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Five More Golden Rules: Knots, Codes, Choas and Other Great Theories of 20th-century Mathematics [Hardcover]

John Casti
4.0 out of 5 stars  See all reviews (1 customer review)

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Amazon Review

Bring more joy to your favourite maths-head with Five More Golden Rules from science writer and national treasure John L. Casti. Though a quick glance through the book will cause an intense fight-or-flight response in the numbers-phobic, Casti's writing is lovely and lucid as ever, explaining not just equations and theorems but their significance in our lives. Having discovered in Five Golden Rules that he couldn't restrict himself to just five important 20th-century mathematical theories, this follow-up explores the intricacies of knot theory, functional analysis, control theory, chaotic systems and information theory. Each of the five lively chapters introduces its subject with a seemingly-unrelated anecdote that is (of course) informed by the theory in question. Then it's headlong into the wonderful details of postulation and demonstration that makes maths so much fun. Unlike a textbook, Five More Golden Rules meanders and breaks away from its proofs to discover relations between the symbols and the real world from the stock market to the coastline of Norway. Besides giving the reader a break, this makes the abstract, almost ethereal concepts concrete and provides a definite advantage to the interested student. Perhaps textbook publishers should take note of this technique; until they do, we'll have to curl up with Casti's Five More Golden Rules if we want to have fun with our higher maths.--Rob Lightner

Review

"I would encourage anyone who would like to discover, or rediscover, the many delights of modern mathematics to read both this book and its predecessor. Jencks′ book is full of enjoyable insights and some impressive eccentricities..." (The Times Higher, September 2000)

"...is full of enjoyable insights and some impressive eccentricities..." (The Times Higher, September 2000)

From the Inside Flap

Five More Golden Rules How did the classic tale of Alexander the Great influence the principal problem facing knot theorists today? What effect did the exalted Kalman filter have on space travel? How did the Shannon Coding Theory make mapping human DNA possible? In this brilliantly conceived follow–up to the highly acclaimed Five Golden Rules, popular science author John Casti treats us to another exhilarating quintet of theoretical gems to answer these and other intriguing questions. Hailed as one of the great science writers of our time, Casti reveals the remarkable origins of some of the most monumental mathematical discoveries of the last century–and sheds light on how these revolutionary ideas have reshaped our lives. Like its critically acclaimed predecessor, Five More Golden Rules takes us on a fascinating journey to the frontiers of modern mathematics, infusing a sometimes intimidating subject with an infectious energy that brings it to thrilling life. Casti draws on such diverse sources as Paul Revere′s plot to warn his fellow patriots of the advancing enemy; the celebrated British play Art; the Greek legend of Gordius and Alexander the Great; the work of nineteenth–century poet William Blake; and a make–believe treasure hunt through New York City′s Central Park to illustrate the power and beauty of the five astonishing theories he illuminates. From explaining how knot theory is contributing to the development of new drugs to revealing how astronomers can predict planetary motions with the Hopf Bifurcation Theorem and highlighting the importance of control theory to space travel, Casti offers an invigorating glimpse into the exciting world of twentieth–century mathematics and how these findings have helped unravel the secrets of the universe. This extraordinary account also introduces us to the legendary figures who made these mathematical contributions possible, including: ∗ James W. Alexander, one of the original Princeton Institute for Advanced Study faculty members who introduced the idea of associating a polynomial with a knot ∗ Claude Shannon, a communications engineer at Bell Telephone Laboratories who pioneered what we now call information theory ∗ Stefan Banach, one of the most prominent mathematicians of his time, whose Scottis Problem Book set the stage for functional analysis With a heady blend of real–life examples, historical anecdotes, and straightforward explanations that will captivate novices and experts alike, Casti probes the mystery and seductiveness of the engrossing tales that lie behind the breakthroughs. Laden with fascinating stories and information, Five More Golden Rules is a spellbinding chronicle that will dramatically transform the way you think.

From the Back Cover

Critical Praise for Five Golden Rules "Five Golden Rules is caviar for the inquiring reader. . Casti′s gift is to be able to let the nonmathematical reader share in his understanding of the beauty of a good theory." –Christian Science Monitor "Casti is One of the great science writers of the 1990s. His new book ranges into exotic fields such as game theory (which played a role in the Cuban Missile Crisis) and topology (which explains how to turn a doughnut into a coffee cup, or vice versa). If you′d like to have fun while giving your brain a first–class workout, then check this book out." –San Francisco Examiner "I urge every reader of New Scientist to get this book." –New Scientist "This book has meat! It is solid fare, food for thought. . Casti′s Five Golden Rules makes math less forbidding and much more interesting." –Hartford Courant "With this volume, Casti has done more than his share of the professional duty of mathematicians. It is one more brick to be used in the construction of a mathematics that is cherished by all people." –Journal of Recreational Mathematics "An excellent exposition of five of the most interesting mathematical theories of the twentieth century that are still undergoing significant study." –Library Journal

About the Author

JOHN L. CASTI, Ph.D., is a professor at the Santa Fe Institute and the Technical University of Vienna and the author of Would–Be Worlds and Five Golden Rules (both published by Wiley) as well as The Cambridge Quintet. --This text refers to an out of print or unavailable edition of this title.

Excerpt. © Reprinted by permission. All rights reserved.

Excerpt from Five More Golden Rules : Knots, Codes, Chaos and Other Great Theories of 20Th-Century Mathematics

By Casti, John L.

Chapter 1: The Alexander Polynomial

Knot History, Mathematics

According to legend, one of the world's first knot theorists was the simple Greek peasant, Gordius, who was named king of Phrygia when he arrived in a public square in an oxen cart around 1200 B.C., thus fulfilling an oracle's pronouncement that the future king would arrive in a wagon. In founding the Anatolian city of Gordium, the peasant gave thanks to the gods by dedicating his oxcart to Zeus, and tying it with an unusual knot. Then, as now, oracles were in ample supply, and another soothsayer came forward to claim that whoever untied the knot would rule all of Asia (at that time, meaning Persia). History records that in 333 B.C., Alexander the Great reached Gordium, was shown Gordius's chariot, and immediately "untied" the intricate knot with one fell swoop of his sword, thus giving rise to the proverbial term "solving a problem by bold action." This ancient story also gives rise to the principal problem facing knot theorists today: How difficult is it to untie a knot?

As with a lot of modern mathematics, the first important study of knots is due to Gauss, who while investigating electromagnetism in the latter part of the nineteenth century, developed methods for calculating properties of knots that he identified with the linking of electric and magnetic fields. A bit later, Lord Kelvin proposed the notion that atoms could be classified according to different types of knots, which led mathematicians such as Tait and Kirkman to produce tables of knot types. These efforts underscore the crucial problem of determining when one knot is the same as another. In what follows, we shall focus mostly on this question, as it remains the most important question in knot theory and is still only partially solved.

Amusing as it is for historical reasons, the Gordian knot is pretty far from the thoughts of theorists today interested in developing methods for distinguishing one knot from another. Knot theorists are motivated by far more important-or at least intellectually more challenging-questions than who is to rule Persia. For example, the strands of DNA residing at the center of every cell of your body are tightly coiled and tangled when packed into genes. But to replicate itself in the process of cell division, a DNA strand must uncoil itself. This unknotting process is carried out by enzymes that serve as microscopic versions of Alexander's sword, slicing through the strands of DNA to unknot them, and then "gluing" the ends back together again. So being able to say how difficult it is to untie a knot can help in determining how fast these enzymes can carry out their activity-as well as help us develop entirely new enzymes that might do things better, faster, or both. Much the same story can be told about questions in polymer chemistry. Even in the rarefied heights of elementary particle physics, there is growing evidence that the constituents of matter itself are nothing more than ultramicroscopic strings of energy coiled up in particular configurations. We'll see these and other examples of ways that knot theory works in practice later in the chapter. But for now let's take a harder look at what mathematicians mean when they talk about a "knotty" problem.

Have Knot, Will Unravel

Intuitively, a knot K is simply a closed curve sitting in ordinary three-dimensional space. Unfortunately, this definition is not quite accept-able because it admits offbeat curves containing an infinite sequence of smaller and smaller knottings, as well as other undesirable pathologies. So it's technically better to define a knot as a simple, closed curve composed of a finite-but possibly very large-number of straight line segments. It can be shown that all the properties of continuous curves we want for studying knots can be recovered from this definition by let-ting the number of segments approach infinity. So when pictures in the chapter show knots as continuous curves, bear in mind the straight-line definition underlying the figures.

A lowly cockroach crawling along any knot will see it as being just a one-dimensional closed curve and will have no sense of the three-dimensional space in which the knot sits (technically, its embedding). So it is the way the curve sits in an embedding space that distinguishes one knot from another. But why must the curve be embedded in three-dimensional space, rather than the plane, or even in four-dimensional space? The answer is that the plane is too small to contain a knot because the places where the curve crosses over itself to form the knot would have to stick up out of the plane. On the other hand, four-dimensional space is too big because any knot can be unraveled there, given the extra degree of freedom. So just as with Goldilock's porridge, ordinary three-dimensional space is not too big and not too small, but just right as the embedding space within which to talk about interesting knots. The simplest possible knot, which is not really a knot at all, but the opposite of a knot-the unknot-is shown in part (a) of Figure 1.1. Part (b) shows the simplest nontrivial knot, the trefoil or overhand knot. And just to show how complicated things can get, part (c) of the figure shows a nonstandard picture of the same trefoil knot. This illustrates nicely why there is a problem in trying to distinguish one knot from another, because it is far from obvious that the two pictures of the trefoil really represent exactly the same knot..... --This text refers to an out of print or unavailable edition of this title.

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