With the resolution of this controversy, the vibrations of a violin string ceased to be a mystery, and the mathematicians went hunting for bigger game. A violin string is a curve-a one-dimensional object-but objects with more dimensions can also vibrate. The most obvious musical instrument that employs a two-dimensional vibration is the drum, for a drumskin is a surface, not a straight line. So mathematicians turned their attention to drums, starting with Euler in 1759. Again he derived a wave equation, this one describing how the displacement of the drums kin in the vertical direction varies over time. Its physical interpretation is that the acceleration of a small piece of the drums kin is proportional to the average tension exerted on it by all nearby parts of the drumskin: symbolically, it looks much like the one-dimensional wave equation; but now there are spatial (second-order) rates of change in two independent directions, as well as the temporal rate of change.

Violin strings have fixed ends. This "boundary condition" has an important effect: it determines which solutions to the wave equation are physically meaningful for a violin string. In this whole subject, boundaries are absolutely crucial. Drums differ from violin strings not only in their dimensionality but in having a much more interesting boundary: the boundary of a drum is a closed curve, or circle. However, like the boundary of a string, the boundary of the drum is fixed: the rest of the drumskin can move, but its rim is firmly strapped down. This boundary condition restricts the possible motions of the drumskin. The isolated endpoints of a violin string are not as interesting and varied a boundary condition as a closed curve is; the true role of the boundary becomes apparent only in two or more dimensions.

As their understanding of the wave equation grew, the mathematicians of the eighteenth century learned to solve the wave equation for the motion of drums of various shapes. But now the wave equation began to move out of the musical domain to establish itself as an absolutely central feature of mathematical physics. It is probably the single most important mathematical formula ever devised-Einstein's famous relation between mass and energy notwithstanding. What happened was a dramatic instance of how mathematics can lay bare the hidden unity of nature. The same equation began to show up everywhere. It showed up in fluid dynamics, where it described the formation and motion of water waves. It showed up in the theory of sound, where it described the transmission of sound waves-vibrations of the air, in which its molecules become alternately compressed and separated. And then it showed up in the theories of electricity and magnetism, and changed human culture forever.

Electricity and magnetism have a long, complicated history, far more complex than that of the wave equation, involving accidental discoveries and key experiments as well as mathematical and physical theories, Their story begins with William Gilbert, physician to Elizabeth I, who described the Earth as a huge magnet and observed that electrically charged bodies can attract or repel each other. It continues with such people as Benjamin Franklin, who in 1752 proved that lightning is a form of electricity by flying a kite in a thunderstorm; Luigi Galvani, who noticed that electrical sparks caused a dead frog's leg muscles to contract; and Alessandro Volta, who invented the first battery.

Throughout much of this early development, electricity and magnetism were seen as two quite distinct natural phenomena. The person who set their unification in train was the English physicist and chemist Michael Faraday. Faraday was employed at the Royal Institution in London, and one of his jobs was to devise a weekly experiment to entertain its scientifically-minded members. This constant need for new ideas turned Faraday into one of the greatest experimental physicists of all time. He was especially fascinated by electricity and magnetism, because he knew that an electric current could create a magnetic force. He spent ten years trying to prove that, conversely, a magnet could produce an electric current, and in 1831 he succeeded. He had shown that magnetism and electricity were two different aspects of the same thing-electromagnetism. It is said that King William IV asked Faraday what use his scientific parlor tricks were, and received the reply "I do not know, Your Majesty, but I do know that one day you will tax them." In fact, practical uses soon followed, notably the electric motor (electricity creates magnetism creates motion) and the electrical generator (motion creates magnetism creates electricity). But Faraday also advanced the theory of electromagnetism. Not being a mathematician, he cast his ideas in physical imagery, of which the most important was the idea of a line of force. If you place a magnet under a sheet of paper and sprinkle iron filings on top, they will line up along welldefined curves. Faraday's interpretation of these curves was that the magnetic force did not act "at a distance" without any intervening medium; instead, it propagated through space along curved lines. The same went for electrical force. Faraday was no mathematician, but his intellectual successor James Clerk Maxwell was. Maxwell expressed Faraday's ideas about lines of force in terms of mathematical equations for magnetic and electric fields-that is, distributions of magnetic and electrical charge throughout space. By 1864, he had refined his theory down to a system of four differential equations that related changes in the magnetic field to changes in the electric field. The equations are elegant, and reveal a curious symmetry between electricity and magnetism, each affecting the other in a similar manner.

It is here, in the elegant symbolism of Maxwell's equations, that humanity made the giant leap from violins to videos: a series of simple algebraic manipulations extracted the wave equation from Maxwell's equations-which implied the existence of electromagnetic waves. Moreover, the wave equation implied that these electromagnetic waves traveled with the speed of light. One immediate deduction was that light itself is an electromagnetic wave-after all, the most obvious thing that travels at the speed of light is light. But just as the violin string can vibrate at many frequencies, so according to the wave equation-can the electromagnetic field. For waves that are visible to the human eye, it turns out that frequency corresponds to color. Strings with different frequencies produce different sounds; visible electromagnetic waves with different frequencies produce different colors. When the frequency is outside the visible range, the waves are not light waves but something else. What? When Maxwell proposed his equations, nobody knew. In any case, all this was pure surmise, based on the assumption that Maxwell's equations really do apply to the physical world. His equations needed to be tested before these waves could be accepted as real. Maxwell's ideas found some favor in Britain, but they were almost totally ignored abroad until 1886, when the German physicist Heinrich Hertz generated electromagnetic waves-at the frequency that we now call radio-and detected them experimentally. The final episode of the saga was supplied by Guglielmo Marconi, who successfully carried out the first wireless telegraphy in 1895 and transmitted and received the first transatlantic radio signals in 1901. The rest, as they say, is history. With it came radar, television, and videotape.

Of course, this is just a sketch of a lengthy and intricate interaction between mathematics, physics, engineering, and finance. No single person can claim credit for the invention of radio, neither can any single subject. It is conceivable that, had the mathematicians not already known a lot about the wave equation, Maxwell or his successors would have worked out what it implied anyway. But ideas have to attain a critical mass before they explode, and no innovator has the time or the imagination to create the tools to make the tools to make the tools that ... even if they are intellectual tools. The plain fact is that there is a clear historical thread beginning with violins and ending with videos. Maybe on another planet things would have happened differently; but that's how they happened on ours.

And maybe on another planet things would not have happened differently-well, not very differently. Maxwell's wave equation is extremely complicated: it describes variations in both the electrical and magnetic fields simultaneously, in three-dimensional space. The violin-string equation is far simpler, with variation in just one quantity-position-along a one-dimensional line. Now, mathematical discovery generally proceeds from the simple to the complex. In the absence of experience with simple systems such as vibrating strings, a "goal-oriented" attack on the problem of wireless telegraphy (sending messages without wires, which is where that slightly old-fashioned name comes from) would have stood no more chance of success than an attack on anti-gravity or faster-than light drives would do today. Nobody would know where to start.

Of course, violins are accidents of human culture-indeed, of European culture. But vibrations of a linear object are universal- they arise all over the place in one guise or another. Among the arachnid aliens of Betelgeuse II, it might perhaps have been the vibrations of a thread in a spiderweb, created by a struggling insect, that led to the discovery of electromagnetic waves. But it takes some clear train of thought to devise the particular sequence of experiments that led Heinrich Hertz to his epic discovery, and that train of thought has to start with something simple. And it is mathematics that reveals the simplicities of nature, and permits us to generalize from simple examples to the complexities of the real world. It took many people from many different areas of human activity to turn a mathematical insight into a useful product. But the next time you go jogging wearing a Walkman, or switch on your TV, or watch a videotape, pause for a few seconds to remember that without mathematicians none of these marvels would ever have been invented.

Chapter 1 : The Natural Order
Chapter 2 : What Mathematics is For
Chapter 3 : What Mathematics is About
Chapter 4 : The Constants of Change
Chapter 5 : From Violins to Videos

Chapter 6 : Broken Symmetry