Walking Wild III

One of the first major uses of probabilistic methods in computers was in calculating the random walks (see "Walking Wild I") of neutrons through different materials—a crucial issue in the design of nuclear weapons and atomic power plants after World War II.

Physicists applied similar techniques to show that a particle of light near the sun's center takes about fifty centuries to stroll in a random walk to the surface before finally escaping the sun and speeding to Earth in about eight minutes.

Mathematicians and scientists also extended the random-walk and Brownian-motion models (see "Quivering Particles II") to encompass other types of phenomena.

The long molecular chain of a polymer floating in a solvent, for example, resembles a miniature, truncated version of the path of a particle undergoing Brownian motion. In other words, you can imagine the chain's small molecular units (called monomers) as points and the chemical bonds between the units as the steps of a three-dimensional random walk.

However, to account for the fact that no two monomers can occupy the same region in space, the random walk has to be modified. A more realistic model is the self-avoiding random walk, which is a path that doesn't intersect itself. Such walks spread out much faster and tend to cover a larger area or volume than their standard counterparts for a certain number of steps.

A short self-avoiding random walk in three dimensions.

Using a self-avoiding random-walk model, polymer scientists can tackle such questions as: How many possible configurations can a long polymer chain adopt? What is the typical distance separating a polymer's ends.

The first question is really the same as asking for the number of different self-avoiding walks that are possible for a given number of steps. That's easy to answer in two dimensions for an ordinary random walk. There are four choices for each step, leading to four one-step walks, 4 ✕ 4 (or 16) two-step walks, and, in general, 4ⁿ n-step walks. Similar formulas can be worked out for other dimensions.

The calculations are a little trickier for a self-avoiding random walk. In two dimensions, there are four one-step walks, 4 ✕ 3 (or 12) two-step walks, and 4 ✕ 3 ✕ 3 (or 36) three-step walks. It turns out that there are 100 four-step walks.

Calculating the number of possible five-step walks is considerably more difficult, and even with computers to help out, no one has ever gone beyond about 39 steps, which has 1.13 ✕ 10¹⁷ possibilities.

Many other problems involving self-avoiding walks—including determination, in different dimensions, of the typical end-to-end distance after a certain number of steps—have turned out to be difficult to solve.

"The Brownian frontier and many other examples of random motions and their interaction properties continue to be an active area of research," commented mathematician Gordon Slade, who worked on self-avoiding random walks as models for polymers.

"Many of the remaining problems are appealing not just because of their relevance to applied fields beyond mathematics but also because the simplicity of their statements has an attraction of its own," Slade added. "This has drawn investigators from diverse backgrounds to study these problems, and there is hope that the progress of the recent past will continue in the coming years."

Next: Money Matters I

Walking Wild II

In the 1920s, Norbert Wiener transformed a random walk (see "Walking Wild I"), which is made up of discrete steps, into a mathematical model suitable to represent Brownian motion (see "Quivering Particles I"). He did it by making the steps or time intervals between steps infinitesimally small.

In Wiener's rigorous approach, once the position of a particle is established at the start, its position at any later time is governed by a Gaussian curve, as it is in Albert Einstein's physical model of Brownian motion (see "Quivering Particles II").

Wiener also proved that although the path of a Brownian particle is continuous, at no point is it smooth. Such a curious, incredibly jagged mathematical curve actually makes physical sense because a particle in Brownian motion can't jump instantaneously from one position to another, so its path must be continuous.

At the same time, as Jean Baptiste Perrin noted, erratic changes in direction appear to take place constantly, so you might expect the path to consist entirely of sharp corners. In fact, a two-dimensional Brownian trajectory wiggles so much that it ends up filling the entire area over which the motion occurs.

Two-dimensional Brownian motion.

With Brownian movement put on a solid mathematical footing by Wiener and others who followed his lead, such abstract formulations began to play a significant role in the creation of models of random phenomena.

Such models were used to represent the diffusion of heat through a metal, the spread of flu epidemics via the random walks of microbes, the structure of background noise and static affecting radio signals, the transport of perfume molecules from source to nose, and even the spread of rumors.

Wiener himself applied his model of Brownian movement to the problem of electronic interference, which disturbs the transmission of radio signals and causes the type of static heard on AM radio between stations.

Because the static has a strongly random character, Wiener was able to use the mathematics of Brownian motion to design an electronic filter to separate the signal from the background noise. Applied to the development of radar during World War II, his results were long kept a military secret.

It is interesting to note that applications of Brownian motion often began with the study of biological processes. The law of diffusion and models of the spread of heat in a material, for instance, initially arose from studies of heat generation in muscles.

Next: Walking Wild III

Walking Wild I

Though physicists began looking at random processes involving large numbers of molecules near the start of the twentieth century (see "Quivering Particles II"), it wasn't until 1920 that mathematicians began to develop a convincing mathematical model of Brownian motion, starting with the work of Norbert Wiener (1894-1964).

At the heart of the mathematics was the difficult problem of making precise mathematical sense of the notion of a particle moving at random.

A Brownian particle suspended in a liquid knows neither when the next shove will occur nor in which direction and how forcefully it will be propelled. Its displacement at any given moment is independent of its past history. These characteristics put Brownian motion in the category of a Markov process, named for mathematician Andrey A. Markov (see "Climbing and Sliding").

One of the simplest examples of such a process is a one-dimensional random walk, in which a "walker" is confined to a long, narrow path and moves forward or backward according to the results of repeatedly tossing a coin. The walker takes a step in one direction if the outcome is heads and in the opposite direction if the outcome is tails.

This graph shows the results of a one-dimensional random walk. The horizontal axis represents the number of steps taken, and the vertical axis shows how many steps you are away from your starting point if you start of 0. Steps in the forward direction are positive (upward) and steps backward are negative (downward).

For a walk along an infinite track, you can calculate a walker's long-term behavior. The resulting trail wanders back and forth along the track, and the probability of the wanderer's being a certain distance away from the starting point after taking a given number of steps is defined by a bell-shaped curve known as a Gaussian (or normal) distribution. The larger the number of steps, the wider would be the curve.

Indeed, the expected average distance of the walker from the starting point after a certain number of equal steps is simply the length of the step times the square root of the number of steps. For infinitely many coin tosses, a random walk confined to a line corresponds to one-dimensional Brownian motion.

One consequence of this type of erratic movement back and forth along a line is that a random walker is certain to return to the origin (or to any particular position on the track)—eventually.

This might sound like a good strategy in the admittedly unlikely situation of someone who's lost on a tightrope: Just take steps at random and you'll end up anywhere you want to be. But it might take longer than a lifetime to get there.

It's straightforward to extend the random-walk model to two dimensions. Take steps to the east, west, north, or south, randomly choosing each direction with equal probability (perhaps by using a tetrahedral die). You can imagine this walk going from vertex to vertex on an infinite checkerboard lattice.

If such a walk continues for an arbitrarily long time, the walker is certain to touch every vertex, including a return visit to its starting point.

In this illustration of a random walk in two dimensions, a walker starts off from a point on the left-hand side of the checkerboard grid, taking random steps to the east, north, west, or south.

The fact that returning to the origin is guaranteed in one and two dimensions suggests that there will be infinitely many returns. Once a walk gets back to the origin, it's like starting from scratch, and there will be a second return, then a third, and so on.

Similarly, such a random walk will visit every point infinitely many times.

Things are a little different in three dimensions. A walker can go up and down as well as in each of the four compass directions, so a standard cubic die serves as a suitable randomizer to determine the movements.

This time, however, even if a walker takes infinitely many steps, the probability of returning to the origin is only about .34. There's so much room available in three dimensions that it becomes considerably harder for a walker to find the way back to the starting point by chance.

In this computer-generated representation of a random walk in three-dimensional space, a walker has an equal probability of moving forward, backward, right, left, up, or down. This particular walk goes for 2,100 steps, beginning with blue, then continuing with magenta, red, yellow, green, cyan, and white segments. Extended indefinitely, the walk has only a 34 percent chance of ever returning to its starting point. G.M. Viswanathan

Indeed, this mathematical result affords an important lesson for anyone who is lost in space. Unless you make it home again within your first few steps, you're likely to end up lost forever (see "Wandering in Space"). No amount of aimless wandering will get you back after such a start. There are simply too many ways to go wrong.

Next: Walking Wild II

Quivering Particles II

Quivering Particles I

In 1905, theoretical physicist Albert Einstein (1879-1955) provided an elegant explanation of how tiny, randomly moving molecules could budge particles large enough to be observable under a microscope.

In his paper "On the Movement of Small Particles Suspended in Stationary Liquids Required by the Molecular-Kinetic Theory of Heat," Einstein used statistical methods to show that a suspended particle would get shoved in different directions by the combined effect of the modest impacts of many molecules.

For particles smaller than about twenty micrometers in diameter, the impact would generally fail to average out equally on all sides, giving the suspended particle a net shove in some direction.

The erratic jiggling of a microscopic particle suspended in water stems from the uneven distribution of impacts by water molecules at any given moment.

Interestingly, Einstein had not been aware of the experimental work on Brownian motion. His paper came about because he had begun to consider what effects might follow from the existence of atoms and molecules moving at high speeds that depended on the temperature.

His main goal, Einstein later wrote, was "to find facts which would guarantee as much as possible the existence of atoms of definite size." But, he continued, "in the midst of this, I discovered that, according to atomistic theory, there have to be a movement of suspended microscopic properties open to observation, without knowing that observations concerning the Brownian motion were already long familiar."

A few years later, physicist and chemist Jean-Baptiste Perrin (1870-1942) confirmed experimentally some of Einstein's key predictions.

In particular, Perrin and his students were able to track the movements of nearly spherical Brownian particles, which they recorded every thirty seconds and plotted on sheets of paper. Armed with these data, the researchers then used a formula derived by Einstein to determine the number of molecules present in a given volume of fluid.

The experiments gave Perrin a sense of the complexity of the path of a Brownian particle. His plots showed a highly irregular track, yet they gave "only a meager idea of the extraordinary discontinuity of the actual trajectory," Perrin noted.

If the researchers could have increased the resolving power of their microscope to detect the effects of bombardment by progressively smaller clusters of molecules, they would have found that parts of a path that originally appeared straight would themselves have had a jagged and irregular structure.

As the number of steps in a computer simulation of Brownian motion increases from 1,000 (left) to 10,000 (middle) to 100,000 (right), the same overall pattern of erratic movements persists, though on increasingly larger scales. G.M. Viswanathan

In fact, Brownian motion isn't the only place where such self-similar patterns occur.

"Consider, for instance, one of the white flakes that are obtained by salting a solution of soap," Perrin wrote in 1906. "At a distance, its contour may appear sharply defined, but as we draw nearer, its sharpness disappears…. The use of a magnifying glass or microscope leaves us just as uncertain, for fresh irregularities appear every time we increase the magnification, and we never succeed in getting a sharp, smooth impression, as given, for example, by a steel ball."

Nowadays such patterns, in which the magnified structure looks similar to and just as complicated as the overall structure, are known as fractals. And the paths of Brownian particles can be modeled mathematically as random walks.

Next: Walking Wild I

Quivering Particles I

Trails of a Wanderer

In 1828, botanist Robert Brown (1773-1858) published a pamphlet titled "A Brief Account of Microscopical Observations on the Particles Contained in the Pollen of Plants."

Brown had traveled widely, exploring the lands around the Cape of Good Hope in Africa and many parts of Australia before returning to London with specimens of more than four thousand plant species for identification. In the course of establishing a major botanical collection at a national museum, he had looked closely at pollen grains obtained from many of the collected plants.

While examining pollen grains of the plant Clarkia pulchella (pinkfairies) under a microscope, botanist Robert Brown observed tiny particles in constant motion within a grain.

Brown noted that, in some cases, "the membrane of the grain of pollen is so transparent that the motion of the…particles within the grain was distinctly visible."

The suspended granules were only a few micrometers in size, less than one-tenth of a typical pollen grain's width. Observed under a microscope, they appeared to be in continuous, erratic motion.

Physician and plant physiologist Jan Ingenhousz (1730-1799), who is credited with discovering photosynthesis, had reported similar particle activity in 1785 when he looked closely at powdered charcoal sprinkled on an alcohol surface.

Brown's experiments with granules extracted from crushed pollen grains, soot particles, and fragments of other materials suspended in water revealed the same type of unceasing, quivering movement.

Some scientists were quick to attribute the effect to minute, heat-induced currents in the liquid surrounding the particles or to some obscure chemical interaction between the solid particles and the liquid. However, the observation that the movements of two neighboring particles appeared to be quite uncorrelated helped rule out currents in the fluid as a cause.

Other characteristics of so-called Brownian motion were just as intriguing. For example, a given particle appeared equally likely to go in any direction, and its past motion seemed to have no bearing on its future course. Particles also traveled faster at higher temperatures, and smaller particles moved more quickly than larger ones. Most striking of all, the jerky movements never stopped.

To explain such observations, some scientists boldly ascribed the phenomenon to molecular movements within the liquid. They pictured the liquid as being composed of tiny molecules whizzing about at high speeds and colliding with each other. When molecules bumped into particles, they would give the particles random pushes.

Although some researchers hailed Brownian motion as visible proof that matter is made up of atoms and molecules, the question of the true structure of matter was at that time still a contentious issue. At the beginning of the twentieth century, for instance, the prominent physical chemist Wilhelm Ostwald (1853-1932) regarded atoms as merely "a hypothetical conception that affords a very convenient picture" of matter.

Next: Quivering Particles II

Time to Relax III

Time to Relax I

Time to Relax II

Several decades ago, John T. Bendler and his colleagues applied the notion of fractal time and defect mobility to understanding the properties of Lexan, a tough polycarbonate resin used for making bullet-resistant windows for automobiles. Defect diffusion turned out to be a good model for how the material responds to stresses and how it ages.

A chunk of Lexan consists of an irregular, three-dimensional network of long polymer molecules, with a precisely defined, repeating pattern of atoms. Experiments indicate that cooling the polycarbonate results in the freezing in of a small population of high-energy kinks in the molecular chains.

Molecular model of a polycarbonate molecule showing the carbonate linkages that act as kinks. The fractal-time motion of such kinks leads to stress relaxation.

The movement of these kinks in a fractal-time process along the molecular chains leads to relaxation. Kink movements reorganize the molecular backbone and effectively absorb mechanical energy, such as the impact of a bullet or sledgehammer.

They are also responsible for aging. As energy-absorbing kinks reach chain ends, the material gradually becomes more brittle and weak. With this insight, researchers looked into the possibility of slowing down aging by modifying chain ends.

The fractal-time, or defect-diffusion, model also helps to explain the stretching of silk and glass threads.

In 1835, physicist Wilhelm Weber noticed that attaching a weight to a long silk thread causes it to stretch to a certain length immediately. But that instantaneous elongation is unexpectedly followed by a gradual further lengthening that depends on how long the weight is applied.

The reason for such behavior lies in the fractal-time movement of defects within the materials.

Silk is a complicated natural polymer and has a variety of amorphous and crystalline forms. Under an applied load, the material tries to rearrange itself to redistribute and minimize stresses. Under these conditions, silk molecules relax by unwinding and changing the hydrogen bonding along their backbones.

In a glass fiber, the mobile defects correspond to imperfections in the distorted, tetrahedral network of oxygen and silicon atoms. Under a load, materials such as silk and glass mechanically reorganize themselves.

Although ceramicists, engineers, and artisans such as glassblowers have long been aware of the peculiar behavior of glasses, polymers, and ceramics and have taken these properties into account when working with the materials, little progress in understanding relaxation phenomena occurred for a long time because the mathematics initially used to describe such processes seemed complicated and difficult.

The concepts of mobile defects and fractal-time motion appear to offer a more tractable, self-consistent picture of the relaxation behavior of supercooled liquids and glassy solids.

One of the chief merits of the defect-diffusion theory is that it's mathematically simple. Researchers can use fractal-time mathematics—the mathematics of intermittent pausing—to model the kind of behavior displayed by almost all amorphous materials.

The theory also suggests ways of modifying in a useful manner the properties of industrially important materials.

Time to Relax II

Time to Relax I

An amorphous material's constituent atoms or molecules lie in random positions rather than at well-defined sites as in an orderly crystal lattice.

In a crystalline material, atoms or molecules sit in an orderly array (left). In amorphous solids, the pattern is more irregular (right).

Moreover, just as crystal structures are rarely perfect and contain dislocations, vacancies, and other imperfections, amorphous materials also contain "defects," in which bonds between atoms or molecules may be strained, distorted, or displaced.

For example, such defects occur during glass formation because molecules find they have too little time during the cooling process to orient themselves into their proper positions to form a closely packed crystal structure. Inevitably, glasses end up containing low-density regions, the analog of vacancies in crystals.

In 1983, physicists Michael Shlesinger and Elliott Montroll proposed that migration, or diffusion, of mobile defects could account for stretched exponential relaxation in the case of an amorphous material relaxing after the application of an electric field. (See An Unbounded Experience in Random Walks with Applications by Michael F Shlesinger.)

They suggested that defects, in order to move, have to overcome different energy barriers scattered throughout the material. Whereas small barriers are easy to hurdle, larger ones significantly reduce mobility.

In the early stages of relaxation, defects that experience low barriers don't have any trouble. There's enough thermal energy for them to jump such barriers, and relaxation ensues. Others, faced with moderate barriers, take longer to get moving.

Thus, a random distribution of energy barriers implies a wide range of relaxation times, leading to the stretched exponential relaxation observed for amorphous materials. Relaxation stretches over a long period of time.

Mathematically, the situation is closely related to the problem of determining the length of a fractal. Magnifying a fractal by any amount reveals a miniature version of the larger form. Finer and finer scales show more and more detail and lead to greater and greater estimates of total length.

For example, measuring the length of a fractal coastline leads to different answers, depending on the scale used.

Scale matters. Taking long steps carries you past a lot of tiny indentations (top). Taking shorter steps means that you end up traveling a longer distance along such an indented shoreline (bottom).

On a world globe the size of a basketball, the eastern coast of the United States looks like a fairly smooth curve, which, according to the globe's scale, may be roughly 3,000 miles long. The same coast drawn on an atlas page showing only the United States looks much more ragged. Adding in the lengths of capes and bays now evident extends the coast's length to 5,000 or so miles.

Piecing together detailed navigational charts to create a giant coastal map reveals an incredibly complex curve perhaps 12,000 miles long. Each change in scale reveals a new array of features to be included in the measurement.

Just as every distance scale occurs in the coastline problem, every time scale occurs for relaxation in amorphous materials. Each shift in time scale—from days to minutes to seconds—adds new features to be included in a relaxation measurement.

Although it isn't as picturesque to think of infinitely many time scales as it is to think of patterns within patterns on different length scales, the analogy is mathematically exact.

This comparison leads to the concept of fractal time. Instead of occurring in a sequence of regular, equally spaced intervals, events that occur in fractal time are clustered.

Instead of occurring at regular intervals, events that happen in fractal time are clustered in a self-similar pattern that features rapid bursts interspersed with long pauses.

Such clusters consist of events that happen rapidly, one after the other, interspersed with long stretches of nothing happening in between.

To support this theoretical picture, researchers have discovered that in polymer relaxation, some phenomena occur within picoseconds whereas other effects aren't apparent for years. Such an astonishing array of time scales shows how tricky it is to do experiments investigating the phenomenon because it's hard to measure physical characteristics over so many orders of magnitude in time.

Time to Relax III

Time to Relax I

The rubber in a pair of boots, retrieved after a long stay in an attic, shows its age in an annoying way. No longer as flexible as it once was, the material (an elastomeric polymer) readily cracks and falls apart. Under the same conditions, many other plastics suffer a similar fate.

One cause of this aging process is chemical. Sunlight or oxygen can initiate chemical reactions that alter the material's properties. But deterioration occurs even when a material is kept in the dark or away from oxygen. The material gradually becomes more dense and brittle, losing its toughness and impact resistance.

The explanation for this behavior lies in the way "defects" within amorphous, or noncrystalline, materials reorganize themselves over long periods of time.

Glass is an example of an amorphous material. PPG Place, Pittsburgh, Pennsylvania.

When expressed in terms of the concept of fractal time, the same mathematical model used to describe polymer aging also applies to the stretching of glass or silk fibers; the recovery, or relaxation, of glassy materials after the removal of a stress; and a wide range of other phenomena in amorphous materials.

In such processes, events occur in self-similar bursts—featuring distinct clusters of activity interspersed with long stretches of inactivity. Some changes in materials occur right away while others take years to show up.

Relaxation is an issue of practical importance. Slow aging processes, both environmental and physical, control the lifetimes of a great many manufactured products, from electronic devices to optical fibers and advanced composite materials. Elucidating of how such processes occur can suggest novel techniques for toughening ceramics and for designing polymers having particular characteristics.

Relaxation processes are common in physical systems. For example, pull on a glass fiber, then let go. The glass first stretches, then shrinks. Apply a strong electric field to a polymer, then turn it off. Areas of positive and negative charge in the polymer line up with the field, then drift out of alignment.

In each case, the material endures a stress, then recovers, or relaxes, when the stress is removed.

Relaxation in a crystalline material typically proceeds at an exponential pace. That type of relaxation follows the same pattern as the decay of a radioactive isotope. Such a process is characterized by a certain time, known in the case of radioactive decay as the half-life.

Normally, you find that relaxation is clustered around a certain time. It might take a second, a day, or a week. But an amorphous solid takes a longer time to relax than would be expected if relaxation simply followed an exponential pattern.

In amorphous systems, some parts relax very quickly. If those parts relax in, say, seconds, other pieces might relax on a time scale of minutes, and still others on a scale of days or weeks. If you were to wait long enough—even years—you would still detect changes taking place. No characteristic time can be defined for such an extended relaxation process.

This type of behavior has come to be known as stretched exponential relaxation. It fits a wide range of relaxation processes in disordered systems, including the way many polymers, glasses, and ceramics respond to stresses caused by changes in pressure and temperature and the imposition of electric and magnetic fields.

Because so many different systems behave in such a strikingly similar fashion, physicists, in their search for an explanation, have concentrated on what these systems have in common. They have found that what's important is not the details of a material's atomic or molecular structure but rather its state of disorder.

Time to Relax II

Water Clock

 

Working model of Su Song's water-powered astronomical clock tower. Science Museum, London, England, 1975.

Photo by I. Peterson

Alpha 'n' Beta I

 

Could It Be?

Could English scientist Michael Faraday have switched on an electric incandescent light bulb in his home? Faraday died in 1867, and the Edison light bulb was not produced until about 1880. Hence, the answer is "no."

Thomas A. Edison's laboratory in West Orange, New Jersey.

Could it have been remotely possible for the following statements to be true?

1. Queen Elizabeth I looked through a telescope and saw Jupiter's moons.

2. Sir Walter Scott lit a Bunsen burner.

3. Theodore Roosevelt was X-rayed.

4. Queen Victoria relieved a headache with aspirin.

5. During the Crimean War, dynamite blew up a Russian fort.

6. Albert Einstein listened to a transistor radio.

7. Galileo wrote to Isaac Newton.

8. Antoine Lavoisier knew of the existence of the planet Neptune.

9. Johannes Kepler read Galileo's Starry Messenger.

10. Charles Darwin used saccharin in his tea.

Answers:

1. Queen Elizabeth I looked through a telescope and saw Jupiter's moons. No.

2. Sir Walter Scott lit a Bunsen burner. No.

3. Theodore Roosevelt was X-rayed. Yes.

4. Queen Victoria relieved a headache with aspirin. Yes.

5. During the Crimean War, dynamite blew up a Russian fort. No.

6. Albert Einstein listened to a transistor radio. Yes.

7. Galileo wrote to Isaac Newton. No.

8. Antoine Lavoisier knew of the existence of the planet Neptune. No.

9. Johannes Kepler read Galileo's Starry Messenger. Yes.

10. Charles Darwin used saccharin in his tea. Yes.

Drums That Sound Alike

The sounds of different types of drums in a marching band are easy to distinguish, even without seeing the instruments.

What makes these sounds so readily identifiable is that each drum vibrates at characteristic frequencies, depending mainly on the size, shape, tension, and composition of its sound-generating drumhead. This spectrum of frequencies—the set of pure tones, or normal modes, produced by a vibrating membrane stretched across a frame—gives a drum's sound its particular color.

Physicists and mathematicians have long recognized that the shape of the boundary enclosing a membrane plays a crucial role in determining the membrane's spectrum of normal-mode vibrations. In 1966, mathematician Mark Kac focused attention on the opposite question.

Kac asked whether knowledge of a drum's normal-mode vibrations is sufficient for unambiguously inferring its geometric shape. His paper, which proved remarkably influential, bore the playful title "Can One Hear the Shape of a Drum?"

Previously, mathematicians had established that both the area of a drum's membrane and the length of its rim leave a distinctive imprint on a drum's spectrum of normal modes. In other words, one can "hear" a drum's area and perimeter.

The question of whether one can infer a drum's geometrical shape from its normal modes remained unresolved until 1991.

That was when mathematicians Carolyn S. Gordon, David L. Webb, and Scott Wolpert came up with two drums that have equal areas and perimeters but different geometric shapes. They proved that the drums, each a multisided polygon, display identical spectra.

The original pair of soundalike, or isospectral, drums discovered by Gordon, Webb, and Wolpert.

In principle, two drums built out of these different shapes would sound exactly alike. Both would generate the same set of normal-mode frequencies.

Since the initial discovery, Gordon, Webb, and others have identified many pairs of soundalike drums. All of the known examples have at least eight corners; typically, each member of a pair consists of a set of identical "building blocks" arranged into different patterns.

However, it's one thing to prove a mathematical theorem and quite another to demonstrate its reality in a physical situation.

When physicist Srinivas Sridhar heard about the Gordon-Webb-Wolpert discovery, he decided to put it to an experimental test—and he had just the right kind of setup to do the necessary experiment.

Sridhar and his coworkers had been investigating aspects of quantum chaos by looking at the patterns created when microwaves bounce around inside thin metal enclosures of various shapes. The same technique could be used to identify normal modes, with microwaves standing in for sound waves and severely squished cavities standing in for membranes.

To test the drum theorem, the researchers constructed two cavities corresponding to one of the pairs of shapes discovered by Gordon and her colleagues. Fabricated from copper and having eight flat sides, each angular enclosure was nearly 8 centimeters long and less than 6 millimeters thick.

Sending in microwaves through a tiny opening and measuring their strength over a range of frequencies at another location enabled the researchers to establish the frequencies of the normal modes of each cavity. They could also map the standing wave patterns inside the cavities.

Remarkably, the frequencies present in both spectra were practically identical. Any discrepancies between the spectra could be attributed to slight imperfections introduced during assembly of the enclosures. See "Experiments on not 'hearing the shape' of drums."

At the time it was done, the experiment provided information that was unavailable mathematically: the shape of standing wave patterns and the actual frequencies making up the spectra in the pair of soundalike drums.

Subsequently, mathematician Toby Driscoll computed the standing wave patterns and frequencies for the same pair of shapes that Sridhar and his colleagues had tested experimentally. His computational results, reported in the paper "Eigenmodes of Isospectral Drums" in SIAM Review, closely matched those obtained by the physicists.

Computed standing wave patterns (first four normal modes) in pairs of polygons having different shapes but identical normal modes. Courtesy of Toby Driscoll.

Driscoll also applied his computational technique to other pairs of isospectral drums. Meanwhile, mathematicians continue to search for additional soundalike doubles. Are there soundalike triples? No one knows.

Originally posted April 14, 1997

See also "Fractal Drum."

Acoustic Residues

There's a surprising mathematical ingredient in the sound of many performing artists and recording stars. It manifests itself in the form of clusters of panels hanging on the walls of recording studios, concert halls, nightclubs, and other venues. Sculpted from wooden strips separated by thin aluminum dividers, each panel consists of an array of wells of equal width but different depths.

Called reflection phase gratings, these panels scatter sound waves. The result is a richer, livelier sound with an enhanced sense of space. Listeners claim that the panels seem to make the walls disappear. A small room takes on the air of a great hall.

The secret lies in the varying depths of a panel's wells. With depths based on specific sequences of numbers rooted in number theory, the wells scatter a broad range of frequencies evenly over a wide angle.

The scientist who pioneered the ideas responsible for this development was Manfred R. Schroeder (1926-2009). In the 1970s, Schroeder and two collaborators undertook a major acoustical study of more than 20 famous European concert halls. One of their findings was that listeners like the sound of long, narrow halls better than that of wide halls. Perhaps the reason for this, Schroeder reasoned, is related to another finding that listeners prefer to hear somewhat different signals at each of their two ears.

In a wide hall, the first strong sound to arrive at a listener's ears, after sound traveling directly from the stage, is the reflection from the ceiling. Ceiling reflections produce very similar signals at each ear. In narrow halls, however, the first reflections reach the listener from the left and right walls, and the two reflections are generally different.

This may be one reason why many modern halls are acoustically unpopular. Economic constraints dictate construction of wide halls to accommodate more seats, and modern air conditioning systems allow lower ceilings. To improve the acoustics of such halls, sound must be redirected from the ceiling toward the walls.

A flat surface by itself can't do the job. It reflects sound in only one direction, according to the same rules that govern light reflecting from a mirror. The ceiling must have carefully orchestrated corrugations that scatter sound so that roughly the same amount of energy goes in every direction.

Schroeder discovered that number theory can be used to determine the ideal depth of the notches, resulting in an acoustic grating that's analogous to diffraction gratings used to scatter light.

One effective acoustic grating is based on quadratic-residue sequences. Such a sequence consists of the remainders, or residues, after squaring consecutive whole numbers, then dividing them by a given prime number.

Suppose, for example, the given prime number is 17. The first sequence member is the remainder, or residue, after the first number, 1, is squared and divided by 17. The answer is 1. Squaring all the numbers from 1 to 16, then dividing by 17 and determining the residue, produces the sequence: 1, 4, 9, 16, 8, 2, 15, 13, 13, 15, 2, 8, 16, 9, 4, 1. For larger numbers, the pattern simply repeats itself.

Finding the depth of a given grating well involves multiplying the appropriate number in the sequence by the longest wavelength for which the grating is designed to scatter sound efficiently and then dividing by a factor that depends on the well's numerical position. Mathematical analysis shows that for such an arrangement, the spectrum of energies scattered into different directions is essentially flat, meaning that roughly equal amounts of energy go in all directions.

Why does number theory work so well? The answer is in the way waves cancel or reinforce each other, depending on whether the crest of one wave meets the trough or crest of another wave. For perfectly periodic waves, destructive interference occurs whenever one wave lags behind the other by half a wavelength, one-and-a-half wavelengths, two-and-a-half wavelengths, and so on. In each case, it's the extra half wavelength that decides when waves cancel each other out.

So, in wave interference, it's not the total path difference between two waves that determines the resulting pattern but the residue after dividing by the wavelength. Hence, modular-arithmetic techniques and quadratic residues are relevant to acoustics.

Architectural acoustics designers have only three ingredients they can use to conjure up every imaginable type of acoustic environment; namely, absorption, reflection, and diffusion. Sound-absorbing surfaces made of foam or fiberglass and sound-reflecting surfaces, such as flat or curved panels, are widely used. Until reflection phase gratings came along, there really were no surfaces designed to spread sound around in both space and time. For designers, it was like trying to type a paragraph without using, say, the letter "d."

Inspired by Schroeder's work, Peter D'Antonio started RPG Diffusor Systems (now RPG Acoustical Systems) in 1983 to bring reflection phase gratings based on quadratic residues and other mathematical constructs to the acoustic marketplace. In recent years, the company has developed novel diffusor designs based on such mathematical concepts as primitive roots and fractals.

Now that improved digital recordings, electronic instruments, and home theater systems are readily available, demand has increased for superior acoustic surroundings for making and listening to recordings. The use of reflection phase gratings to diffuse sound helps create a listening environment in the home and elsewhere that allows a listener to experience an old-fashioned concert-hall ambiance.

Number theory makes an important contribution to the sound of music.

Originally posted July 9, 2001

Meteorite Speckles

Sericho pallasite meteorite, Meteorite Museum, University of New Mexico, Albuquerque, New Mexico, 2020.

Photo by I. Peterson

Navajo Meteorite

Navajo meteorite. Meteorite Museum, University of New Mexico, Albuquerque, New Mexico, 2020.

Photo by I. Peterson

Physics Demonstrations

For more than 25 years, Richard B. Minnix (1933-2018)  and D. Rae Carpenter Jr., physics professors at the Virginia Military Institute (VMI) in Lexington,Virginia, offered summer courses for high school teachers interested in perfecting the art of presenting physics demonstrations.

The programs, funded in part by the National Science Foundation, gave participants the chance to spend nearly two weeks sharing methods of demonstrating physical principles, learning new techniques for enlivening physics lectures, and building equipment in the well-equipped machine shop to take back to their own classrooms.

As a high school physics teacher at Kingston Collegiate and Vocational Institute in Kingston, Ontario, I participated in the program in 1977, spending the first two weeks of August at VMI. To me, no physics lesson was complete without some link to the real world and everyday experience, and the course provided a wealth of opportunities to engage in that vision.

Attendees at the 1977 "Lecture Demonstration Methods in Physics Instruction" summer course, held at the Virginia Military Institute. Instructor Dick Minnix is on the left side of the second row; Rae Carpenter is on the right side of the top row.

Ivars Peterson getting the point: sitting on a bed of nails to experience the relationship between pressure and surface area.

Field trips took course attendees to the radio telescopes at Green Bank, West Virginia, Thomas Jefferson's Monticello and University of Virginia, and a farm, where they could see vivid demonstrations of physical principles in action, harnessed for human use.

Thanks to gravity and careful design, a wooden millrace delivers water to a mill at Halcyon Farm.

Standing beside a massive radio telescope at the National Radio Astronomy Observatory in Green Bank, West Virginia.

Thomas Jefferson's Monticello, Charlottesville, Virginia, 1977.

Statue of Thomas Jefferson in front of the Rotunda at the University of Virginia, Charlottesville,Virginia, 1977.

Course completion certificate.

In 1993, Minnix and Carpenter published The Dick and Rae Physics Demo Notebook, which contains 650 of their favorite physics lecture demonstrations.

Row Your Boat

MSRI Journal
The Mark of Zeta
The Return of Zeta
Solitaire-y Sequences
A Song About Pi

Row Your Boat

Designed for speed, a racing shell has a distinctive shape. The boat's slim, needle-like profile allows it to skim the water at a rapid rate, propelled by oar.

In varsity and Olympic competition, races may involve boats with one, two, four, or eight rowers. Interestingly, although a shell with eight rowers is much larger than one with a single rower, all the boats have roughly the same proportions (at least for the surface area over which the shell makes contact with water).

Data from 2000-meter world and Olympic championship races show that the larger boats go faster than the smaller ones. In the late 1960s, that fact caught the eye of Thomas A. McMahon (1943-1999), a professor of applied mechanics and an expert in animal locomotion at Harvard University. He wondered why that might be true. How does the speed depend on the number of rowers?

Race data are from the 1964 Summer Olympics in Tokyo, the 1968 Summer Olympics in Mexico City, the 1970 World Rowing Championships in Ontario, Canada, and the 1970 Lucerne International Championships.

McMahon recognized that because racing shells seating one, two, four, or eight rowers happen to be built with roughly the same proportions, it may be possible to use a simple mathematical model to predict the speed as a function of boat size, even though the physics, in all its gory detail, is quite complex.

The total mass of the rowers plus that of the boat equals the mass of the water displaced by the boat. Because the boats are geometrically similar, this displacement is proportional to the volume, or the cube of the boat's length. The length is, in turn, proportional to the number of rowers.

Two main effects produce the drag experienced by a boat moving through water: wave generation and friction between hull and water. The long, thin shapes of the boats minimize the part of the drag due to wave-making, so that component can be neglected. Skin-friction drag is proportional to the product of the wetted area and the square of the speed. The wetted area itself is proportional to the square of the boat's length. So the drag force is proportional to speed squared times length squared.

McMahon then made the additional assumption that the power available to drive a boat is proportional to the number of rowers. That power is used to overcome the drag force and provide speed. Hence, the power is proportional to the drag force times the speed, or the speed cubed times the length squared.

Putting these proportionalities together algebraically allows you to deduce a relationship between speed and the number of rowers. This model predicts that the speed should be proportional to the number of rowers raised to the one-ninth power. McMahon found that plotted data from various races fit that theoretical relationship quite nicely. It would be interesting to see if it still holds for more recent sculling events.

McMahon's study of rowing is a striking example of how a relatively simple mathematical model can capture essential features of a complex physical phenomenon and yield insights into what is going on. The trick is to come up with an appropriate model. That task requires not only a firm grasp of the relevant areas of mathematics but also an understanding of physical law and behavior.

The summer program in "dynamics of low-dimensional continua," held in 1999 at the Mathematical Sciences Research Institute (MSRI) in Berkeley, Calif., introduced students to tools and concepts for developing and analyzing mathematical models applicable to fluid flow, crystal growth, and many other phenomena encountered in materials science, chemistry, biology, engineering, and physics.

Conducted by L. Mahadevan and Anette Hosoi, the two-week course gave the students some insights into the art of building mathematical models relevant to the real world. The students ended up doing computational projects on such topics as fluid mixing, vortex formation, turbulent convection, and dendritic crystal growth.

McMahon's rowing study was just one of a number of examples Mahadevan cited to illustrate how scaling arguments and dimensional analysis can provide a good starting point. You get good guesses for extremely complicated problems, he said. [L. Mahadevan lecture video: Scaling and Dimensional Analysis]

Once the basic framework is in place, you can then focus on details and study deviations from an initially derived theoretical relationship. In the rowing example, boats with more rowers actually perform a little better than the one-ninth power relationship suggests. It's possible, for instance, that the longer, more heavily laden boats are better at overcoming wave-making drag, which was neglected in the initial model.

In a 1915 paper published in Nature, Lord Rayleigh (1842-1919) extolled the value of the principle of similitude (now called dimensional analysis) in providing physical insight and deducing physical laws. "It happens not infrequently that results in the form of 'laws' are put forward as novelties on the basis of elaborate experiments, which might have been produced a priori after a few minutes' consideration," he wrote.

Originally posted July 19, 1999.

Juggling By Design
Averting Instant Insanity
Matrices, Circles, and Eigenthings
Lunar Shadows
MSRI Reflections

Rust Wedge

Rust wedge. Exploratorium, San Francisco, California, 2018.

Photo by I. Peterson

Geneva Drive

Geneva drive model. Exploratorium, San Francisco, California, 2018.

Photo by I. Peterson