young English mathematic ian I met years ago who did some crazy things. For instance, he wrote and illustrated COMIC BOOKS about fractals and chaos! And since, I guess, he couldn ' t get them publ i shed in English, he pub- lished them in France, IN FRENCH! He even gave me copies of those two won- derful works of his.

That wasn ' t all that he did which some would have considered ill-ad- vised. A "hot" new specialty in mathe- matics appeared with a wonderful name: "Catastrophe Theory." My English friend became an active worker in this new field, and an active public advo- cate of it, even though everyone knew that it was controversial. Many influen- tial senior mathematicians disliked it, considered it a shallow fad, vastly over- publicized because of its exciting name. Many of his older friends must have had doubts whether this was really the wis- est career move he could make.

Of course, he also did fine work in other noncontroversial specialties. But he also did something much more un- wise. He knew, of course, that many re- search mathematicians don ' t have the highest admiration or respect for jour- nalists. And that to many people high up in mathematics, popular books that can be understood by anybody are hardly above journalism. And yet, what did he do but write lots of popular books! Over 20 are listed in the front matter of "Let- ters to a Young Mathematician."

I would say that the career of Ian Stewart is the career of a supremely suc- cessful mathematician whose first con- cern does not seem to have b e e n "play- ing it safe."

So, n o w I am ready to offer my own advice. I have tried to tell you about Stewart's advice. You may know, from experience or from hearsay, about a parent who admits that he isn't in all ways an ideal role model for his child. He may admit to some weaknesses or even vices. But, "Child," says he, "do as I say, not as I do." I am e n o u g h of an optimist to look at things the opposi te way. Read Stewart's book, enjoy it, but do as he Does, not as he Says.

1000 Camino Rancheros Santa Fe New Mexico 87505 USA e-maih rhersh@gmail.com

FOnf Minuten Mathematik by Ehrhard Behrends

WIESBADEN, FRIEDR. VlEWEG & SOHN VERLAG,

2006, 256 PP, 22.90, ISBN-IO: 3834800821,

ISBN-13:978-3834800824

REVIEWED BY VAGN LUNDSGAARD HANSEN

D inner is served: O ne hund r e d well assorted German mathe- matical "tapas" onl ine. Each of

them takes five minutes to consume, and what an en joyable five minutes! The recipes for the mathematical tapas are collected in a book by Ehrhard Behrends based on the first one hun- dred of his co lumns "FQnf Minuten Mathematik," writ ten for the German daily newspaper Die Welt in the years 2003 and 2004. The tapas are easily di- gestible and well prepared, indeed.

Behrends is an exper ienced popu- lariser of mathematics. His co lumns in Die Welt have attracted much attention, and by reading his book you can see why. There is someth ing for everybody with the slightest interest in mathemat- ics in the book. And Behrends is not compromising with his presenta t ion of mathematical results and methods. One gets real and valuable informat ion about mathematics in small doses. This is, I think, a fruitful way of reaching out to the general public. Having read the col- u m n one week, I am sure that m a n y readers have looked forward to seeing what fascinating mathematics the next co lumn would bring the fol lowing week.

It is a pleasure to notice that Behrends covers almost all subjects of mathemat- ics: number theory, algebra, geometry, analysis, probability theory, stochastics, and pure as well as appl ied topics, like coding theory. The book demonstra tes that, with the right care, almost anyth ing from mathematics can be presen ted so that a lay person can get some feeling for it.

Space prevents me from describing the 100 columns in detail; I can only present a few of my o w n personal favourites. In a nice c o l u m n on mathe- matics and music, Behrends explains the Pythagorean and the chromatic tone scales, revealing his insights and strong

interest in music. His mathematical re- search field is probabi l i ty theory, and there are several interest ing co lumns on topics from combinator ics and the the- ory of g a m e s - - a m o n g others, a co lumn on mathematics a nd chess. I also l iked co lumns on the old classics from geom- etry about construct ions with ruler and compass. In a c o l u m n titled 'How much mathematics do h u m a n beings need?' Behrends shows that he is also prepared to take up this k ind of discussion, which I suppose all mathemat ic ians engage in from time to time.

The book is attractively prepared by Vieweg, and several illustrations are in colour. The b o o k wou ld make a good gift for a lay pe r son with an interest in mathematics. The book is also valuable for the mathemat ic ian w ho needs ideas for the occasional, unp repa red mathe- matical conversat ions at d inne r parties. Given the opportuni ty , why not offer your guests some mathematical tapas from Behrends ' b o o k "FOnf Minuten Mathematik"? They will p robab ly be sur- prised but they will enjoy it. And those five minutes spent with mathematics will truly be remembered!

Department of Mathematics Matematiktorvet, Building 303 Technical University of Denmark DK-2800 Kgs. Lyngby Denmark e-maih vlh@dtv.dk

Images of a Complex World The Art and Poetry of Chaos by Robin Chapman and

Julien Clinton Sprott

SINGAPORE, WORLD SCIENTIFIC PUBLISHING CO.

PTE. LTD., 2005, 175 PP. PLUS CD, $34.00

ISBN-13:978-9812564016

REVIEWED BY JOHN PASTOR

T he goal of both mathemat ic ian and poet is to seek clarity and beauty of express ion about the

world a round us through e legant use of their respective languages. While nature

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is a c o m m o n source o f inspira t ion for bo th ma thema t i c i an and poet , the poe t e x a m i n e s the h u m a n r e sponse to na ture whi le the ma themat i c i an exp lo res the logical o r d e r of nature.

Despi te these similarities, creative use of both mathemat ics and poet ry together is u n c o m m o n . Several mathematicians have wri t ten poe t ry - - J . J. Sylvester and James Clerk Maxwell somet imes incor- po ra ted p o e m s into their papers , al- though these are n o w forgotten (perhaps wi th g o o d reason) while their mathe- matics cont inues to inform n e w research. Many of us have en joyed the light verse of Ralph Boas, Jr. [1] and poe t ry has of- ten g raced the pages of The Mathemat- ical Intelligencer. But few poe ts have da red to incorpora te mathematical themes in their explora t ion of the hu- m a n condi t ion, a l though Anne Michaels has cap tured Kepler ' s life and thought superb ly in her long first-person poem, A Lesson f rom the Earth [2], which be- gins "I b e g g e d scraps from the Rudol- ph ine T a b l e - - t h e rinds of orbits, stars scattering like p ips spat from Tycho 's chewing m o u t h . . . " and cont inues wi th " . . . We mus t learn this lesson from the earth, that the greater must make room for the small, just as the earth attracts the smallest s t o n e . . . " and " . . . I u sed to th ink that w e escape t ime by disap- pear ing into beauty. Now I see the op- posite. Beauty reveals time."

In Images of a Complex World, a p o e t and research psychologis t (Chapman) and a physicist and dynamicis t (Sprott), bo th at the Universi ty of Wisconsin, col- laborate on exp lor ing the beautiful wor ld of dynamica l systems and nature through poems, illustrations, and thumbnai l es- says. Al though bi l led as an addi t ion to your coffee table, this b o o k really be- longs in you r c lassroom instead.

By a d d i n g d e p t h and d i m e n s i o n to m a n y dynamica l ideas and concepts , C h a p m a n ' s p o e m s enr ich our and our s tudents ' unde r s t and ing of them. The a c c o m p a n y i n g poem, Fixed Point, is f rom a set o f p o e m s t i t led Stillness (in the s e c o n d column) .

The clarity and dep th of her p o e m is not, of course, the pi thy clarity and dep th of ~ = f (x*). But just as w e teach our s tudents h o w to unpack an equat ion to discover its h idden meaning, so does Chapman u n p a c k the concept of a fixed poin t to uncover its h idden poet ic beauty.

The p o e m s in this b o o k also exp lo r e

the h u m a n condi t ion at the same t ime they illustrate mathematical ideas. H o w m a n y of us have had difficulty ex- p la in ing to s tudents the meaning beh ind the p r o p e r t y of nonlinear systems that f ( x + y) ~ f ( x ) + f(y)? Here ' s Chap- man ' s i l lustration of nonl inear systems, po ignan t not only for captur ing the e s sence of this proper ty but also for de- scr ib ing an all-too c o m m o n condi t ion of c o n t e m p o r a r y human existence:

Of course , the w h o l e thrust of mathe- matical teaching for the pas t 150 years or so has b e e n to d i s e m b o d y mathe- matics, to get away from this "human d imension ," but w h e n used with dis- cret ion and taste as in the hands of ski l led pract i t ioners and teachers such as Chapman and Sprott, I s ee no harm and even much g o o d in exp lo r ing the poet ic d imension.

Jul ien Sprott, an accompl i shed au- thor [3] and scientific educa to r (see his webs i t e at h t tp : / / sp ro t t . phys ic s .wi sc . edu/ ) , p rov ides clear expos i t ions of the mathemat ica l concepts that are the sp r ingboards of the poems, serving as p rose counterpoin ts to Chapman ' s verse, including entropy, state space, basins of attraction, and the th ree -body p rob lem, a m o n g many others. Here is his exp lana t ion of hysteresis, in a sec- t ion cal led Time's Arroz~. "Hysteresis is a form of m e m o r y . . . . Hysteres is re- quires a nonl inear i ty in the govern ing equat ions , and the nonl inear i ty is often in the form of a th reshold w h e r e the behav io r switches abrupt ly from one form to ano the r as a pa r ame te r is changed (a bifurcation)." This neat ly ba lances the first two lines of the next p o e m , ca l led Hysteresis: "Whatever w e expec ted w o u l d h a p p e n next / it wasn ' t this . . . " O n e could s p e n d a mathe- matical or poe t ic lifetime exp lo r ing the d e e p mean ing beh ind these descr ip- tions.

Sprott 's o ther contr ibut ion to the b o o k are the s tunning il lustrations of at- tractors (s t range and otherwise) , iter- a ted funct ion systems, and Julia sets. At this point , the reader might w o n d e r w h y she or he needs to see yet m o r e ex- amples of these, but the cleanliness, clarity, and visual beauty of these illus- t rat ions are truly marvelous. All of the at tractors are in color, with the co lor de- pic t ing the va lue in the z-dimension, and most are on a white background , wh ich gives them a cr ispness lacking in many o ther books . Because m a n y non- mathemat ic ians f ind the unst ructured a p p e a r a n c e of s trange attractors difficult to apprecia te , Sprott in t roduces sym- metry into some of these i l lustrations by an interest ing t ransformat ion of the x and y coordina tes to polar coordinates , a t echn ique which he has p rev ious ly pub l i shed in [4]. This gives the at tractor image the shape of a wedge , and sev- eral w e d g e s are then a s sembled into a

THE MATHEMATICAL INTELUGENCER

radially symmetric image, like the arms of snowflakes or bursts of fireworks. I suspect that under ly ing these polar transformations there may be some in- teresting ideas in the mathematics of symmetry which may make a nice stu- dent project or undergradua te thesis. There is an Append ix for the Mathe- matically Incl ined that discusses these and other mathematics beh ind the im- ages.

This collection of Chapman ' s poems with Sprott's visual art and mathemati- cal exposit ions is a we lcome compila- tion and nicely shows the breadth of their work, both separately and in col- laboration. The book comes with a CD with these and m a n y other images,

readings of some of the poems by Chapman herself, weblinks, and a vari- ety of other classroom resources. Im- ages of a Complex World should set a standard for collaborations be tween mathematicians or scientists and artists: indeed, the l ine separating which au- thor is a scientist from which is an artist is often blurred in this book. Buy it or have your library purchase it, share it with your students, and dip into it w h e n your creative p u m p needs priming.

REFERENCES

[1] Gerald Alexanderson and Dale Mugler, ed-

itors. Lion Hunting & Other Mathematical Pursuits. A Collection of Mathematics, Verse, and Stories by Ralph P. Boas, Jr.

The Mathematical Association of America, Dolciani Mathematical Expositions, Volume 15, Washington, DC, 1995.

[2] Anne Michaels. Poems. Alfred A. Knopf, New York, 2001.

[3] Julien Clinton Sprott. Chaos and Time-Se- ries Analysis. Oxford University Press, Ox- ford, 2003.

[4] Julien Clinton Sprott. Strange attractor symmetric icons. Computers & Graphics 20: 325-332, 1996.

Department of Biology University of Minnesota Duluth Duluth, MN 55812 USA e-mail: jpastor@d.umn.edu

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