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The Mathematical Tourist
A map is a picture of both the known and the unknown. Ancient mariners carried maps showing major cities and well-traveled trade routes alongside mysterious territories marked by fanciful names and decorated with mythical creatures. What wasn't known was guessed at. It was these unknown regions, with their promise of adventure and vague hints of fabulous wealth, that attracted early explorers.
A map of modern mathematics would reveal a similar mix of the familiar, the exotic, and the unknown. Algebra, trigonometry, and euclidean geometry, familiar to high-school students, lie in well-settled areas. Newer settlements, such as calculus, establish their spheres of influence nearby. Young upstarts--computer science, for one--nibble at old boundaries. Beyond the familiar are vast regions of mathematics still to be discovered.
MAPS OF A DIFFERENT COLOR
A deceptively simple mathematical problem lurks within the brightly colored maps showing the nations of Europe or the patchwork of states in the United States. It's the sort of problem that might worry frugal mapmakers who insist on decorating their maps with as few colors as possible. The question is whether four colors are always enough to fill in every conceivable map that can be drawn on a flat piece of paper so that no countries sharing a common boundary are the same color.
An additional definition and a condition turn this mapmakers' conundrum into a well-defined mathematical problem. A single shared point doesn't count as a shared border. Otherwise, a map whose countries are arranged like the wedges of a pie would need as many colors as there are countries. Moreover, countries must be connected regions; they can't have colonies scattered all over the map (see Figure 1.1, top left and middle).
First proposed in 1852 by British graduate student Francis Guthrie in a letter to his younger brother, the four-color problem has since intrigued and stumped professional and amateur mathematicians alike. Early on, mathematicians realized that three colors are certainly not enough for every possible map. It's easy to draw a map that needs four colors (see Figure 1.1, top right). English mathematician Augustus De Morgan also proved that it was impossible for five countries to be placed so that each one of them borders the other four. That led him to believe that five colors would never be needed. However, the proof that exactly four colors suffice remained elusive despite years of map sketching and attempted proofs.
In 1976, two mathematics professors finally proved the four-color theorem. It was the kind of news that mathematicians greet with champagne toasts. When Kenneth Appel and Wolfgang Haken of the University of Illinois announced their success, there were great celebrations in mathematical circles. They had scaled one of the Mount Everests of mathematics.
But there was a shock awaiting anyone curious about how the four-color theorem had been proved. The proof was unlike any other mathematical proof that had ever been done. Its complex text--strewn with diagrams and filling hundreds of pages--daunted all but the most determined readers. Even more dismaying to many mathematicians was that for the first time, a computer had been used as a sophisticated accountant to verify certain facts needed for the proof.
The computer allowed Haken and Appel to analyze a large collection of possible cases, a collection shown by mathematical analysis to be sufficient to prove the theorem. That task took 1,200 hours on afast computer; it would have taken practically forever by hand. That also meant the proof could not be verified without the aid of a computer. Furthermore, Appel and Haken had tried various strategies in computer experiments to perfect several ideas that were essential for the proof.
Even now, doubts still linger about the validity of the proof of the four-color theorem. Although most mathematicians accept the idea that the Haken-Appel approach works, persistent rumors insist there is something wrong with the proof. Some people hint that the computer program was faulty; others, that the method itself is wrong in some way. A few complain that the computer program can't be verified properly. Nevertheless, the proof continues to hold up. Despite the discovery of such minor flaws as typographical and copying errors, the proofs basic strength has not been threatened.
To Appel and Haken, their proof of the four-color theorem represents a new and interesting facet of mathematics. Although a shorter, more elegant proof may someday be found, it's also possible that no such proof exists. The theorem may be one for which there never will be a proof short enough to be readily understood at a glance. It's an example of a curious occurrence in mathematics: the coupling of a simple, succinct problem with an incredibly complicated proof.
More than a decade after Haken and Appel's proof of the four-color theorem, computers are still scarce among mathematicians and seldom used for serious mathematical research. One mathematician, Stanford University's Joseph Keller, even remarked in 1986 that at his university, the mathematics department has fewer computers than any other department, including French literature. Many mathematicians have the feeling that using a computer is akin to cheating and say that computation is merely an excuse for not thinking harder.
Still, computers are beginning to creep into mathematics. By providing vivid images that suggest new questions, they are helping to mend a rift that had developed between pure and applied mathematics. They are linking mathematical questions in computer science with computational questions in mathematics. These changes are enriching a subject that many outsiders have regarded as an abstract, even useless, pursuit.
The availability of very fast computers with large memories has led to two important developments in science and technology, both ofwhich rely on mathematics. The first development is the use of mathematical equations to represent a physical situation, such as the air rushing over an airplane's wing or the chemical reactions leading to the formation of acid rain. When they are good enough, such mathematical models allow researchers to replace many of their wind-tunnel and test-tube experiments with computer manipulations.
The second development arises because computers are invaluable for extracting relationships and patterns hidden deep within vast amounts of data. For example, they can pinpoint the location of an oil field using seismic data or the site of a brain tumor using x-ray absorption measurements. The storage and processing of data require subtle, sophisticated techniques. More often than not, a piece of abstract mathematics worked out years before--and believed to be totally without practical value--finds a role in the "real" world.
One particularly striking example of the value of computer simulations and of the interaction between mathematics and science is in the work of applied mathematician Charles Peskin of New York University. He has spent more than a decade perfecting a computer model for blood flow in the heart. Peskin's aim is to create a tool that designers of mechanical heart valves can use to test prototypes without having to rely entirely on animal experiments and clinical trials with human patients.
Peskin has concentrated on modeling the heart's left side and on the movement of one particular valve, known as the mitral valve. With every heartbeat, the mitral valve's thin, flexible flaps of tissue smoothly slip out of the way when blood pushes forward. When the heart contracts, the valve snaps shut to keep blood from flowing back the wrong way. Peskin's mathematical model includes the characteristics of both the blood flow and the heart chamber's muscle tissue. Like an elastic band, the modeled tissue responds flexibly to the blood's pressure while it also exerts a force on the flowing blood.
In Peskin's heart model, the blood is represented by a large number of discrete points, each with a specific velocity and pressure. These velocities and pressures change as the fluid elements interact with their neighbors in ways that can be calculated using equations from physics. The heart's muscle fibers are modeled by a collection of moving particles joined by tiny springs. The properties of these elastic links change over time to simulate the change in tissue stiffness during a heartbeat.
Coupling flexible boundaries with a moving fluid made up of particles was perhaps one of the greatest mathematical challenges that came up during the development of this heart model. The mathematical and computational methods that Peskin and his collaborators developed to solve the problem also apply in many other situations--from the movement of fish in water to the flow of suspended particles in a liquid.
The net result of the millions of computations required to make Peskin's model work is a sequence of pictures that can be strung together to produce a dramatic movie of a beating heart (see Figure 1.2). The vivid, two-dimensional images clearly show where flow is uneven, how differently shaped artificial valves respond, and even the likely performance of a weakened or diseased natural valve.
To an increasing number of practitioners, computer simulations represent a third way of doing science, alongside theory and experiment. In the past, a physical theory often consisted of a set of differential equations, which describe how a system changes over time or varies from place to place (as defined in Chapter 6). A theory today may consist of a computer program that models the way a system is supposed to evolve. Once experience shows that a given computer program accurately simulates a real system, then experiments may be done on the computer instead of in the lab. Of course, the computer model may produce results that fail to mimic reality. Then it's back to the drawing board.
The use of computer models is spreading rapidly to fields as diverse as astronomy, chemistry, and psychology and is illuminating the dynamics of pinwheel galaxies, the tremors that shake protein molecules, and the mechanisms that underlie human memory. In many cases, mathematical models of physical situations save time and money. In aerodynamics, for example, typing in a few values at a computer terminal and seeing a graphics display of the results can save days otherwise spent crafting a wooden or metal model and testing it in a wind tunnel. Laboratory experiments and observations are still necessary, but now the researcher can choose which experiments are most likely to be useful and may even have a better idea of what behavior to observe and which variables to measure in a given experiment.
When an experiment produces masses of data, mathematical and statistical techniques are available to help researchers make sense of the jumble. The development of x-ray tomography and other techniques for probing the human body provides a good example of how much mathematics, both old and new, is needed to make such techniques work.
In computerized x-ray tomography, hundreds of x-ray beams, finer than needle points, zip through a slice of the human body. As each beam passes through tissue, bone, and blood, it weakens by an amount that depends on what it encounters. In mathematical terms, firing an x-ray beam through an object and seeing what happens to the beam is equivalent to finding "the projection of a function along a given line." This mathematical operation is called a Radon transform, named for a French mathematician who worked on such transforms early in the twentieth century. Radon's work appears to have been pure mathematics carried out for its own sake.
In the case of tomography, researchers must sift through their x-ray absorption data to compute the tissue density at a particular spot. In other words, from patterns hidden within all the x-ray intensity measurements, they need to work out the tissue arrangement that would give that particular set of intensities. That's not unlike looking at a complex pattern of ripples on a water surface and trying to decide where the ripples started and how many sources there were. Mathematically, the operation means computing the original set of values when only a set of averages of those values is known. It means performing a Radon transform in reverse.
Radon proved the basic theorems that specify under what conditions a function can be reconstructed from various projections or averages. Radon's ideas, however, apply only to continuous functions, which can be represented as smooth curves. In tomography, an infinite number of x-ray beams would have to be used to sample the entire cross section before the tissue densities at every spot could be accurately computed. Because only a finite number of x-ray measurements are actually made, mathematicians have had to work out approximatemethods that allow cross sections of human organs to be reconstructed with a minimum of error. They try to make sure that an error isn't right where a tumor may be. New mathematical work is leading to faster algorithms for doing tomography computations so that x-ray images can be generated almost instantaneously.
Work on the discrete Radon transform and its inverse as applied to tomography has also suggested some interesting mathematical questions to investigate. Mathematician Ron Graham at AT&T Bell Laboratories and statistician Persi Diaconis at Stanford University are taking a close look at what can be deduced in situations in which certain averages are known but the original data are missing. Graham wonders, for example, to what extent secret data contained in confidential files can be uncovered if the right questions are asked.
Cracking a confidential data base may be something like the old parlor game of twenty questions. A player, receiving only yes-or-no answers yet asking the right sequence of pertinent questions, can often deduce the identity of some hidden object or person. In the same way, the answers to a series of general questions addressed to a particular data base could add up to a revealing portrait of something that is supposed to be secret.
A simple example shows how such a scheme might work. Suppose someone wants to find out Alice's salary. The inquisitor has access to information revealing that the average of Alice's and Bob's salary is $30,000, the average of Alice's and Charlie's salary is $32,000; and the average of Bob's and Charlie's salary is $22,000. This provides enough information to deduce that Alice's salary is $40,000.
Researchers often face a situation in which certain averages are known but the original data are missing. If eight data points happen to be identified with the eight vertices of a cube and each of the eight numbers is the average of its three nearest neighbors, then it's possible to deduce the actual but currently hidden value associated with each vertex (see Figure 1.3). In this situation, the actual value at each vertex is equal to the sum of the nearest-neighbor averages minus double the average at the corner farthest from the point of interest. Curiously, the point that makes the largest contribution to the answer is the one that's farthest away.
Diaconis and Graham have developed a mathematical theory, based on the idea of discrete Radon transforms, that helps to decide how many and which averages are needed to crack a data base or to analyze statistical data. At the root of their exercise is the mathematical concept of how completely a bunch of averages captures the mathematical relationship underlying a data set.
To many people, mathematics--unchanging, reliable, dusty with age--has an aura of authority and rests on a firm foundation of pure logic. It promises certainty. Schoolchildren learn strict, seemingly infallible rules, ranging from the mechanics of addition to the intricacies of factoring algebraic expressions. Engineers routinely calculate specifications, counting on reference volumes filled with mathematical formulas. Stockbrokers use the logic built into elaborate computer programs to help make decisions about when to buy and sell.
But behind the apparently stolid, pristine, immutable public face of mathematics lies the exciting, turbulent, ever-changing world of mathematical research. Just as physics and other sciences go through episodes of both revolution and evolution, mathematics, too, changes and grows, not only in the way it is applied but also in its fundamental structure. New ideas are introduced; intriguing connections between old ideas are discovered. Chance observations and informed guesses develop into whole new fields of inquiry.
The territory of mathematics can be divided into three large chunks. The first, called algebra, involves the study of number systems. In general, an algebra consists of a number of mathematical entities (such as integers, matrices, vectors, or sets) and operations (such as addition or multiplication) with formal rules expressing the relationships between the mathematical entities. It includes, for example, the rules needed for adding, subtracting, multiplying, anddividing the ones and zeros (or binary digits) that zip through a computer's mind and memory.
Number systems and the operations that are performed within them can be classified in much the same way that animals can be divided into families and species. Just as zoologists can place cats in the family of mammals, mathematicians have places for algebraic systems that obey certain rules. One particularly important and useful category is the "group," which shows up in all branches of mathematics and in crystallography, particle physics, and other sciences.
Analysis, the second major piece of the mathematical continent, concerns functions. Functions express relationships. Simply put, a function is any rule that assigns a fixed output to a given input. For example, if the function is squaring, the number 3 is paired with the number 9, the number 7 with the number 49. The development of calculus, discovered independently in the seventeenth century by Isaac Newton and Gottfried Leibniz, is one of the central achievements of analysis.
The third region, geometry, is the study of the properties of shapes and spaces. Most people are familiar with the rigid forms of euclidean geometry--squares and cubes, circles and spheres, congruent triangles and parallel lines. But geometries can take on many different guises, reaching into higher dimensions and obeying various rules. Topology focuses on geometrical features that remain unchanged after twisting, stretching, and other deformations are imposed on a geometric space. Problems such as coloring maps, distinguishing knots, and classifying surfaces, or manifolds, not just in one, two, and three dimensions, but in higher ones as well, all fall within topology.
Lying a short distance offshore from the mathematical mainland are the islands of number theory and set theory. Number theory, at one time considered the purest of pure mathematics, is simply the study of whole numbers, including prime numbers. This abstract field, once a playground for a few mathematicians fascinated by the curious properties of numbers, now has considerable practical value. Identifying primes and finding the prime factors of a composite number play a crucial role in many cryptographic schemes--systems designed to keep secrets secret.
A set is any collection of entities, including numbers, that belong to a well-defined category. A "dog," for example, is a member, or element, of the set of all "four-legged animals." Numbers such as 57, --4. and 6.897 belong to the set of integers, whereas numbers such as 4.67 and p do not. Set theory concerns the study of the structure and size of sets, as defined by various rules or axioms.
Somewhere near the mathematical continent lie the burgeoning landmasses of statistics and computer science. Both have close ties with mathematics, and the links are becoming increasingly important.
Computer science, for example, can be thought of as the study of algorithms: the methods or procedures used to solve given classes of problems. In cooking, a recipe is the algorithm that guides a cook in transforming a motley collection of ingredients into a scrumptious cake. Mathematicians also need recipes. In classical geometry, the ancient Greeks devised a slew of procedures, employing only ruler and compass as tools, for performing a variety of geometric feats, including the bisection of angles and the drawing of regular figures such as hexagons. Later mathematicians spent much of their time looking for algorithms for efficiently computing p, finding logarithms, identifying primes, and performing countless other mathematical tasks.
Today's explosive growth in computer use adds urgency to the investigation of algorithms. Because computers operate on the basis of a small, built-in set of operations, the programs that instruct them, which are essentially algorithms, must be made as efficient and reliable as possible. The mathematical analysis of algorithms has spawned the field called computational complexity.
Although mathematical methods play important roles in statistics and computer science, the focus in these fields is on accomplishing particular goals in an efficient, practical manner. In contrast, pure mathematics delves more deeply into the existence and nature of mathematical objects, even when these objects can't be computed or constructed.
Even what is traditionally called pure mathematics isn't immune to experiment and observation. To explore the properties of prime numbers, those special whole numbers divisible only by themselves and 1, mathematicians centuries ago compiled lengthy tables, using them to look for trends, to guess which properties such numbers have, and to see how primes are distributed among all whole numbers. The use of computers today merely facilitates this kind of list making to identify trends. Such computations, although not always an integral part of the final proof, often suggest what ought to be proved.
Numerical experiments, in which computers are used as untiring accountants and bookkeepers, have already suggested important ideas about the behavior of algebraic expressions and differential equations. One result of such experiments is the discovery of chaotic regions coexisting with islands of stability when certain dynamicalsystems are simulated: Out of this comes the disturbing news that some mathematical procedures--for instance, those used to solve equations--may not be as reliable as people had thought.
Experiments with soap films show the tremendous variability in the shapes of surfaces. Computer-generated pictures of four-dimensional forms reveal unusual geometric features. The crinkly edges of coastlines, the roughness of natural terrain, and the branching patterns of trees point to structures too convoluted to be described as one-, two-, or three-dimensional. Instead, mathematicians express the dimensions of these irregular objects as decimal fractions rather than whole numbers. Experiments and observations provide the heuristic hints vital for progress in mathematics.
But experiments and observations are not the whole story. Mathematics also involves proof. Once proposed, mathematical conjectures, or guesses, go through a trial by fire before they emerge as carefully defined theorems with a permanent place in the structure of mathematics. Mathematical ideas, once conceived, seem to have a life of their own and often wander far from their origins. For instance, the abstract concept of minimal surface, inspired by visions of soap films stretched across wire frames, now includes forms that would, as soap films, be too fragile or convoluted ever to be observed.
Mathematical research in modern times is an extensive enterprise. Because thousands of mathematicians publish hundreds of thousands of pages of new mathematical findings every year, most mathematicians find it difficult to keep up with what's happening across the field. Most of the time, they have to be content with being up-to-date in only a narrow furrow in the field. Consequently, mathematics tends to stay tightly packed into segregated compartments.
Nevertheless, many of the most striking mathematical results of the last decade are notions developed in one field that turn out to be a key element in solving outstanding problems in another, seemingly unrelated field. For example, in 1984, Dutch mathematician Hendrik Lenstra, out of curiosity, decided to study elliptic curves: equations of the form y2 = x3 + ax + b, where values for a and b are chosen arbitrarily. To his surprise, Lenstra serendipitously noticed a connection between elliptic curves and the age-old problem of determining the factors of an integer. His insight led to a new, speedier method of factoring large numbers. In another case of mathematical transfer, mathematician Vaughan F. R. Jones discovered a connection between operator algebras and knot theory, two topics that didn't have an obvious link. The result was an improved method for distinguishing different types of knots. British mathematician Simon Donaldson took the theory of Yang-Mills fields, which plays a role in the study of electromagnetic effects, out of physics and brought it to bear on aproblem in topology. The result was the startling and unexpected discovery of a particularly bizarre geometry in four-dimensional space. All these examples attest to the essential unity of mathematics.
Even a brief glance at modern mathematical research reveals a dynamic enterprise of provocative ideas and questions. Far from being a domain of largely settled questions, mathematics is truly a wilderness. The well-mapped settlements lie few and far between, scattered across the continent and linked by a still skimpy network of highways and trails, some better traveled than others.
It's an exciting adventure to explore some of the newest paths that penetrate the mathematical wilderness--to pursue primes, to untangle twisted spaces, to delve into higher dimensions, to wander in labyrinths, to battle with chaos, and to puzzle over tiling patterns. These images may seem fantastic and metaphorical, but they're not. They are the objects studied by mathematicians--objects that we will see much more of on our tour. Novel landscapes, new vistas, and unexpected pathways lure the mathematical tourist to the frontier, where only questions and conjectures fill the scene.
Copyright © 1988 by Ivars Peterson