Glimpses of Benoît B. Mandelbrot (1924–2010) · PDF fileGlimpses of Benoît...

Glimpses ofBenoît B. Mandelbrot(1924–2010)Edited by Michael F. Barnsley and Michael Frame

Michael F. Barnsley

IntroductionBenoît B. Mandelbrot died in Cambridge, Mas-sachusetts, on Thursday, 14 October 2010. He waseighty-five years old and Sterling Professor Emeri-tus of Mathematical Sciences at Yale University. Hewas also IBM Fellow Emeritus (physics) at the IBMT. J. Watson Research Center. He was a great andrare mathematician and scientist. He changed theway that many of us see, describe and model, math-ematically and geometrically, the world around us.He moved between disciplines and university de-partments, from geology to physics, to computerscience, to economics and engineering, talkingexcitedly, sometimes obscurely, strangely vain,about all manner of things, theorizing, speculat-ing, and often in recent years, to the annoyanceof others, pointing out how he had earlier donework of a related nature to whatever it was thatsomeone was explaining, bobbing up and down tointerrupt, to explain this or that. He was an un-forgettable, extraordinary person of great warmthwho was also vulnerable and defensive.

Michael F. Barnsley is a professor at the Mathematical Sci-ences Institute, Australian National University. His emailaddress is [email protected] Frame is professor adjunct of mathematics at YaleUniversity. His email address is [email protected].

DOI: http://dx.doi.org/10.1090/noti878

Two roads diverged in a wood, and I— I took the one less travelled by,And that has made all the di!erence.(Robert Frost, “The Road Not Taken”)

Figure 1. Benoît Mandelbrot next to JohnRobinson’s sculpture Intuition outside the Isaac

Newton Institute, Cambridge, during theMathematics and Applications of Fractals

Program in 1999. (Photo: Findlay Kember/IsaacNewton Institute.)

Looking back, Benoît saw his life as a rough path.In [7] he recounted how his father escaped fromPoland and the Nazis with a group of others and, ata certain point, went a di!erent route through thewoods, which saved his life. Benoît saw his own lifein similar terms: he too took the path less travelledby, and that made him very di!erent from mostmathematicians. What he did that was di!erentwas to work in many areas, following where hisgeometrical intuitions led, regardless of academicboundaries. This path repeatedly risked failureand embarrassment because each discipline has

1056 Notices of the AMS Volume 59, Number 8

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Figure 2. Benoît, agefour, in Poland.

its entrenched bigguns, experts sur-rounded by well-constructed defenses,peer groups armed withstacks of citations.

Prior to writing boththis article and [1], weemailed colleagues toask for memories andcomments on Benoît’scontributions to math-ematics, his influence,and personal recol-lections. We receivedreplies from many—notonly mathematicians,

but artists, physicists, biologists, engineers, andso on. Our goal has been to put together a pairof memorial articles, something special, using thewords of everyone who wrote, but, in general, edit-ing and shortening to avoid repetitions of themes.The second article, [1], is centered on Benoît’sinfluence and contributions to mathematics. Thepresent article is more directed to the man himselfin a personal manner. Both Michael Frame and Iknew and loved Benoît: Michael Frame was hissidekick at Yale for many years, and I, MichaelB., first met him in 1981 and a number of timesduring the following fifteen years, mainly duringthe early 1980s. In 1988 he came to my home inAtlanta for dinner during the Siggraph conference,together with Richard Voss, Heinz-Otto Peitgen,and others. His magical personality filled the din-ing room that hot summer evening, contrastedwith his house in Scarsdale, where I first met himand Aliette on a February day in 1981, with snowand light gleaming o! it into the windows of hisbook-and-Xerox-piled o"ce.

There is a large body of written materials,available online, that are easily accessed andwhich recount aspects of Benoît’s life, times,research, quotations, and opinions. But here wetry to capture afresh the fact that he was one ofus, a mathematician, and to give a glimpse andfeeling, for the time that you read this, of the realand amazing man that he was.

Ian Stewart

He Began His Lecture by Shu!ing His SlidesMy first contact with Mandelbrot was when hephoned me to say that he’d been asked to write apopular article on fractals, didn’t have time, andwondered whether I’d be interested. I accepted the

Ian Stewart is emeritus professor of mathematics at theUniversity of Warwick. His email address is [email protected].

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Figure 3. Benoît, agethirteen or fourteen,in Paris.

invitation. From then on, hewould occasionally call me whensome unusually striking devel-opment in fractal geometry hadhappened. When The Economistasked me to write a feature ar-ticle on applications of fractals,Benoît mailed me a stack of pho-tocopies six inches thick, savingme weeks of work.

I met him a few times. Hevisited the University of Warwickand began his lecture by shu#inghis slides to make sure it wasdi!erent from previous talks. Mywife and I had breakfast withhim at a conference on financialmathematics in Santa Fe. He was in great spiritsbecause a conjecture of his had just been proved,but he knew that my wife was not a mathematician,so he took care to avoid talking shop.

David Mumford

He Opened a Door and Let in a Gale of WindI met Benoît when he came to Harvard as a visitingprofessor in 1979. At that time, the Harvard mathdepartment was an insulated place, a temple ofpure math. His appearance opened a door andlet in a gale of wind. He was a large man andhis presence was large too. He gave lectures in adozen departments, and every lecture dealt witha di!erent phenomenon of nature. He seemed tohave studied everything and picked up grist forhis mill in every corner of the world.

I had some wonderful times socially with Benoîtand his wife, Aliette. They were warm and fasci-nating hosts who seemed to know everyone too. Iremember especially talking about Gadjusek, thediscoverer of the link between cannibalism andprion diseases, who was a good friend of theirs. Ilast saw him at the birthday celebratory meetingin his honor at Bad Neuenahr. Surrounded byhis hosts who had contributed so much to histheories, he gave a moving speech on the factthat this was his first visit to Germany since theHolocaust. Benoît was a completely unique personand scientist who cannot be pigeon-holed and hisinfluence has been vast. I count myself very luckyto have known and worked with him.

David Mumford is professor emeritus of mathe-matics at Brown University. His email address [email protected].

September 2012 Notices of the AMS 1057

Kenneth Falconer

Benoît Told Me…Everyone Would Be Merryafter Food and WineThe first time that I met Benoît was at the WinterWorkshop on Fractals at Les Houches in 1984.This was the first time that I had encounteredthe “fractal community”. It was an eye-openingmeeting, as I realized the wealth of ideas thatwas emerging from mathematicians and scientistsinterested in fractals. I have a vivid memory ofthe friendliness and encouragement shown tome by Benoît at that meeting. The highlight ofthe week’s cuisine was the fondue, and I wasscheduled to give the evening talk immediatelyafterwards. Benoît took me aside and told me thatthere was no need to be nervous, as everyonewould be merry after the food and wine, somy talk was bound to be appreciated! In fact, Ithink that the talk did go well. This was the firstoccasion on which I presented my “digital sundial”theorem—that there exists a fractal such that itsorthogonal projections can be essentially anythingone wishes, for example, the thickened digits ofthe time. Along the same lines I also proposedthe construction of a space station that wasplainly visible to Western countries and e!ectivelyinvisible to Eastern countries, to the amusementof many present. Benoît told me afterwards thathe liked these examples because they gave a“visual” interpretation of an abstract mathematicaltheorem.

The year 1984 also saw the publication of TheGeometry of Fractal Sets [4], which was one of

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Figure 4. Benoît, atage twenty-eight

or twenty-nine,was living in

France.

the earliest books on fractals, apartfrom those of Benoît himself. Benoîtprovided very helpful commentson my manuscript, but in a re-view he did refer to “…the usualdry mathematics”, though I don’tthink he meant it unkindly. In lateryears Benoît coauthored a numberof papers employing similar formalmathematics, so I think that once hisideas had been accepted by the math-ematical community, he became lessconcerned about the “dryness”.

I saw a great deal of Benoît dur-ing the program Mathematics andApplications of Fractals at the IsaacNewton Institute in Cambridge in

1999; indeed he, along with Aliette, occupied ano"ce adjacent to mine. The four-month programwas organized by Robin Ball and me, and we were

Kenneth Falconer is professor of pure mathematics at theUniversity of St. Andrews in Scotland. His email address [email protected].

delighted that Benoît stayed in Cambridge forthe entire time. A number of young researchersand research students took part, and Benoît madea point of taking time to encourage them bytalking to them all and discussing ideas with themindividually.

I was delighted when Benoît accepted an hon-orary degree from the University of St. Andrewsin 1999. It was a pleasure to entertain him andAliette in my hometown, and it was clear thatreceiving such an honor from a Scottish univer-sity meant a great deal to him. I recall that, aswe crossed the Firth of Forth on the way fromthe airport, he commented that the Forth RailwayBridge, constructed in 1890, displayed fine fractalfeatures in its hierarchical structure!

For many years I met Benoît regularly at con-ferences; he was rarely absent from any meetingon fractals. He once paid me the biggest compli-ment that my lecturing has ever received: “I reallyliked your talk, Ken; you have such a wonderfultheatrical style!”

Ron Eglash

“That Is Not Criticism; That Is a Tribute toYour Work”

I am best known for my book African Fractals[3]. Needless to say, that would have been im-possible without Benoît! Much of the research inethnomathematics had been things like “how tocount to 10 in Yoruba” or “African houses areshaped like a cylinder.” But when I first saw aerialphotos of African villages, their fractal structure

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Figure 5. Benoît atage thirty-nine or

forty when he wasworking at IBM.

was immediately obvious.That gave me a basis fora Fulbright fellowship toAfrica, and once I wasthere I found that that re-cursive scaling cropped upin all sorts of artifacts andknowledge systems, fromsculpture and textiles todivination and cosmology.The NSF has allowed us todevelop software for teach-ing math and computingusing fractal algorithms[13]. This work has alsocaught the eye of archi-tects; for example, there are now plans for anentire university in Angola to have a fractallayout. Benoît leaves behind a legacy on manycontinents.

Ron Eglash is associate professor of mathematics atRensselaer Polytechnic Institute. His email address [email protected].


Figure 6. Benoît Mandelbrot, Kenneth Falconer,and Keith Ball outside the Isaac NewtonInstitute, Cambridge, in 1999. (Photo: FindlayKember/Isaac Newton Institute.)

The first time I spoke to Benoît was when hevisited UCSC, where I was in graduate school inthe late 1980s. After his lecture, I asked himwhy some fractals show Euclidean shapes—theSierpinski gasket, for example—and others showonly mush or globs that show no recognizableshapes. To my surprise he said, “I have beenasking myself that question for over a decadeand have yet to find a satisfactory answer.” Thesecond time was a phone call; he wanted to knowif I had been given tenure—he had written arecommendation for my case. We got to chattingabout African Fractals, and he asked me if I wasgetting any criticism for it. So I described someof the hate mail I was receiving from critics whoinsisted that black people had genetically inferiorbrains and could not have created fractals ontheir own. He said, “That is not criticism; that is atribute to your work!”

Harlan Brothers

Ways in Which Music Could Manifest FractalStructureBenoît’s most important contribution to educationwas the work he did in conjunction with MichaelFrame and Nial Neger in conducting the FractalGeometry Workshops at Yale. Related collabo-rations included the book Fractals, Graphics, &Mathematics Education [5], the DVD Mandelbrot’sWorld of Fractals [8], and the vast Yale website onfractal geometry [12], which contains the collec-tion of labs called Kitchen Science Fractals. Thanksto Benoît’s vision, countless young minds aroundthe globe have come to appreciate mathematicsthrough their exposure to fractal geometry.

Harlan Brothers is an inventor, mathematician, and musi-cian based in Branford, Connecticut. His email address [email protected].

Figure 7. This photo, taken by Richard Bernt,shows, from left to right, Richard Voss, Benoît B.Mandelbrot, Dietmar Saupe, and Heinz-OttoPeitgen. Richard Bernt believes it was taken inMay 1987. It was the week Dietmar and Vosswent sailing on his catamaran. Mandelbrot andhe had talked about the fractal dimension ofocean waves and how it was similar to music.The photo was taken at UC Santa Cruz by theApplied Science Building, which had picnic tablesnearby. It was near the o!ces of Heinz-Otto,Dietmar, and Ralph Abraham, near room 366,where they, with Richard Bernt, generatedfantastic fractal images on SGI machines.

It was Benoît who set me on the path ofestablishing some mathematical rigor for the term“fractal music”. Prior to the summer of 2003,someone had given him and Michael Frame aCD of what purported to be fractal music. Theypassed it on to me. When I explained that thecomposer did not seem to have a solid grasp ofthe fundamentals, Benoît agreed, saying, “Yes, Ithink you are right. If you would like to look intothis subject, that would be wonderful.”

In collaboration with Michael Frame, the fol-lowing summer we did a presentation and lab onfractal music. I have since continued to publishand present on the subject and last year appearedin the BBC documentary Bach & Friends [2] dis-cussing fractal geometry and its relationship tothe music of Bach. I regularly receive email fromstudents around the world, high school throughgrad school, who are working on projects or havequestions about fractal music.

I had been recommended to Benoît by a formerstudent of his, Miguel Garcia, who was my profes-sor at Gateway Community College. I will alwaysremember our first meeting at Yale. Benoît wasseated, his hands pointed in and resting on hislegs. He began by saying, in his inimitable accent,“So, Miguel tells me you are not the average cookie-cutter student…” I shared some of my research,


Figure 8. Benoît Mandelbrot in an iconic pose,fingers together, appearing in the documentaryClouds Are Not Spheres [9]. The shot was taken

in Paris in 1999. (Photo courtesy of NigelLesmoir-Gordon.)

and by the time I left, he had shared everythingfrom the lesser-known work of John Venn to thesociopolitical history of Budapest dating back tothe sixteenth century. Since then, over the years,through the Fractal Geometry Workshops and innumerous phone calls, Benoît continued to sharehis overwhelming expertise, his humor, and hiswisdom in practical matters. His generosity ofspirit and fundamental good nature have inspiredme and helped to define who I am.

Nigel Lesmoir-Gordon

Benoît Was Superb, Inspiring and Lucid

I made the film The Colors of Infinity [6] in 1992.After we had finished filming Arthur C. Clarke inSri Lanka, I interviewed Benoît at his home. Hewas, in essence, happy with the questions thatI proposed to ask. I commented uneasily thathe looked a bit formal in his suit and tie andsuggested that he should dress more casually. Helaughed and said that he had a blue jacket hecould wear and that he would drop the tie. Whenhe came back into the room, instead of the brightblue jacket I was expecting, he was wearing a coatof subtle grey. He took his seat, and I sat downby the camera, clutching my notes and my list ofquestions. He looked good. Lights, camera, action!Then everything popped. We had blown a fuse.Our electrician rerouted the lights, and we startedagain. After ten minutes the lights blew again. I

Nigel Lesmoir-Gordon is a filmmaker; his works includeThe Colors of Infinity, a documentary about the Mandel-brot set. His email address is [email protected].

started to get very agitated, but Benoît remainedas cool as a cucumber.

We had another co!ee break while the electri-cian did some serious rewiring. Everyone stayedcalm except for me. It was essential that I madea good job of the interview for the sake of ourinvestors, the crew, and most of all for Benoît. Wemanaged to get started again and, save for a half-hour lunch break, worked on into the afternoon.Benoît was superb—inspiring and lucid.

It took many months to complete the postpro-duction; then we sent tapes o! to the contributors.Benoît was generous with his praise and his ex-pressions of gratitude. We found a distributor forthe film. It went on to sell in over forty territoriesworldwide and has been subtitled in three for-eign languages. It was shown on eighty-two PBSstations in the United States.

When Benoît was recalling his research work atIBM, he told me:

For me the first step with any di"cult math-ematical problem was to program it and seewhat it looked like. We started programmingJulia sets of all kinds. It was extraordinarygreat fun! And in particular, at one point, webecame interested in the simplest possibletransformation: z ! z

2+c. ... And after a few

weeks we had this very strong, overwhelm-ing impression that this was a kind of bigbear we have encountered!

This discovery was named after me. It iscalled the Mandelbrot set. I think the mostimportant implication is that from very sim-ple formulas you can get very complicatedresults.…

Benoît, Michael Frame, and I went on to makethe educational DVD production [8], which wascommissioned by the National Science Foundationthrough Yale University. This DVD concludes withBenoît addressing the camera:

I’ve spent most of my life unpacking theideas that became fractal geometry. This hasbeen exciting and enjoyable, most times. Butit also has been lonely. For years few sharedmy views. Yet the ghost of the idea of fractalscontinued to beguile me, so I kept lookingthrough the long, dry years. So find the thingyou love. It doesn’t so much matter what itis. Find the thing you love and throw yourselfinto it. I found a new geometry; you’ll findsomething else. Whatever you find will beyours.


Javier Barrallo

Not Only Should the Toy Be Built, But WeShould Know How to Play with ItAs Johannes Kepler used his toy, the ellipse, toexplain our solar system, so did Benoît Mandelbrotuse his toy, the fractal, to interpret the geometryof nature. Once Benoît explained to me: “Not onlyshould the toy be built, but it should also beknown how to play with it.”

My first contact with Benoît was the invitationI sent to him to chair the international fractal artcontest that bears his name. When attending thefirst exhibition contest at Conde Duque in Madrid,he was surprised to see a long line of people. “Iam an inveterate optimist, but I never expectedto see a crowd standing in a long line to admiremathematics in any of its forms,” he said thatnight. I remember having to wait for him for overforty minutes while he signed autographs andtook pictures with fans. He was more like a rockstar or a Hollywood actor.

I remember while walking one beautiful au-tumn morning in San Sebastian, Benoît noticed asculpture in the rocks of La Concha Bay. It wasThe Comb of the Wind by Eduardo Chillida. He im-mediately recognized the artist, then proceededto tell me that born just a few miles away wasIgnatius of Loyola, founder of the Jesuits; next heinformed me that in the nearby port of Guetaria,the explorer who completed the first circumnav-igation of the world, Juan Sebastian Elcano, wasborn. For a nonnative, he had remarkable culturalknowledge. He could talk about Hokusai style andimmediately illustrate the Japanese character byrelaying an anecdote that took place while he wasdining with the empress of Japan. He told meonce that Eugène Delacroix used to instruct hisstudents that to paint a tree it was necessaryto draw inside another smaller tree, and insideanother, and another.

Benoît chaired three of the International FractalArt Contests. In each case, twenty-five imageswere selected for exhibition. The results of thethird contest, by artists of seventeen di!erentnationalities, were exhibited in Bilbao (Spain),Buenos Aires (Argentina), and Hyderabad (India).Benoît guided our e!orts to discover new waysto express fractal art. Thus, the typical filamentsand spirals were reduced to an aesthetic closerto contemporary art rather than the usual fractalstructures. Looking at the last exhibition contest,he said, “Many will prefer the old images, butcompared with these, they look like antiques.”

Javier Barrallo is professor of mathematics at The Univer-sity of the Basque Country, San Sebastian, Spain. His emailaddress is [email protected].

Figure 9. Picture by Kerry Mitchell, a panelmember image for the 2007 Benoît MandelbrotArt Contest. Javier Barrallo noted that Benoîtreally enjoyed this picture.

The Benoît Mandelbrot International Fractal ArtContest also gave him the opportunity to partici-pate for the first time in the International Congressof Mathematicians (ICM). He entered through theback door as honorary director of the fractal artcontest. But when his presence became known,he raised unexpected excitement—well above anyother guest speaker. Thus, Benoît Mandelbrot wasinvited to give the closing lecture of ICM2006, withseveral thousand people attending in the main au-ditorium. In his speech he congratulated WendelinWerner for being a recipient of the Fields Medalas well as for being able to demonstrate one ofhis conjectures. In fact, he said, “This is the thirdtime a Fields Medal was awarded for proving oneof my conjectures.”

For some people this may portray a smugman, but this is not true. I remember the nighthe turned down an invitation to a prestigiousdinner with some of the best mathematicians inthe world to join my group of young colleagueswho had planned a beer and tapas tasting in thebustling Plaza of Santa Ana. “Could we join you?”he asked. That night we drank and laughed butmostly listened to Benoît tell fantastic stories andanecdotes from his life, science, history, and art…it was an unforgettable moment that revealeda much more approachable and intimate personthan one might think.

My last conversations with Benoît dealt withthe Mandelbrot set in 3D, also called Mandel-bulb. Although he truly admired the gorgeousanimations of the Mandelbulb and other graphicsexperiments, he never entered the debate on them.His era was ending and a new one was beginning.


Figure 10. Close-up and zoom on the attractor of an iterated function system comprising fourprojective transformations. Mandelbrot’s work built on classical geometry and leads to simple

mathematical models for natural objects, such as forests and leaves.

Benoît was not a conventional mathematician,but he was certainly the most brilliant mind I’vehad the chance to meet.

Sir Michael Berry

It Is Winter and the Trees Are Bare of LeavesWhen Benoît visited the UK, he and Aliette oc-casionally stayed with us. My abiding memory isof his nonnegligible bulk dominating our kitchenamid a whirl of culinary activity. Fortified by a con-tinuous supply of orange juice, he entertained andentranced us with his monologues about mathe-matics, his wartime experiences, his opinions ofpublishers and colleagues.…As I write, it is winterand the trees are bare of leaves. Frost on thebranches dramatically enhances their fractility,and I remember Benoît, who taught us to see it.

Michael Frame

Epilogue: November 12, 2010Benoît and I worked together for twenty years. Wewrote papers, edited a book on fractals and edu-cation, ran summer workshops for teachers, andspent hours upon hours discussing . . . everything.These conversations were exhilarating, among thevery best moments of my life. Benoît collaborated

Sir Michael Berry is professor of physics at the Univer-sity of Bristol, UK. His email address is [email protected].

with many scientists, all much brighter than I, butour relationship was di!erent. Deep inside, I re-main an eleven-year-old kid, inhabiting a simplerworld filled with mysteries, where the job of everykid is to explore. Benoît was more complicated,but with me he followed his sense of innocentwonder at the wide world. During our first freelyroaming conversation, I had an image that hasstayed with me through the years, that gives mesome small comfort at his loss. Benoît and I weretwo little kids running around in a big field undera bright sky, showing each other what we found.Friends sharing the unalloyed joys of discovery.

Benoît was fascinated by complex things. Hislife’s work revolved around finding a featurecommon to examples from mathematics, physics,economics, art, and music: patterns that keptrecurring as he looked ever closer. Others hadnoticed some aspect of this before, but Benoît sawso much more: that complicated shapes can beunderstood dynamically as processes, not objects.Continuing to astound each new generation ofstudents, the power of this view is remarkable.

I’ll end with two more points: some of ourfinal conversations, and what I really learned fromBenoît.

When Benoît called to share the news of hisdiagnosis, at first he asked me to tell no one. All hewanted to discuss was how to try to finish the workthat remained undone: his memoirs and projectson negative dimensions and lacunarity were muchon his mind. Working from Benoît’s notes, Aliette


Figure 11. “I, Javier Barrallo, took this picture inDonostia-San Sebastián, a city in the BasqueCountry in northern Spain. The date wasNovember 15, 2007. Benoît and Aliette are inOndarreta Beach in front of a famous sculptureby Eduardo Chillida named ‘The Combs of theWind’. I remember it was a very happy day forthem.” (Photo courtesy of Javier Barrallo.)

Mandelbrot and Merry Morse finished the memoirs[11]. Through their considerable e!orts, Benoît’sstory will be told.

In addition to unfinished projects, we contin-ued to discuss some general scientific questions.Despite ample reason to think only of himself,curiosity—one of our very finest traits, the onlything that might save the species, the only thingthat could make us worth saving—burned in Benoîtwith the brilliance it did in his youth eight decadesearlier. These feelings would persist until the end.

Benoît and Aliette were very kind to Jean andme, but I cannot understand why he broughtme into his world. Hundreds and hundreds ofconversations, just he and I. Why? This made nosense. Surely he had better things to do with histime. But these talks have given me a detailedpicture of Benoît.

What do I know for sure about Benoît? Inhis mind, shapes were fluid, bending, twisting,and turning without e!ort. He read everything,remembered everything, but dynamically, lookingfor connections in combinations both expectedand unlikely. Familiarity with so many topicsallowed Benoît to converse with anyone. With myfather, a machinist, Benoît had a long discussionabout annealing. Benoît loved music, especiallyopera, knew Charles Wuorinen’s work long beforeCharles contacted Benoît to talk about fractalaspects of music. During the Yale memorial forBenoît, Ralph Gomory characterized Benoît ascourageous, refining and extending his ideas aboutscaling across many disciplines, following thepaths and practices of no field, ignored for years.Early in his life, Benoît wanted his own Keplerianrevolution. This he achieved, but at a cost. Manyyears later Benoît lamented not having a large

group of assistants; so much more would havebeen finished if the path he’d taken had not beenso lonely. Still, that path got him to where he was,gave fractals to us all.

Years ago, when asked if he was a mathemati-cian, a physicist, or an economist, Benoît repliedthat he was a storyteller. After Benoît died, I sawanother interpretation of his answer. By empha-sizing how an object grows, a fractal descriptionof the object is a story. Twists and turns of asnowflake in a cloud, rough waves sculpting ajagged coastline, my lungs growing before I wasborn, the spread of galaxies throughout the deepdark of space. These share something? Benoît toldus they have similar stories. Benoît told us scienceshould tell more stories.

Did Benoît’s stories change how we understandthe world? Yes, indeed.

References[1] Michael F. Barnsley, Michael Frame (eds.), The

Influence of Benoît B. Mandelbrot on Mathematics,Notices, Amer. Math. Soc., to appear.

[2] Bach and Friends, Michael Lawrence Films,http://www.mlfilms.com/productions/bach_project.

[3] R. Eglash, African Fractals, Rutgers University Press,1999.

[4] Kenneth Falconer, The Geometry of Fractal Sets,Cambridge University Press, 1985.

[5] Michael Frame, B. Mandelbrot (eds.), Fractals,Graphics, and Mathematics Education, MAA, Washing-ton, DC, 2002.

[6] N. Lesmoir-Gordon, Fractals: The Colors of Infinity,Film for the Humanities and Sciences, 1994.

[7] Richard L. Hudson, in the Prelude to [10].[8] N. Lesmoir-Gordon, M. Frame, B. Mandelbrot,

N. Neger, Mandelbrot’s World of Fractals, KeyCurriculum Press, 2005.

[9] N. Lesmoir-Gordon, Clouds Are Not Spheres, Jour-neyman Films, UK, 2009.

[10] B. Mandelbrot, with R. Hudson, The (Mis)Behavior ofMarkets, Basic Books, New York, 2003.

[11] , The Fractalist: Memoir of a Scientific Maverick,Pantheon, 2012.

[12] http://classes.yale.edu/fractals/.[13] http://www.csdt.rpi.edu.


The Influence ofBenoît B. Mandelbroton MathematicsEdited by Michael F. Barnsley and Michael Frame

Michael F. Barnsley

Introduction

We begin this article, which deals largely withBenoît B. Mandelbrot’s contributions to and influ-ence upon mathematics, with a quotation fromthe introduction to Fractals: Form, Chance, andDimension [16]. This essay, together with manypictures and numerous lectures in the same vein,changed the way science looks at nature and hada significant impact on mathematics. It is easy forus now to think that what he says is obvious; itwas not.

Many important spatial patterns of Natureare either irregular or fragmented to such anextreme degree that Euclid—a term used inthis essay to denote all classical geometry—is hardly of any help in describing their form.The coastline of a typical oceanic island, totake an example, is neither straight, norcircular, nor elliptic, and no other classicalcurve can serve, without undue artificial-ity in the presentation and organization ofempirical measurements and in the searchfor explanations. Similarly, no surface in Eu-clid represents adequately the boundaries ofclouds or rough turbulent wakes.…

Michael F. Barnsley is a professor at the Mathematical Sci-ences Institute, Australian National University. His emailaddress is [email protected] Frame is adjunct professor of mathematics at YaleUniversity. His email address is [email protected].

DOI: http://dx.doi.org/10.1090/noti894

In the present Essay I hope to show thatit is possible in many cases to remedy thisabsence of geometric representation by us-ing a family of shapes I propose to callfractals—or fractal sets. The most usefulamong them involve chance, and their irreg-ularities are statistical in nature. A centralrole is played in this study by the conceptof fractal (or Hausdor!-Besicovitch) dimen-sion.…Some fractal sets are curves, othersare surfaces, still others are clouds of discon-nected points, and yet others are so oddlyshaped that there are no good terms forthem in either the sciences or the arts. Thevariety of these forms should be sampled bybrowsing through the illustrations.…

—Benoît B. Mandelbrot [16, pp. 1–2]

As with the now familiar principle that grav-itational force tethers the earth to the sun, ithas become hard to imagine what it was likenot to know that many physical phenomenacan be described using nondi!erentiable, roughmathematical objects.

Important fractals such as the Cantor set, theSierpinski triangle, and Julia sets were well knownto some mathematicians, but they were neithervisible nor promoted to any practical purpose. Tome, looking back, it seems that these beautifulthings were hidden behind veils of words andsymbols with few diagrams, certainly no detailedpictures; for example, the long text (in French)of Gaston Julia failed to reveal to most people,including most mathematicians, the full wonder of


the endless arabesques and intricate visual adven-tures in the boundaries of Fatou domains. It was asthough such objects were guarded by the priestsof mathematics, occasionally to be displayed, likethe monstrance at Benediction, to the inner core oftrue believers. I was ritually inducted to calculusin my first year at Oxford by Hammersley, whotook us through a full proof of the existence ofa Weierstrass nowhere di!erentiable continuouscurve from first principles. Half an hour with pic-tures would have saved a lot of time and wouldnot have tainted our logical skills.

Benoît not only wrested these abstract objects,these contrary children of pure mathematics, outfrom the texts where they lay hidden, but healso named them and put them to work to helpto describe the physical observable world. Hesaw a close kinship between the needs of puremathematics and the Greek mythological beingAntaeus. In an interview [6] Benoît said, “Theson of Earth, he had to touch the ground everyso often in order to reestablish contact withhis Mother, otherwise his strength waned. Tostrangle him, Hercules simply held him o! theground. Separation from any down-to-earth inputcould safely be complete for long periods—butnot forever.” He also said, “My e!orts over theyears had been successful to the extent, to take anexample, that fractals made many mathematicianslearn a lot about physics, biology, and economics.Unfortunately, most were beginning to feel theyhad learned enough to last for the rest of theirlives. They remained mathematicians, had beenchanged by considering the new problems I raised,but largely went their own way.”

John Hutchinson is an example of a pure math-ematician who was strongly influenced by Benoît’swork.

In 1979 I was on study leave from theAustralian National University, visiting FredAlmgren at Princeton for 6 months, as a re-sult of my then interest in geometric measuretheory. While there, Fred suggested I readMandelbrot’s book Fractals: Form, Chanceand Dimension and look at putting it, or someof it, into a unified mathematical framework.As a result, we organised a seminar in whichI spoke about six times as my ideas de-veloped. Participants included, besides Fredand myself, Bob Kohn, Vladimir Schae!er,Bruce Solomon, Jean Taylor and Brian White.Out of this came my 1981 article “Fractalsand self-similarity” [7] in the Indiana Uni-versity Math. Journal, which introduced theidea of an iterated function system (thoughnot with that name) for generating fractalsets, similar ideas for fractal measures, and

Figure 1. An outlier Mandelbrot set (M-set)(surrounded by yellow, then red) connected via abranch of a tree-like path to the whole M-set.The connectivity of the M-set was conjectured byBenoît in 1980 and established by AdrienDouady and John Hubbard in 1982.

Figure 2. Picture of F16(S)F16(S)F16(S) where S ! R2S ! R2S ! R2,F(S) = f1(S)" f2(S)F(S) = f1(S)" f2(S)F(S) = f1(S)" f2(S), and f1, f2 : R2 # R2f1, f2 : R2 # R2f1, f2 : R2 # R2 are a!necontractions. The sequence (Fn(S))(Fn(S))(Fn(S)) convergesin the Hausdor" metric to a self-similar set, afractal, with Hausdor" dimension less than two.This article has been decorated with pictures, inthe spirit of Benoît.

various structure theorems for fractals. In-terestingly, this paper had no citations for afew years, but now it frequently gets in theAMS annual top ten list.

Mandelbrot’s ideas were absolutely es-sential and fundamental for my paper. I still

October 2012 Notices of the AMS 1209

have my original copy of his book, signed byBenoît, on the one occasion at Princeton thatwe met.

—John Hutchinson

Iterated function systems (IFSs) are now astandard framework for handling deterministicfractals, self-similar sets and measures. They werenamed by this author and Stephen Demko [1],though Benoît thought we should have calledthem “map bags”. He was fascinated by models ofleaves with veinlike internal structures made byinvariant measures of IFSs.

Hutchinson’s paper and the work of many oth-ers influenced by Mandelbrot ended a long periodwhere geometry and the use of pictures playedlittle role in mathematics. Mandelbrot believedpassionately in pictorial thinking to aid in the de-velopment of conjectures and formal proofs. Hisadvocacy has enabled it to be okay once again formathematicians to do experimental mathematicsusing pictures.

Mandelbrot’s ideashave inspiredahugeamountof research, from pure mathematics to engineer-ing, and have resulted in deep theorems; a newacceptance of geometry and pictures as having arole to play in experimental mathematics; and var-ious applications, including image compressionand antenna design. The notion of a fractal nowforms part of good preuniversity mathematics ed-ucation, while the mathematical study of fractalshas its own specialist areas, including, for exam-ple, analysis on fractals [8] and noncommutativefractal geometry [9].

One important idea of Mandelbrot was thatvarious random phenomena, such as stock marketprices, are governed by probability distributionswith “fat tails”. This led him to warn in 2004 that“Financial risks are much underestimated. I thinkwe should take a strongly conservative attitudetowards evaluating risks.” The subsequent globalfinancial crisis underlined his point.

Prior to editing both this article and [3], weemailed colleagues to ask for memories and com-ments on Benoît’s contributions to mathematics,influence, and personal recollections. We receivedreplies from many: not only mathematicians butartists, physicists, biologists, engineers, and so on.Using these replies we have produced two articles:this one and [3], which is more focused on recol-lections of the man himself. Our goal has been toput together something special using the wordsof everyone who wrote but, in general, editing andshortening to avoid repetition of themes.

From early on, Mandelbrot was driven by adesire to do something totally original, to look atproblems that others found too messy to consider,and to find some deep unifying principles. As the

Figure 3. Superposition of the attractors, coloredusing fractal transformations (see [2]) of two

simple bi-a!ne iterated function systems.

words in the following contributions show, hesucceeded.

Roger Howe

Participating in a Conversation That TakesPlace over Long Spans of TimeOne pleasure of doing mathematics is the senseof participating in a conversation that takes placeover long spans of time with some of the smartestpeople who ever lived. Benoît’s work on fractalsprovides a good example of this kind of long-termdialogue.

A significant factor in the invention of calculuswas the idea of representing a curve by the graphof a function and, reciprocally, of representing thetime variation of a quantity by a curve. This back-and-forth identification allowed one to connectthe drawing of tangent lines with finding the rateof change of quantities that vary in time.

When calculus was invented in the seventeenthcentury, the concept of function was not veryprecise. Work during the eighteenth century onsolving the wave equation using sums of sine andcosine functions led to a sharpening of under-standing of the essential properties of functionsand of their behavior. This led in the first half ofthe nineteenth century to the isolation by Cauchyof the notion of continuity, which made clearfor the first time the distinction between con-tinuity and di!erentiability. During the rest ofthe nineteenth century, mathematicians exploredthis di!erence, which contributed to the generalunease and insecurity about the foundations ofmathematics. Hermite is quoted as “recoiling inhorror from functions with no derivatives.” Theearly twentieth century saw the production of

Roger Howe is professor of mathematics at Yale University.His email address is [email protected].


Figure 4. Two illustrations of IFS semigroup tilings. The triangle on the left is tiled with the orbit of asix-sided figure under a system of two a!ne transformations. The limit set of the set of triangulartiles on the right is the attractor of a system of three a!ne transformations. A theme of Benoît’s workwas that the iteration of simple rules (e.g., elementary geometrical transformations) can producenondi"erentiable (rough) objects. Figure from [2].

a menagerie of striking examples (the Cantorset, the Koch snowflake, the Sierpinski carpet,etc.) illustrating the di!erence between continuityand di!erentiability. However, for several decadesthese examples were regarded as exotica, mon-sters with no relation to the physical world. Theywere objects only a mathematician would inves-tigate. They were liberated from this marginalstatus by Mandelbrot, who said, “Wait a minute.A lot of things in the world—clouds, river sys-tems, coastlines, our lungs—are well described bythese monsters.” Thus started the use of thesemathematical objects to study complicated, messynature.

Ian Stewart

No Lily-White HandsI first learned about fractals from Martin Gar-dener’s Scientific American column. I promptlybought a copy of Fractals: Form, Chance, and Di-mension [16]. Despite, or possibly because of, itsunorthodoxy and scope, it seemed to me thatBenoît Mandelbrot had put his finger on a brilliantidea.

I’m pleased that, towards the end of his life, hereceived due recognition, because it took a longtime for the mathematical community to under-stand something that must have been obvious tohim: fractals were important. They were a gamechanger, opening up completely new ways to thinkabout many aspects of the natural world. But fora long time it was not di"cult to find professional

Ian Stewart is emeritus professor of mathematics at theUniversity of Warwick, UK. His email address is [email protected].

research mathematicians who stoutly maintainedthat fractals and chaos were completely uselessand that all of the interest in them was pure hype.This attitude persisted into the current century,when fractals had been around for at least twenty-five years and chaos for forty. That this attitudewas narrow-minded and unimaginative is easy toestablish, because by that time both areas werebeing routinely used in branches of science rang-ing from astrophysics to zoology. It was clear thatthe critics hadn’t deigned to sully their lily-whitehands by picking up a random copy of Nature orScience and finding out what was in it.

To be sure, Mandelbrot was not a conventionalacademic mathematician, and his vision oftencarried him into realms of speculation. And it waseasy to maintain that he didn’t really do muchthat was truly novel—fractal dimension had beeninvented by Hausdor!, the snowflake curve wasa century old, and so on. Mathematicians wouldhave cheerfully gone on employing Hausdor!-Besicovitch dimension to consider such questionsas finding a set of zero dimension that coversevery polygon, but they would not have figuredout that quantifying roughness would make itpossible to apply that kind of geometry to clouds,river basins, or how trees damp down the energyof a hurricane.

Mandelbrot’s greatest strength was his instinctfor unification. He was the first person to real-ize that, scattered around the research literature,often in obscure sources, were the germs of acoherent framework that would allow mathemat-ical models to go beyond the smooth geometryof manifolds, a reflex assumption in most areas,and tackle the irregularities of the natural world


Figure 5. A Julia set associated with the firstcascade of period doubling bifurcations of thelogistic equation. Julia sets for quadratic maps

are intimately related to the Mandelbrot set.Benoît was one of the first to use computers to

make pictures of mathematical objects:computations which took hours to run on

expensive mainframes can now be performed inseconds on handheld devices. This image and

Figures 1 and 13 were computed using freesoftware (Fractile Plus) on an iPad.

in a systematic fashion. It took many years be-fore these ideas began to pay o!, but that’s howpioneering mathematics often goes.

The conjecture whose proof so pleased him(see [3, contribution by Ian Stewart]) was the workof Gregory Lawler, Oded Schramm, and WendelinWerner in their paper “The dimension of theplanar Brownian Frontier is 4/3” [10]. It is partof the work for which Werner received a FieldsMedal, and it shows that fractals have given riseto some very deep mathematics. I suspect thatonly now are we beginning to see the true legacyof Mandelbrot’s ideas, with a new generation ofresearchers that has grown up to consider chaosand fractals to be as reasonable and natural asperiodic motion and manifolds. Mandelbrot wasa true pioneer, one of the greatest mathematicalvisionaries of the twentieth and early twenty-firstcenturies.

David Mumford

Benoît Told Me: “Now You Can See TheseGroups and See Teichmüller Space!”Benoît Mandelbrot had two major iconoclasticthemes. First, that most of the naturally occurringmeasurements of the world were best modeledby nondi!erentiable functions, and second, thehistograms of these measurements were bestmodeled by heavy-tailed distributions. Even ifhe did not bring a new unifying law like Newton’s

David Mumford is emeritus professor of mathematics atBrown University. His email address is [email protected].

F = ma and even if he did not have the deep andsubtle theorems that make waves in the pure mathcommunity, this vision was revolutionary. Whathis lectures made clear was that fractal behaviorand outlier events were everywhere around us,that we needed to take these not as exceptions butas the norm. For example, my own work in visionled me later on to express his ideas about out-liers in this way: that the converse of the centrallimit theorem is true, namely, the only naturallyoccurring normal distributions are ones which areaverages of many independent e!ects.

Benoît’s immediate e!ect on my work was toreopen my eyes to the pleasure and mathematicalinsights derived from computation. I had playedwith relay-based computers in high school andwith analog computer simulations of nuclear reac-tors in two summer jobs. But at the time I thoughtthat only white-coated professionals could han-dle the IBM mainframes and puzzled over whatin heaven’s name my colleague Garrett Birkho!meant when I read “x = x + 1” in some of hisdiscarded code. But Benoît told us that complexiterations did amazing things that had to be seento be believed. These came in two types: the lim-iting behavior of iterations of a single analyticfunction and the limiting behavior of discretegroups of Möbius transformations. The second ofthese connected immediately to my interests. Iwas always alert to whatever new tool might beavailable for shedding any sort of light on modulispaces, whether it was algebro-geometric, topolog-ical, characteristic p point counting, or complexanalytic. I had sat at the feet of Ahlfors and Bersand learned about Kleinian groups and how theyled to Teichmüller spaces and hence to modulispaces. Benoît told me, “Now you can see thesegroups and see Teichmüller space!”

I found an ally in Dave Wright, learned C, andwith Benoît’s encouragement, we were o! andrunning. When he returned to his position at theIBM Watson Lab, he set up a joint project with us,and we visited him and his team there. Later, CurtMcMullen, who also appreciated the power andinsight derived from these experiments, joinedus. It turned out that, in the early hours of themorning, their mainframes had cycles to spare,and we would stagger in each morning to see whatthese behemoths had churned out. There was noway to publish such experiments then, but Daveand I astonished the summer school at Bowdoinwith a live demo on a very primitive machine ofa curvy twisting green line as it traced the limitpoint set of a quasi-Fuchsian group. Ultimately,we followed Benoît’s lead in his Fractal Geometryof Nature [18] and, with Caroline Series, publishedour images in a semipopular book, Indra’s Pearls[27]. One anecdote: We liked to analyze our figures,


estimating, for example, their Hausdor! dimen-sion. We brought one figure we especially liked toWatson Labs and, thinking to test Benoît, askedhim what he thought its Hausdor! dimension was.If memory serves, he said, “About 1.8”, and indeedwe had found something like 1.82. He was indeedan expert!

Hillel Furstenberg

He Changed Fundamentally the Paradigmwith Which Geometers Looked at Space

Let me begin with some words of encouragementto you on this project, dedicated to memorializingan outstanding scientist of our times and one wecan be proud of having known personally.

What do you see as Benoît’s most important con-tributions to mathematics, mathematical sciences,education, and mathematical culture?

Benoît Mandelbrot sold fractals to mathemati-cians, changing fundamentally the paradigm withwhich geometers looked at space. Incorporatingfractals into mainstream mathematics rather thanregarding them as freakish objects will certainlycontinue to inspire the many-sided research thathas already come into being.

Kenneth Falconer

It Was Only on the Fourth or Fifth OccasionThat I Really Started to Appreciate What HeWas Saying

Benoît’s greatest achievement was that he changedthe way that scientists view objects and phenom-ena, both in mathematics and in nature. Hisextraordinary insight was fundamental to this,but a large part of the battle was getting his ideasaccepted by the community. Once this barrier wasbroken down, there was an explosion of activity,with fractals identified and analyzed everywhereacross mathematics, the sciences (physical andbiological), and the social sciences.

Benoît realized that the conventional scien-tific and mathematical approach was not fittedto working with highly irregular phenomena. Heappreciated that some of the mathematics neededwas there—such as the tools introduced by Haus-dor!, Minkowski, and Besicovitch—but was onlybeing used in an esoteric way to analyze spe-cific pathological sets and functions, mainly as

Hillel Furstenberg is professor of mathematics at Bar-Ilan University, Israel. His email address is [email protected].

Kenneth Falconer is professor of pure mathematics at theUniversity of St. Andrews in Scotland. His email address [email protected].

Figure 6. A self-similar fractal of Hausdor"dimension (4ln2)/ln5$ 1.72(4ln2)/ln5 $ 1.72(4ln2)/ln5$ 1.72 associated with thepinwheel tiling.

counterexamples that illustrated the importanceof smoothness in classical mathematics.

Benoît’s philosophy that such “fractal” objectsare typical rather than exceptional was revolu-tionary when proposed. Moreover, he argued thatthe mathematical and scientific method couldand should be adapted to study vast classesof fractals in a unified manner. This was nolonger mathematics for its own sake, but math-ematics appropriate for studying all kinds ofirregular phenomena—clouds, forests, surfaces,share prices, etc.—that had been ignored to alarge extent because the tools of classical smoothmathematics were inapplicable.

Benoît also realized that self-similarity, broadlyinterpreted, was fundamental in the genesis, de-scription, and analysis of fractals and fractalphenomena. Given self-similarity, the notion of di-mension is unavoidable, and “fractal dimension”in various guises rapidly became the basic mea-sure of fractality, fuelling a new interest in theearly mathematics of Hausdor!, Minkowski, andothers.

Benoît had many original ideas, but his presen-tation of them did not always follow conventionalmathematical or scientific styles, and as a result itoften took time for his ideas to be understood andsometimes even longer for them to be accepted.A case in point is that of multifractal measures.Multifractals are, in many ways, more fundamen-tal than fractal sets. Many of the now standardnotions of multifractals may be found in his 1974paper in the Journal of Fluid Dynamics [14], butthis is not an easy paper to fathom, and it was notuntil the 1980s that the theory started to be ap-preciated. Benoît suggested that “the communitywas not yet ready for the concept,” but I think thedelay was partly because of the way the ideas werepresented. I heard Benoît’s talk on multifractalsmany times in the 1980s; he was charismatic, buthis explanations were such that it was only onthe fourth or fifth occasion that I really started toappreciate what he was saying.


Figure 7. The left-hand picture illustrates thepoints in the orbit of a set; the flower picture atcenter left, under a Möbius transformation. The

picture at center right reveals that it is a “tiling”,where the initial tile is shown on the right.

Mandelbrot caused many to look anew at naturalobjects in geometrical terms. Figure from [2].

I am one of many whose life and career havebeen influenced enormously by Benoît and hiswork, both directly and indirectly. We miss him,but the legacy of his ideas and work will remainwith us all and with those who follow.

Bruce J. West

The Intermittent Distribution of the Stars inthe Heavens

Benoît’s idiosyncratic method of communicat-ing mathematical ideas was both challengingand refreshing. The introduction of geometri-cal and statistical fractals into the scientificlexicon opened up a new way of viewing na-ture for a generation of scientists and allowedthem to understand complexity and scaling ineverything from surface waves on the ocean to theirregular beating of the heart to the sequencingof DNA. This accelerated the early research doneby biologists, physicians, and physicists on theunderstanding of complex phenomena.

The line between what was proven and whatwas conjecture in Benoît’s work was often ob-scure to me, but in spite of that, or maybe evenbecause of that lack of clarity, I was drawn intodiscussions on how to apply the mathematicsof fractals to complex phenomena. Fractals be-gan as descriptive measures of static objects,but dynamic fractals were eventually used to de-scribe complex dynamic phenomena that eludeddescription by traditional di!erential equations.Culturally, fractals formed the bridge betweenthe analytic functions of the nineteenth- andtwentieth-century physics of acoustics, di!usion,wave propagation, and quantum mechanics to the

Bruce J. West is adjunct professor of physics at Duke Uni-versity. His email address is [email protected].

twenty-first-century physics of anomalous di!u-sion, fractional di!erential equations, fractionalstochastic equations, and complex networks.

Benoît identified some common features ofcomplex phenomena and gave them mathemati-cal expression without relying on the underlyingmechanisms. I used this approach to extract thegeneral properties of physiological time series,which eventually led to the formation of a newfield of medical investigation called Fractal Phys-iology, the title of a book [28] I coauthored in1995 and the subject of an award-winning book[29] on the fractional calculus. Later, in 2010, I be-came founding editor-in-chief of the new journalFrontiers in Fractal Physiology, which recognizesthe importance of fractal concepts in humanphysiology and medicine.

I first met Benoît when I was a graduate studentin physics at the University of Rochester. ElliottMontroll, who had the Einstein Chair in Physicsand who had been a vice president for researchat IBM, was friends with Benoît and would invitehim to come and give physics colloquia. In thelate 1960s, before the birth of fractals, I heardBenoît conjecture as to why the night sky wasnot uniformly illuminated because of the inter-mittent distribution of stars in the heavens, whythe price of corn did not move smoothly in themarket but changed erratically, and why the timebetween messages on a telephone trunkline werenot Poisson distributed as everyone had assumed.These problems and others like them struck meas much more interesting than calculating per-turbation expansions of a nuclear potential. SoI switched fields and became a postdoctoral re-searcher in statistical physics with Elliott. I haveinteracted with many remarkable scientists, andBenoît is at the top of that list. I am quite surethat my decision to change fields was based inlarge part on Mandelbrot’s presentations and thesubsequent discussion with him and Montroll.

Marc-Olivier Coppens

Engineering Complexity By ApplyingRecursive RulesAs a chemical engineering researcher who workedwith Benoît since the middle of the 1990s, Ibenefited a lot from his mentorship. I also misshim a lot as a friend. In 1996, while completing myPh.D. thesis, I worked closely with him for severalmonths at Yale, sharing an o"ce with MichaelFrame. I developed, with Benoît, a new way to

Marc-Olivier Coppens is professor and associate direc-tor of the Multiscale Science and Engineering Center,Rensselaer Polytechnic Institute. His email address [email protected].


Figure 8. A right-angle Sierpinski triangle. Benoîtrealized that such objects were not freaks andbelonged in mainstream mathematics. Analysison fractals is now a fascinating area ofmathematics.

generate multifractals by taking the product ofharmonics of periodically extended functions.

Fractals in chemical engineering have a!ectedthe modeling and characterization of variousporous materials. As Mandelbrot liked to sayin later years, fractals are an ideal way to mea-sure “roughness”, and roughness is prevalent inchemical engineering and materials science. Theroughness of porous media a!ects transport andreactions in them and hence has a significant im-pact on chemical engineering. For example, in mythesis I showed how molecular-scale roughnessof porous catalysts influences chemical productdistributions up to industrial scales.

In my research I have used fractal trees tointerpolate e"ciently between the micro- andthe macroscale, as in nature. Scaling up fromthe laboratory to the production scale requirespreservation of small-scale, controlled featuresup to larger scales. This challenge is met bydistributing or collecting fluid in a uniform way,as is realized by scaling fractal architectures innature, such as trees, lungs, kidneys, and thevascular network. Specifically, I proposed a fractal,treelike injector to uniformly distribute fluids overa reactor volume, so that the fluids can mix andinteract with the reactor contents. This patentedfractal injector has proven very e"cient for gas-solid fluidized beds. My laboratory is currentlydeveloping a fractal fuel cell design, inspired bythe structure of the lung.

Benoît has had a major influence on my think-ing. To a large extent, thanks or due to the advance

Figure 9. An invariant measure on a fractalattractor of a system of three similitudes hashere been rendered in shades of green. (Brightgreen === greater “density”, black=least “density”.)

of massively parallel, high-performance comput-ers, chemical, biological, and materials sciencesare increasingly atomistic, deconstructing andconstructing matter out of individual elements inwhich the details of each component and its inter-actions are more and more explicitly accountedfor. This atomistic treatment is very powerful andfacilitates the study of specific properties of mat-ter. However, sometimes the importance of theforest tends to be lost in looking too closely at onetree. The complementary, holistic view is, in myopinion, extremely powerful as well, as it allows usto see essential features in a phenomenon withoutthe need to resolve every detail. Fractals are anexample of this idea, where complexity emergesfrom the combination of simple rules. A marriagebetween the holistic and atomistic views can leadus beyond the deficiencies of each one separately.

Nathan Cohen

Complexity Was Well Modeled by FractalsMathematicians spar in an uncomfortable matchbetween the pure and applied, in which migrationfrom one to the other is one way, and no one isallowed to do both. But Benoît Mandelbrot did.

My interest in fractals stems from needing tosolve real-world problems. In 1985 I was a newlyminted Ph.D. in Cambridge (MA). There the generalview was that fractals were a “flavor” of the month,and they were treated as an a posteriori paradigmwith no evidence of solving problems unsolvedin other ways. But I read The Fractal Geometry

Nathan Cohen is the founder of Fractal Antenna Systems,Waltham, Massachusetts. His email address is [email protected].


Figure 10. A self-a!ne fractal provides a simplemodel for the geometry of a fern.

of Nature and landed a consulting job on stockoptions pricing. I concluded, as Mandelbrot hadsurmised decades earlier, that the stock price isnot a “random walk”, that complexity and noiseare often indistinguishable, and that complexitymay be modeled by fractals. Market pricing isessentially deterministic, not random. At thattime, on a daily basis, traders would run theirBlack-Scholes models, which assume pricing is arandom di!usion process, and bring the resultsto the floor each morning like racing forms atthe horsetrack. They trusted these cheat-sheets totell them when to buy and sell. But I was able toexploit the limitations of the Black-Scholes modelusing fractals and made a decent little fortune forsomeone who had recently been a poverty-strickenstudent.

The notion of “fractals as antennas” occurredto me in 1987 while attending a lecture by Man-delbrot. I went home and explored this curiousidea, which has subsequently become a majortheme of my e!orts and a field in its own right.Some years later I saw Benoît again at a fractals-in-engineering conference. This was finally theopportunity to converse with him and the firstof several lunch meetings and subsequent phoneconversations in the last dozen years of his life.No one who had such conversations can forget thebrilliant, witty joy of Benoît the polymath. In par-ticular, they helped me to realize that Maxwell’sequations require self-similarity for frequency in-variance, a fundamental and what should havebeen obvious result. Now I see many problemsthat benefit from fractals: metamaterials, a newform of radiative transport, optimization, andfluid mechanics and drag reduction. I only regretthat I can’t share these with Benoît anymore.

Figure 11. Various pictures constructed from theorbit of a leaf picture under a system of three

a!ne transformations. The limit set of thesemigroup is illustrated in red and yellow.

Figure from [2].

Stéphane Ja!ard

Parts of Mathematics Are Totally Bathing inthe Ideas That Benoît IntroducedBenoît was one of the first to apply computergraphics to mathematical objects. He used them todevelop intuitions and to make either discoveriesor deep conjectures.

He also put forward particular entities suchas Mandelbrot cascades, the Mandelbrot set, Lévydusts, and so on as beautiful objects, worthy ofstudy in their own right. At that time, this wasorthogonal to the main direction of mathematicstowards generalizations and abstract structures. Ibelieve that Benoît’s influence on the mathemat-ical community was very helpful in that respect:mathematics was able to admit a down-to-earthcomponent. Some parts of mathematics are nowtotally bathing in the ideas that Benoît introduced.For example, the idea of scale invariance is every-where present in the mathematics of signalprocessing, my area.

More broadly, the notion of fractal prob-ability has been one of the most importantunifying concepts in science introduced in thelast fifty years. It has allowed scientists with di-verse specializations to draw connections betweenseemingly unrelated subjects and has created un-expected cross-fertilizations. This was driven bythe mesmerizing and enthusiastic personality ofBenoît.

Note that fractals are one of the few parts ofmathematics that can be “shown” to the general

Stéphane Ja!ard is professor of mathematics at Univer-sité Paris Est (Créteil Val-de-Marne). His email address [email protected].


Figure 12. “…[E]ighty students in my fractal geometry course learn in a single class how to generatethe fractals pictured here.…”

public. As a teenager, I was influenced by Benoît’sfascinating books. They explained a part of math-ematics that was under construction yet could bereadily understood.

My thesis was on the then-new topic of“wavelets”. I worked at École Polytechnique underthe supervision of Yves Meyer. Once Benoît visitedÉcole Polytechnique, and he heard that a Ph.D.student was working on systems of functions thatcould be decomposed into elementary blocks,related inter alia by dilations and translations. Hecame to my o"ce, and we had long conversationsabout new possibilities o!ered by wavelet anal-ysis. For me, this was the start of interactionswhich influenced me considerably; it certainlypushed me towards specializing in multifractalanalysis, a part of fractals where Mandelbrot’sideas are prevalent. Our interactions resulted intwo joint papers on Polya’s function, whose graphis space-filling and multifractal (its Lipschitzregularity index jumps everywhere). The interestthat Benoît showed in this example, which wasquite forgotten at that time, was typical of hisfascination for beautiful mathematical objectsand the art with which he managed to drawa correspondence between their mathematicalbeauty and their graphical beauty. In all the con-versations that we shared, I was always amazedby the uninterrupted flow of original and brilliantideas that he very generously shared.

Sir Michael Berry

How to Model…a Surface With No Separationof Scales

In the early 1970s, I was studying radio-waveechoes from the land beneath the ice in Antarc-tica. Existing theories separated the “geography”,supposedly measured by the start of the echo,from the “roughness”, indicated by the disorderly

Sir Michael Berry is professor of physics at the Univer-sity of Bristol, UK. His email address is [email protected].

echo trail. The separation was modelled by a flatsurface (“geography”) superimposed on what wassingle-scale randomness (“roughness”), typicallygaussian. I found this not only unappealing butalso scientifically absurd: in a natural landscape,any apparent dichotomy must be an illusion, anartifact of the wavelength used to interrogate it.But how to model, or even describe, a surface withno separation of scales? I had no idea until I readPhilip Morrison’s review of the English edition ofMandelbrot’s Fractals: Form, Chance and Dimen-sion [16]. I cannot remember being so excited by abook review. It was immediately clear that fractaldimension was the key idea I needed, and this wasconfirmed by the book itself.

Quickly came the identification of a new class ofwave phenomena: “di!ractals”, that is, waves inter-acting with fractal objects. In the echo-soundingof landscapes, the interaction is mainly reflec-tion. Later, a grim consequence of an absorptioninteraction emerged: we realized that the pro-longed winter predicted to occur after a nuclearwar, because of the absorption of sunlight bysmoke, would be significantly intensified by thefact that smoke particles are fractal (it would alsobe prolonged, because smoke’s fractality slowsthe particles’ fall). From the development of quan-tum chaology in the late 1970s came a conjectureabout the spectra of enclosures (“drums”) withfractal boundaries: the “surface” correction to the“bulk” Weyl eigenvalue counting formula wouldscale di!erently with frequency and depend onthe fractal dimension. This generated considerablemathematical activity.

In di!ractals it is the objects interacting withthe waves, not the waves themselves, that arefractal. But in some phenomena the wave intensityis fractal on a wide range of scales down to thewavelength. One such, unexpected in one hundredfifty years, is the Talbot e!ect, associated with lightbeyond di!raction gratings whose rulings havesharp edges: the fractal dimensions of the waveacross and along the beam direction are di!erent.All this sprang from Benoît Mandelbrot’s insight,


Figure 13. “Zoom in a few times…mysteriousspirals of spirals of spirals appear.”

meshing perfectly with my preoccupations at thetime. For further details, see my earlier tribute [4]or my home page [5].

Michael Frame

I Believe the Classroom Is an AppropriateStage for a Final View of Benoît’s Work

Here I’ll give a sketch of the remarkable breadthand depth of Benoît’s work, setting most examplesin the world I know best, the classroom. Thatstudents in college, high school, and elementaryschool study the concepts Benoît developed filledhim with happiness. In his memoirs [26], Benoîtdescribes his reaction to student comments afterhis lecture, “Uncanny forms of flattery! Each liftedme to seventh heaven! Truly and deeply, eachmarked a very sweet day! Let me put it morestrongly: it is occasions like that that make mylife.” For this reason, I believe the classroom is anappropriate stage for a final view of Benoît’s work.

In September 2010, a few days after Benoît toldme of his diagnosis, I watched the eighty studentsin my fractal geometry course learn in a single classhow to generate the fractals pictured in Figure 12just by looking at the images and understandinga few attributes of plane transformations.

Their surprise and satisfaction are what Benoîtgave me, gave the mathematical world. To thosewho doubt the value of this approach, I saycompare a standard geometry class lesson onplane transformations with this day in any fractalsclass. The combination of visually complex imagesand the ability to decode these images by a fewsimple rules explains why fractals are a wonderfultool for teaching geometry.

A few weeks later in the course, I showedthese pictures again and asked the class to

find their dimensions. Immediately, they answeredlog(3)/ log(2) and log(6)/ log(3) for the first two,and after a moment, log((%1 +

&3)/2)/ log(1/2)

for the third. That thousands, maybe tens ofthousands, of students know how to computeand interpret dimensions and that dimensionmeasures complexity and roughness of objectsmathematical (Julia sets, Kleinian group limitsets), physical (aggregation clusters, the distribu-tion of galaxies), biological (pulmonary, nervous,and circulatory systems), and artistic (Pollock’sdrip paintings, at least according to some) aredue to Benoît. Some knew bits of the picture;Benoît assembled the whole and got many, manyothers working on measuring and interpretingdimensions.

For the teacher of a fractals class, the bestmoment occurs during the day the Mandelbrotset is introduced. The formula zn+1 = z2

n + c issimplicity itself. Describe the iteration processand the color coding, start the program running(seconds now for images that burned hours ordays with the personal computers of the mid-1980s), and wait. (See Figure 1.) Startling baroquebeauty, but from a class jaded by CGI e!ects,only a few polite “Oohs” and “Ahhs”. Zoom in afew times near the boundary; mysterious spiralsof spirals of spirals appear. (See Figure 13). A bitmore emphatic exclamations of surprise, and then,“You do remember this is produced by iteratingzn+1 = z2

n+ c, don’t you?” Expressions of disbeliefand occasional profanity follow.

Another day or two describing the known ge-ometry of the Mandelbrot set, the arrangementof the cyclic components, the infinite cascade ofever smaller copies of the whole set, and thiscomplicated object starts to seem familiar. Thenstate the hyperbolicity conjecture and point out itremains a conjecture despite two decades of workby brilliant mathematicians. Beautiful pictures forsure; deep, deep mathematics, you bet.


Some Key Events in the Life of Benoît B. Mandelbrot

1924 Born in Warsaw, Poland, 20 November1936 Moved to Paris1939 Moved to Tulle1947 Ingenieur diploma, École Polytechnique1948 M.S. aeronautics, CalTech1952 Ph.D. mathematics, University of Paris1953 Postdoc at MIT, then IAS postdoc of von Neumann1955 Married Aliette Kagan1958 Moved to the U.S., joined IBM Thomas J. Watson1963 Publication of “On the variation of certain speculative prices”, [11] and

“The stable Paretian income distribution, when the apparent exponent is near two” [12]1967 Publication of “How long is the coast of Great Britain?” [13]1972 Visiting professor of physiology, Albert Einstein College of Medicine1974 Publication of “Intermittent turbulence in self-similar cascades:

Divergence of high moments and dimension of the carrier” [14]1975 Publication of Les Objets Fractals: Forme, Hasard et Dimension [15]1977 Publication of Fractals: Form, Chance, and Dimension [16]1979 Began studying the Mandelbrot set; formulated the MLC (Mandelbrot

set is locally connected) conjecture1980 Publication of “Fractal aspects of the iteration of z ! !z(1" z)

for complex ! and z” [17];formulated the question that the Mandelbrot set is connected

1982 Publication of The Fractal Geometry of Nature [18];Fellow of the American Academy of Arts and Sciences;formulated the 4/3 conjecture and that the inside and outside of theBrownian boundary curve are statistically self-similar; connectivity of theMandelbrot set proved by Douady and Hubbard

1984 TED lecture; formulated the n2 conjecture, proved byGuckenheimer and McGehee

1985 Barnhard Medal, U.S. National Academy of Sciences;formulated the conjecture that the boundary of the Mandelbrotset has dimension 2

1986 Franklin Medal, Franklin Institute; D.Sc., Syracuse University1987 Foreign associate, U.S. National Academy of Sciences;

Abraham Robinson Adjunct Professor of Mathematical Sciences at Yale;D.Sc., Boston University

1988 Steinmetz Medal, IEEE; Science for Art Prize, Moet-Hennessy-LouisVuitton; CalTech Alumni Distinguished Service Award;Humboldt Preis, Humboldt-Sti!tung;honorary member, United Mine Workers of America;D.Sc., SUNY Albany, Universität Bremen

1989 Chevalier, National Legion of Honor, Paris;Harvey Prize for Science and Technology, Technion;D.Sc., University of Guelph

1990 Fractals and Music, Guggenheim Museum, with Charles Wuorinen1991 Nevada Prize1992 D.Sc., University of Dallas1993 Wolf Prize in Physics;

D.Sc., Union College, Universitè de Franche-Comtè, UniversidadNacional de Buenos Aires


1994 Honda Prize; J.-C. Yoccoz awarded the Fields Medal, in part for hiswork on MLC; Shishikura proved the Mandelbrot set boundaryhas dimension 2

1995 D.Sc., Tel Aviv University1996 Médaille de Vermeil de la Ville de Paris1997 Publication of Fractals and Scaling in Finance [19]1998 Foreign member, Norwegian Academy of Sciences and Letters;

C. McMullen awarded the Fields Medal, in part for his work on MLC;D.Sc., Open University London, University of Business and Commerce Athens

1999 Sterling Professor of Mathematical Sciences at Yale; John Scott Award;publication of Multifractals and 1/f Noise [20]; publication of“A multifractal walk down Wall Street” [21]; D.Sc., University of St. Andrews

2000 Lewis Fry Richardson Award, European Geophysical Society2001 Member, U.S. National Academy of Sciences;

publication of “Scaling in financial prices, I – IV”2002 Sven Berggren Priset, Swedish Academy of Natural Sciences;

William Proctor Prize, Sigma Xi; Medaglia della Prezidenza dellaRepublica Italiana; publication of Gaussian Self-A!nity and Fractals [22]and of Fractals, Graphics, and Mathematics Education [23];D.Sc., Emory University

2003 Japan Prize for Science and Technology; Best Business Book of the YearAward, Financial Times Deutschland, for The (Mis)Behavior of Markets [25]

2004 Member, American Philosophical Society; publication of Fractals andChaos. The Mandelbrot Set and Beyond [24], and (with R. Hudsonof The (Mis)Behavior of Markets [25]

2005 Sierpinski Prize, Polish Mathematical Society; Casimir Frank NaturalSciences Award, Polish Institute of Arts and Sciences of America;Battelle Fellow, Pacific Northwest Labs; D.CE., Politecnio, Torino

2006 O!cer, National Legion of Honor, Paris; Einstein Public Lecture, AMSAnnual Meeting; Plenary Lecture, ICM; W. Werner awarded the FieldsMedal for proving (with G. Lawler and O. Schramm) the 4/3 conjecture;Doctor of Medicine and Surgery, University degli Studi, Bari, Puglia

2010 D.Sc., Johns Hopkins University; TED lecture; S. Smirnov awarded theFields Medal for work on percolation theory and SLE related to the4/3 conjecture.Died in Cambridge, MA, 14 October

References[1] M. F. Barnsley and S. G. Demko, Iterated function

systems and the global construction of fractals, Proc.Roy. Soc. London Ser. A 399 (1985), 243–275.

[2] Michael F. Barnsley, Superfractals: Patterns ofNature, Cambridge University Press, 2006.

[3] Michael F. Barnsley, Michael Frame (eds.),Glimpses of Benoît B. Mandelbrot (1924–2010), AMSNotices 59 (2012), 1056–1063.

[4] M. V. Berry, Benefiting from fractals (a tribute toBenoît Mandelbrot), Proc. Symp. Pure Math., vol. 72,Amer. Math. Soc., Providence, RI, 2004, pp. 31–33.

[5] http://www.phy.bris.ac.uk/berry mv/publications.html.

[6] John Brockman, A theory of roughness: a talk withBenoît Mandelbrot (12.19.04), Edge, http://edge.org/conversation/a-theory-of-roughness

[7] J. E. Hutchinson, Fractals and self-similarity,Indiana Univ. Math. J. 30 (1981), 713–747.

[8] J. Kigami, Harmonic calculus on p.c.f. self-similarsets, Trans. Amer. Math. Soc. 335 (1993), 721–755.


[9] M. L. Lapidus, Towards a noncommutative fractal ge-ometry? Laplacians and volume measures on fractals,Contemporary Mathematics, vol. 208, 1997, Amer.Math. Soc., Providence, RI, pp. 211–252.

[10] G. Lawler, O. Schramm, W. Werner, The dimensionof the planar Brownian frontier is 4/3, Math. Res. Lett.8 (2001), 401–411.

[11] B. Mandelbrot, The variation of certain speculativeprices, J. Business (Chicago) 36 (1963), 394–419.

[12] , The stable Paretian income distribution whenthe apparent exponent is near two, InternationalEconomic Review 4 (1963), 111–115.

[13] , How long is the coastline of Great Britain?Science (New Series) 156 (1967), 636–638.

[14] , Intermittent turbulence in self-similar cas-cades: Divergences of higher moments and dimen-sion of the carrier, J. Fluid Mechanics 62 (1974),331–358.

[15] , Les Objets Fractals: Forme, Hasard etDimension, Flammarion, Paris, 1975.

[16] , Form, Chance and Dimension, W. H. Freeman,San Francisco, CA, 1977.

[17] , Fractal aspects of the iteration z % !z(1 % z)for complex ! and z, Ann. New York Acad. Sci. 357(1980), 249–259.

[18] , The Fractal Geometry of Nature, W. H.Freeman, San Francisco, CA, 1983.

[19] , Fractals and Scaling in Finance, Springer-Verlag, New York, 1997.

[20] , Multifractals and 1/f Noise, Springer-Verlag,New York, 1999.

[21] , A multifractal walk down Wall Street,Scientific American, Feb. 1999, 70–73.

[22] , Gaussian Self-A"nity and Fractals, Springer-Verlag, New York, 2002.

[23] , Fractals, Graphics, and Mathematics Educa-tion, MAA, Washington, DC, 2002.

[24] , Fractals and Chaos: The Mandelbrot Set andBeyond, Springer-Verlag, New York, 2004.

[25] , The (Mis)Behavior of Markets, with R. Hudson,Basic Books, New York, 2003.

[26] , The Fractalist: Memoir of a Scientific Maverick,Pantheon, 2012.

[27] D. Mumford, C. Series, D. Wright, Indra’s Pearls,The Vision of Felix Klein, Oxford University Press,2002.

[28] James B. Bassingthwaighte, Larry S. Liebovitch,

Bruce J. West, Fractal Physiology, Oxford UniversityPress, New York, 1994.

[29] B. J. West, M. Bologna, P. Grigolini, Physics ofFractal Operators, Springer-Verlag, New York, 2003.


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