Benoit Mandelbrot’s The Fractalist: Chapter by Chapter, Page by Page Review of an Excellent Autobiography

Mandelbrot’s breadth of work, language skills, and clarity make his autobiography a pleasure to read. Highlighting my favorite sentence on a page was my first step in an effort to improve my own writing. This led to page by page, and then chapter by chapter summaries – Mandelbrot’s The Fractalist has been my favorite so far.

Mandelbrot’s life is full of challenge and urgency. He is born into World War 2 on November 20, 1924, demonstrating academic prowess from an early age while fleeing a wave of Nazi persecution from Poland to Vichy France in Part 1. With Part 2, the eminent threat of death is removed, but the threat of distraction and a lack of clarity in the value of the new field he was developing – previously unconnected study of roughness which he would brand as ‘fractals’. Part 3 establishes a cadence of success, as Mandelbrot achieves and extends his ‘Keplerian Dream’ with the broad success of fractals. He ties together multiple fields of study, creates unique insights, creates an educational culture around his new field of study, achieves penetration in the popular press with his ideas, and is widely recognized for his contributions before his death. Several rules, habits and talents are used through his life, chronologically they are:

  1. Discuss risks and odds.
  2. Survive. Flee when you must.
  3. Split up to survive.
  4. Build on your strengths.
  5. Have a big dream.
  6. If it isn’t going to work, don’t force it.
  7. If your capability is broad, be broad.
  8. Connect – Tie things together.
  9. Look for talent.
  10. Use deadlines.
  11. Write clearly.
  12. Brand.
  13. Note the missing items.
  14. Measure, math, numbers.
  15. Enjoy it.

These rules for life are simple, and they are supported many times throughout Mandelbrot’s life as he highlights in the book:

  1. Discuss risks and odds, Mandelbrot on his parents, “They never shouted at each other but argued constantly about strategy, and they taught me very early that before taking big risks, one must carefully figure the odds.” The family discussed risk at a ‘War Council’ when it was time to choose Benoit’s schooling.
  2. Survive. Flee when you must. When other families worried about the silverware or where to go – the Mandelbrots got out.
  3. Split up to survive. Benoit and his brother split from his parents, probably saving their lives.
  4. Build on your strengths. Benoit knew math. Then he knew Zipf. Then he built a career on both.
  5. Have a big dream. Benoit was determined to find his Keplerian dream – unifying and describing some integral form of how the world works. He brought order to roughness.
  6. If it isn’t going to work, don’t force it. In Chapter 22, when fractals change the understanding of finance, Benoit notes, “I was acutely aware that my findings would have devastating consequences for the accepted standard theory of speculation.” Benoit misses several academic appointments, but he moves on – he does not dwell on the missed opportunities. Onward.
  7. If your capability is broad, be broad. Because of his perspective seeing many problems at IBM, he was able to develop the early work with Zipf to explain how a broad array of problems had a common underlying explanation – fractals.
  8. Connect – Tie things together. The patterns from Zipf followed the patterns in finance, followed the patterns of the early Internet. Rather than learning anew with each problem, he looked for ways to apply a common set of tools.
  9. Look for talent, “The Revolution succeeded because Carnot hired men such as the Corsican Napoléon Bonaparte.” IBM was undergoing change. The French academy was not. Mandelbrot sought out areas of study and entities that would appreciate who he was.
  10. Use deadlines. Mandelbrot did not shy away from the opportunity to present a summary of his findings in France in Chapter 24, “It forced me to gather all I had achieved and fit it into an hour. This effort started me on my 1975 book.”
  11. Write clearly (Chapter 26):
    “Many scientific articles are completely flat because they are written for people who do not have to be convinced.” 
    “… one must know how to enter into a subject quickly ….”
  12. Brand. Mandelbrot’s breadth was a liability, until he used his strength in linguistics to coin a neologism – FRACTAL, “I had not a single identifying brand name for my activity. Ten more years went by until I gave up and coined the word “fractal.””
  13. Note the missing items. “A fractal is defined as well by what has been removed as it is by what remains.”
  14. Measure, math, numbers. Mandelbrot embraces the math and numeracy of every situation he encountered.
  15. Enjoy it. Mandelbrot loves his wife, Aliette, his sons, his grandchildren. He finds fractals in music and artwork. His French appreciation for the finer things in life is clear in the way that he built friendships that spanned the globe.

Part by Part, Chapter by Chapter, Page by Page Review – 91 Benoit Mandelbrot Quotes

Part One – “How I Became a Scientist”

Chapter 1 Highlights

Mandelbrot’s book tells the story in a fractal nature – bouncing big and small and returning to reinforce important topics.  From this first family photo Mandelbrot introduces us to the adults that shaped his young life, helped him survive World War II, and put him in position to excel when life provided him with options.  What a family!

Best Quote(s)

“As observed by a writer native to that part of Europe, Woe to the poet born in an interesting piece of geography in a violent time.”

Mandelbrot’s fractals visual replication of geographic features – coast lines and elevations, was one of the first clear victories of the new field.  Mandelbrot’s life would move about the globe, making big leaps and small.

“Other war survivors describe being in a herd on the way to the death camps, noticing a way out, and taking it instantly. That is the kind of man Father was.”

Little opportunities lead to big changes.  Mandelbrot’s family would face many opportunities where following the herd would have led to a very different life – or even death.  Instead, they found the way out.

“They never shouted at each other but argued constantly about strategy, and they taught me very early that before taking big risks, one must carefully figure the odds.”

Discussion of risks helps mitigate risks.  Identify risks, then address them.

“Like many social customs, it could be defied, but at a cost: not being part of a system of patronage that is pervasive in intellectual and professional groups.”

Mandelbrot’s career would lack the rudder of mentorship – but he made up for it by staying focused on finding a larger calling and harnessing his intellectual capabilities and breadth of interests with hard work.

Chapter 2 Highlights

Warsaw shaped Mandelbrot as his first home, site of his early education and as a way of life that would be completely erased.  His family focused on staying alive and ahead of the Nazi advance at a time when others who were less urgent were murdered.

Best Quote(s)

“For these and other truly unavoidable reasons, Polish history from 1919 to 1939 was rough.”

Rough – just like the fractals that Mandelbrot invented.

“Since diversity cannot be avoided, one may as well like it (as I came to) or at least learn to live with it.”

This is a delightfully modern approach to diversity that is all too common in more integrated, international cultures.

“Before everything they had dreaded became horribly concrete in Poland, my parents’ bold scheme had worked.” Chapter 2, Location 659

His parents craft successful strategies again and again, allowing their family to survive and innovate on behalf of mankind.

“Of the people we knew, we alone moved to France and survived. Most procrastinated—until times turned awful. Only two Warsaw friends survived:…” Chapter 2, Location 663

The eradication of this world is frightening and it pervades Mandelbrot’s view of the world for the rest of his life.  How could it not?

“Others had been detained by their precious china, or inability to sell their Bösendorfer concert grand piano, or unwillingness to abandon the park view from their windows. Mother was horrified by their stories but listened stone-faced.” Chapter 2, Location 668

When the Mandelbrot family needed to act, they did so – unencumbered by the weight of their past.  They moved with focus to achieve their goal, survival.

Chapter 3 Highlights

Pages – ; Locations 675 – 860

Mandelbrot’s youth is about promise and his family’s focus on survival.  In Chapter 3 we find our young student in France with a need to exceed in education, but with the reader knowing that war and atrocity lie in his future.

Best Quote(s)

“By pulling up their deep roots in a community that only a few years later vanished in smoke, my lucid and decisive parents saved us all and earned the utmost gratitude.” Chapter 3, Location 681

Chapter two told us of the wonderful upbringing, education and life lessons that Mandelbrot experienced in Poland – all of which were wiped away with the Nazi invasion and Holocaust.  The lessons of his parents’ focus on survival is repeated throughout the book.

“Each time I recall that successful exam, my heart rejoices. Lady Luck is blind and needs assistance. In 1936, my parents assisted by moving out of Poland. In 1937, I was called to assist—and I did.”  Chapter 3, Location 795

From Mandelbrot’s mind, “My parents kept us alive – and to make the most of it I had to nail that test.  What fortune!”

“A belated benefit from my years of Latin is that they helped me correctly coin new words—like “fractal.”” Chapter 3, Location 833

Mandelbrot’s writing follows his study of roughness – we know where the story is going, and still the book plays out like a mystery.  We know the ending – otherwise who would read this autobiography?  Throughout he foreshadows his contribution to science and math.

Chapter 4 Highlights

Mandelbrot’s family continues to survive as the violence of World War II escalates.  Hard work provides a way out.  Luck breaks for the family again.

Best Quote(s)

“Xenophobia lost, meritocracy won, and she deliberately misplaced my family’s files.” Chapter 4, Location 879

The family survived many challenging times in WW2 – and they did so with some lucky breaks and a commitment to hard work.

“When I was nearing forty, my work became devoted to the phenomenon called intermittence, present in both nature and the financial markets.” Chapter 4, Location 885

Again, we see Mandelbrot foreshadowing his focus on fractals and the study of roughness.

“The final examination included two very easy problems, which I saw instantly to be a single problem stated in two different ways. Apparently, few students noticed.” Chapter 4, Location 1030

Mandelbrot manages to describe his academic and scientific prowess directly without boasting – that is a communication skill to envy.

Chapter 5 Highlights

Mandelbrot continued to excel in his studies despite the oppression of the Nazi occupation of France.  He studied shapes, and used the basic of geometry in novel ways to solve problems more fitting for older students.

Best Quote(s)

“True to our antiherding instinct, our family decided it was best to split up: the boys on their own, and the parents on theirs.” Chapter 5, Location 1068

It is remarkable that parents would make this decision – it had to have been agonizing.  And it was the right one.

“Not unlike sports, the bulk of training consisted of mastering a single but extremely arcane gesture.” Chapter 5, Location 1079

When brilliant minds comment on sport and athletics, the observations are always fascinating.

“Oradour-sur-Glane is a little town where the Waffen SS committed a horrible massacre in 1944, herding 642 villagers into a church and setting it on fire.” Chapter 5, Location 1118

Referring back to the family decision by the Mandelbrot’s to split up – the acts of violence by the Nazis defy the civility of modern life.  As we will see in a later chapter, the Mandelbrot decision saved the lives of their children.

“In a way, I was learning to cheat.”  Chapter 5, Location 1197

By learning to think in shapes, Mandelbrot would teach himself ways to think that even he considered cheating.  His life’s great accomplishment is taking that way of thinking, using it to study roughness and sharing it with the rest of humanity.

Chapter 6 Highlights

Mandelbrot deploys his genius in order to help an aristocrat maintain the family horse farm while a war rages in the background.  Carnegie uses animals to endear the reader to people – and the olfactory descriptions of a young Dr. Mandelbrot amidst equestrians is powerful imagery.

Best Quote(s)

“Nobody ever listens to me, but you did. And you remembered everything. You can’t be altogether bad.” Chapter 6, Location 1242

Mandelbrot takes a page out of the Carnegie playbook from How to Win Friends and Influence People.

“Horse owning Gentry thought Germany would win, “I cajoled them, first to listen to Swiss radio in French, then to France Libre in London.”” Chapter 6, Location 1298

From a persuasion standpoint, Mandelbrot was pacing his gentrified hosts to understand there were more possibilities about what might happen with the war.

Chapter 7 Highlights 

Locations 1312 – 1347

We wrap up the first part of Mandelbrot’s life – with a wonderful preview of the 2nd and third parts.  The family made a bold bet to split up, and it was truly necessary based on the gruesome description of French resistance youths meeting their death hanging from street lamps.

Best Quote(s)

“The bold plan our parents had devised—bless their hard-won survivor skills—had let them and their sons cope with events separately. This bet, the riskiest of our complicated lives, worked better than any realist could have hoped.” Chapter 7, Location 1335

The plan had worked, all four were alive – while other families in similar situations were never to meet again.

“During this second, twelve-year stage of my life, I was not going to manage elegantly—as will be seen. So, in time, I deliberately provoked a belated third stage.” Chapter 7, Location 1347

Mandelbrot’s self reflection is impressive.

Part 2: Chapters 8 – 20, “My Long and Meandering Education in Science and in Life”

In Part 1, we watched Mandelbrot’s family survived World War 2 intact. They had a strategy – splitting when others stayed together. Tough decisions were made and they stayed ahead as a wave of Nazi persecution washed over Europe. Strengths were developed, hardships endured.

Part 2 is an incubation period for Mandelbrot. His stories are full of consistent life strategies and goals, but all throughout he feels unfulfilled. There are several components to Mandelbrot’s life strategy that lead to the crescendo that is Part 3, when fractals are a field of study and his Keplerian dream is realized. The accomplishments of Part 3 are built on the foundation laid here in Part 2.

The Habits of Mandelbrot

  1. Having a goal and continually re-affirming what he does want to do. He names his goal – The Keplerian Dream.
  2. Small activities are consistent with his larger ambition.
  3. Associating with the best and brightest – and being open about his ambitions
  4. Building on his strengths in mathematics.
  5. Enjoying life – Mandelbrot exhaults in the bureacratic interludes, he finds love with Aliette, rejoices in music.
  6. Figuring out what he doesn’t want to do. Reversing bad decisions.
  7. Being comfortable with who he is.

Chapter 8: Paris: Exam Hell, Agony of Choice, and One Day at the École Normale Supérieure, 1944–45

Mandelbrot has now survived World War II, but with survival he is now quickly forced into major life decisions.  Which school to attend?  His decision to attend Normal – which he does for 1 day before realizing he has made a mistake – is met with a quick reversal.  He attends Polytechnique.  Uncle Szolem is embarassed, but what courageous behavior for a young man to exhibit at a hectic time of life.

Best Quote(s)

“The high stakes terrified us all, and my parents did not trust my teachers. So a family “war council” was called to help:…” Location 1430

This is more of a statement about me – as the writer of this chapter summary and blog, than it is about Mandelbrot.  I’d never heard of a family doing such a formal review before helping a young member make a major life decision.  The examples that Mandelbrot shares of the other great scientists – who went on to win Noble prizes and more, shows the wisdom of such a process.

“Good wine or cheese must not be rushed. So why rush good humans by pressing a cookie cutter on a malleable young mind?” Location 1536

By attending Polytechnique, Mandelbrot would enter into a French bureaucratic educational system that would force him to pause his scientific career several times to follow the rules of the state.  Rather than seeing this idle time as a waste, he embraced it and believed it to help his ultimate achievements in developing a theory of roughness and fractals.

Chapter 9: A (Then Rare) Foreign Student at the École Polytechnique, 1945–47

As noted in his quote below, Mandelbrot enters Polytechnique literally in rags and is one of the few international students.  He devours the opportunities in front of him, while also realizing that the degree will provide him golden handcuffs that tie him to France – which may not be the best geography for someone with his Keplerian dream.

Best Quote(s)

“… my whole life’s orbit was to show that professional authority did not awe me either.” Chapter 9, Location 1645

Mandelbrot listens to authority – and is politely interested in its guidance.  However, he does not blindly obey.

“Plus, all generally valid rules suffer from deviant exceptions, and I went on to prove that a person profoundly rooted in classics may very well be a successful, yet troublemaking, maverick.” Chapter 9, Location 1746

Mandelbrot was classically trained at Polytechnique, and he had a deep education in geometry – albeit learned in nontraditional ways.  It was this domain expertise in these fields that led him to create the study of fractals.

Chapter 10: Pasadena: Student at Caltech During a Golden Age, 1947–49

This chapter documents the many remarkable minds and concepts that Mandelbrot encountered in his studies, all while he was searching for a remarkable topic to develop fully as his own.  His self reflection while wandering in a field of geniuses makes for great reading.

Best Quote(s)

“But freedom of choice was a negative asset; it set me on a wide sea without sufficient guidance.” Location 1762

Mandlebrot has succeeded and is at the top of the pecking order in France with his success at Polytechnique – and that success has brought freedom.  Freedom after the travails of WW2 is paralyzing and gives too much freedom to operate.

“I wanted to feel the excitement of being the first to find a degree of order in some real, concrete, and complex area where everyone else saw a lawless mess.” Location 1767

Mandlebrot curtails his immense freedoms by teasing out a personal and professional goal – and an audacious one at that.  This ‘Keplerian’ dream that he has outlined is bold and big.  In his autobiography, written later in life, he starts to see visions of this goal early.  As the reader, I wonder how much he felt early in life, and how much he portrayed backwards as he enjoyed success in later years.  Mandelbrot allows us to feel this mystery along with him without forcing the development of fractals as predetermined.

Chapter 11: French Air Force Engineers Reserve Officer in Training, 1949–50

Chapter 11 could serve as an excerpt from Heller’s masterpiece, Catch-22. Mandelbrot attempts to conform to his military service requirements while the military bureaucracy struggles to make use of a loyal, but atypical, young man.

Best Quote(s)

“A BLESSING THROUGHOUT LIFE: I never wonder who I am. To the contrary, many successive bureaucracies wondered endlessly.” Location 1995

Mandelbroth may have had uncertainty in life as he wandered in pursuit of his ‘Keplerian Dream’ which he would find in his study of roughness, but he knew who he was. He knew he was in pursuit of that goal – even as he knew that the precise goal was not known.

Mandelbrot could tolerate that lack of clarity – and as we’ll see in the chapters about his early family, his wife and children could too. He created a life that let him search for it. Bureaucracies – here the French Air Force, but later IBM – served as stewards of his income, but were themselves confused.

It took a lot of commitment for a young graduate, and later a young father to:

  • Admit he did not know the goal with precision.
  • Harness the bureaucratic energy of these organizations while it was clear that they were confused about who he was.

Chapter 12: Growing Addiction to Classical Music, Voice, and Opera

Opera and classical music becomes a passion for some – but it has never been so for me. Synthetic, computer generated music on its own has never sounded good to me – but I do like how it can be used to augment the beat or rhythm in dance or techno music. My favorite artist, Bob Dylan, made a legendary transition to the electric guitar and took folk music with him. Perhaps someone is using that with fractal or other digital music now.

Best Quote:

“What brought the three of us together was a special development—the observation that music has a fractal aspect.”

Mandelbrot, Chapter 12

Chapter 13: Life as a Grad Student and Philips Electronics Employee, 1950–52

“Unlike Szolem, I enjoy intellectual fencing and occasionally showing off. Otherwise—like Szolem—I absolutely stopped having patience for their games.”

Mandelbrot, Chapter 13

Like Tversky and Kahneman – there had to be a base rate. Mandelbrot needed to engage with his peers and barb with them in order to realize that the games they were playing were silly. Without the base rate – what it was those peers were going to accomplish – then the prodigy was unable to see that he wanted more. Mandelbrot couldn’t articulate a clear destination, but he could put himself among an exceptional group of peers, hold his own intellectually, and while evaluating their goals say, “More than that!”

Mandelbrot recognized he did not fit, and he developed a system – albeit a loose one – to help refine what he wanted to do to truly excel. He knew what not to do. He trained himself to find the things he should do.

Best Quote(s)

The role of a mentor who has done what you want to do is highlighted by Kastler’s advice in Mandelbrot’s choice of thesis;

“This man was attuned to nuance and comfortable with living between two cultures.” Mandelbrot, Chapter 13 – Location 2324

As Mandelbrot reflects on the lives of his parents following the 1951 passing of his father;

“Their explanation was that less fortunate persons would have perished early in one of the catastrophes that they had managed to sail through.” Mandelbrot, Chapter 13 – Location 2331

Chapter 14: First Kepler Moment: The Zipf-Mandelbrot Distribution of Word Frequencies, 1951

Our scientist-hero has identified a goal – creating a ‘Keplerian’ field of study which he can pioneer. Mandelbrot has found a mentor, Kastler, who understood the need to drift between worlds and excelled at it. He has a method to pursue this need – surrounding himself with brilliant peers, and then finding out how to outshine them and realizing that their bureaucratic games of hierarchy hold no attraction for him.

Best Quote(s)

Path dependency matters. Many had noted the trends and methods that Mandelbrot would make into a formal field, but under ‘Important’ areas of study, it was hard to take the intellectual risks necessary to openly discuss their use. Failure would be too painful. By starting in an atypical area that threatened no one, he was allowed room to develop and grow. (On a personal note – this feels like my time in nanofibers.)

“My luck was to begin with the distribution of word frequencies—a thoroughly atypical example without any important consequences, and uniquely easy to handle.” Mandelbrot, Chapter 14, Location 2387

Mandelbrot, Chapter 14, Location 2387

Scaling – ‘being fractal’ – in your activities is one of my personal big take aways from reading this book. Mandelbrot declares that if something is not smooth, then it is rough – and things that are rough have common traits in how they behave. Apply this to life – if things are not smooth, then they must be rough. Do things that scale up and scale down. Much in business is focused on ‘can it scale up’ – but in many ways this avoids the hard challenges of finding ways for activities to be worthwhile if they must scale down. To win in a world of roughness, scaling – both up and down – must be part of the strategy.

“The language—English, French, Latin, whatever—does not matter. Neither—quite oddly—does the writer’s degree of literacy. This is an example of what physicists were soon to call a universal relationship. Another notion in physics, called scaling, is one that underlies fractals.”

Mandelbrot, Chapter 14, Location 2408

If ever a movie is made of Mandelbrot’s life – then the scene where he reads this paper, a gift from Uncle Szolem, and hits this Eureka moment would be a highlight.  

“In one of the very few clear-cut eureka moments of my life, I saw that it might be deeply linked to information theory and hence to statistical thermodynamics—and became hooked on power law distributions for life.”

Mandelbrot, Chapter 14, Location 2425

Zipf’s paper was not new – it had first been published 16 years earlier. Mandelbrot would be in a tenuous position – he believed that his mathematical capabilities provided him a unique perspective, but he was alone with this view point. Only as he was able to add to this perspective with advances in topics far afield from word theory would the strength of Mandelbrot’s position improve.

“My good fortune resided in an unfair advantage. I was to be the first—and for an interminable time, the only—trained mathematical scientist to take Zipf’s law seriously.” Mandelbrot, Chapter 14, Location 2425

Mandelbrot, Chapter 14, Location 2425

Chapter 15: Postdoctoral Grand Tour Begins at MIT, 1953

Mandelbrot heads to MIT (images above from this MIT group), studies under a pioneer and namer of ‘cybernetics’ and continues on his quest to uncover and master a Keplerian field of study.

Best Quote(s)

“Cybernetics” was a word Wiener had just coined…”

Mandelbrot, Chapter 15 Location 2493

Mandelbrot’s definition of fractal and creation of the word as the systematic study of roughness is important to him, and it was something he explored diligently.

“… grammar is like the chemistry or algebra of language.”

Mandelbrot, Chapter 15 Location 2575

This is slipped in as a throw away line about language and Zipf, but it is an elegant way to think of how words are used.

“These two men were the only living proof that my Keplerian dream was not an idle one—that it was possible to put together and develop a new mathematical approach to a very old, very concrete problem that overlapped several disciplines.”

Mandelbrot, Chapter 15 Location 2499

In using Mandelbrot to explore good habits for achieving a goal – here he has set a Goal and identified two living benchmarks. It may be only two – but at least there is someone who has completed his objective!

Chapter 16: Princeton: John von Neumann’s Last Postdoc, 1953–54

The Chapter starts with the tale of a public lecture where Mandelbrot is excoriated – but then experiences a classic ‘Oppenheimer’ explanation. Following a fractal pattern, we then find Von Neumann providing another level of explanation on top of Oppenheimer. It is amazing how Mandelbrot’s career was so closely entwined with these brilliant titans of discovery.

Best Quote(s)

“No major turn in my entire life proceeded more smoothly.

Mandelbrot, Chapter 16 Loc 2664

If things are not smooth – then they are rough. Mandelbrot hints here at a ‘Fractal Strategy’ for life. If things are going smooth – then there are a certain set of rules. But when things are rough, be sure that your actions can scale up – and that they can also scale down. Big things build from small patterns repeated with consistency.

“As soon as he heard a field had become hot, he made himself an expert with a competitive edge and identified several key issues he could solve.”

Mandelbrot, Chapter 16, Loc 2631

In Finance this would be a convex strategy, where you follow the winners and keep winning provided they continue to be the winners in the next period.

Chapter 17: Paris, 1954–55

Young Mandelbrot has completed his PhD and looking for employment, mentors and other problems that will all work towards his goal of a Keplerian type view of a major problem. Zipf put him on the path to Fractals, but Fractals have not yet come to his life.

Best Quote(s)

“…. the unreasonable effectiveness of mathematics in the natural sciences.”

Mandelbrot, quoting Wigner in Chapter 17

“… his fear of being a mere survivor of the last century, and his feeling of being a mathematician unlike all the others.”

Mandelbrot, describing Levy’s autobiography in Chapter 17

Chapter 18: Wooing and Marrying Aliette, 1955

Mandelbrot pursues love and family life – and does it with the usual cast of exceptional and brilliant people.

Best Quote(s)

“Without her willingness to let me gamble my life—and hers and our children’s—the odd career I undertook would have been unthinkable.”

Mandelbrot, Chapter 19

Mandelbrot’s approach to risk, and his life’s dream, is linked.  He knows that his family took this risk with him, and that without their involvement, he could not have succeeded.

On Citroen, “He tamed front-wheel drive for mass production, and his brilliant engineers rethought every part from scratch so that even some key parts could be duplicated, if needed, in a home garage.”

Mandelbrot, Chapter 19

In a way Citroen was fractal in its approach to manufacturing. The parts were assembled into a vehicle, but then the parts themselves could be easily assembled with ordinary tools.

Chapter 19: In Geneva with Jean Piaget, Mark Kac, and Willy Feller, 1955–57

Mandelbrot was able to assist Piaget’s early work to transform the study of human behavior and human development and also learn from him how to take his own ideas farther.

Best Quote(s)

“… Jean Piaget (1896–1980). He was pleased to hear that I was aware of his fame in trying to bring rationality to child psychology.”

Benoit Mandelbrot, The Fractalist Chapter 19

The number of brilliant minds that Mandelbrot was able to connect with was staggering. Piaget is one of many who was the founder of a major modern field of science.

Piaget, “He promptly changed fields and set out on a lifelong effort to extend proper scientific principles to human behavior.”

Benoit Mandelbrot, The Fractalist, Chapter 19

Before Piaget, the use of data, analytics and rigor was weak in human development. Mandelbrot was to take this same approach to a theory of roughness.

“… telling me that, instead of more papers that looked unrelated, I must write a book.”

Benoit Mandelbrot, The Fractalist, Chapter 19

Mandelbrot put himself near so many people with great skill and was able to position them as coaches and make use of their advice.

Chapter 20: An Underachieving and Restless Maverick Pulls Up Shallow Roots, 1957–58

In this final chapter before heading into Book 3, we read how an older Mandelbrot looks back on his decisions to pursue his goal and the risks that were required to succeed.

Best Quote(s)

Mandelbrot has a dream – he has a goal. He sees that his life to this point is not getting him to this goal. Without this goal – the Keplerian dream – Mandelbrot has no guiding vision. It is the yardstick by which he measures success.

“Unfortunately, my various enterprises up to 1957 had not gone very far to further my aging but still vibrant Keplerian dream.”

Mandelbrot, Chapter 20

Mandelbrot isn’t sure what he should do – but he is increasingly sure that what he is doing is not a fit for his goals in life. Pursuing a career in French academia will block his goal.

“I saw no compatibility between a university position in France and my still-burning wild ambition and dreams.”

Mandelbrot, Chapter 20

Mandelbrot may not know what to do – but he does know clearly what not to do. Such simple steps as having a goal and avoiding bad decisions keep the dream alive and his pursuit a rationale one.

Part 3: Chapters 21 – 29, “My Life’s Fruitful Third Stage”

There is a grace and tact to Mandelbrot’s writing instilled from his upbringing – he has not written a ‘final’ part, simply ‘Part 3.’

Fractals For When Things Are Not Smooth

“What I’m asserting, very strongly, is that when some real thing is found to be unsmooth, the next mathematical model to try is fractal or multifractal. Since roughness is everywhere, fractals are present everywhere.” Location 4460

Dr. Benoit Mandelbrot, Chapter 29, Location 4460

Before Mandelbrot, mathematics and models dealt with a world that was smooth. Interest rates compounded continuously, and graphs were full of straight lines, slopes and curves. But this did not fit the world as it was seen. Mandelbrot’s creation of fractals allowed models and graphics that created images that reflected the roughness shown in nature, finance, and so many other fields.

For a ‘Fractal Life Strategy’, we could say:

  • You know how things behave if the world is smooth.
  • When the world is not smooth, it is rough. And when it is rough, we anticipate things using our knowledge of fractals.

Wild, Mild and Slow

“The three states of chance—wild, mild, and slow—can be compared to the three states of matter. Are not solid and gas separated by liquid? Absolutely. In my view, the same is true of chance—the counterpart of liquid being “slow” randomness.”

Mandelbrot’s The Fractalist, Chapter 22

Any form of enumeration is enlightening – Mandelbrot’s study of fractals established that probabilities can shift between three states – mild, wild and slow.


“I had not a single identifying brand name for my activity. Ten more years went by until I gave up and coined the word “fractal.””

Benoit Mandelbrot, The Fractalist, Chapter 23

Mandelbrot contributed to many different fields, all building from his first work with Zipf’s Law. As he matured, he realized that he needed a name for this field of study – he needed to market his accomplishments. ‘Fractals’ were born.


“Many scientific articles are completely flat because they are written for people who do not have to be convinced.” 4189

Benoit Mandelbrot, The Fractalist, Chapter 24

Mandelbrot succeeded in his Keplerian dream because he was outside of the academic establishment; by being with IBM he could see many problems, and by having a broad focus for his work, he was forced to make his writing more accessible than his peers who had a narrower, more learned, target audience.

Team Building

“The Revolution succeeded because Carnot hired men such as the Corsican Napoléon Bonaparte.”

Mandelbrot, Chapter 21

Mandelbrot joins IBM at a time when the business is going through a massive post World War 2 hiring binge to promote a much needed culture changes. They hire for brilliance, ignoring quirks, and in so doing develop a world class research center. The camaraderie and stability of IBM enable the birth of fractals and fulfillment of Mandelbrot’s Keplerian dream.

Opportunity, Impact

“I had often demonstrated the capacity to formulate big dreams that everyone else held to be odd and unreachable—but that I managed to fulfill.” 4253

Mandelbrot’s The Fractalist, Chapter 27

Mandelbrot had a grand dream – to impact to humanity along the lines of what Kepler brought by studying, calculating, and predicting the structure of the universe. His dreams were big. Others though them unattainable, but he continued to work and was successful in reaching them.

Part 3, Chapter by Chapter Summary – 34 Mandelbrot Quotes

Chapter 21

Mandelbrot would spend the next 35 years employed by an associated with IBM Research. He would serve as a professor at times, but there was always a tie back to industry, giving him flexibility in his pursuits.

Best Quote(s)

IBM was going through a change, and with it they needed to change the kind of people that they were hiring.

“For one thing, relaxed hiring rules brought in many individuals for whom other institutions did not compete: “oddballs,” “wild geese,” scientists whose high-class record was marred by some fault or another or by disputes with faculty advisers.”

Mandelbrot, The Fractalist, Chapter 21

IBM had to change. It had to go through a step change in personnel to create a step change in its own business. It had need of a fractal ‘big change’, which could only occur if it made a series of correct ‘small change’ decisions with its hiring. Mandelbrot again shows examples of fractal activity in his life; if things are smooth – we know how they will behave, but when things are rough, there is a different set of rules.

“The Revolution succeeded because Carnot hired men such as the Corsican Napoléon Bonaparte.”

Mandelbrot, Chapter 21

Looking back Mandelbrot sees that this was the beginning of the ‘Fruitful third stage’ in pursuit of Keplerian dream of pioneering a new field of study. But he at the time the day-to-day activities that were asked of him by IBM didn’t necessarily fit that vision. Again, he shows us a fractal pattern in action – the small steps would come together to create a big shift.

“They appeared at first sight to clash badly—but they really didn’t clash at all. It soon emerged that I was working on the building blocks of my soon-to-be fractal geometry of nature.”

Mandelbrot, The Fractalist, Chapter 21

Chapter 22

My first exposure to Mandelbrot was through his joint work with Richard L. Hudson in The (Mis)Behavior of Markets, which was published in 2004 and stands as a testament to the contributions he made to finance beginning with this work in 1962. Mandelbrot follows the rules that led him through Part 2 of his life, and in doing so is presented with a unique opportunity to extend humanity’s understanding of financial markets.  He takes full advantage of the opportunity, but then draws on the lessons learned in Part 1 to realize that the political inertia of this industry will blunt the impact of his magnificent contribution.  Rather than dwell on this, he collects on the cache and karma of his contributions and continues on his life.  

There are several impressive things that Mandelbrot does not do:

  • He doesn’t sever his ties with IBM to become a financier. 
  • He doesn’t dig his heels in to fight the financial establishment – despite his convictions that he is right.
  • He refines his work on his terms, rather than trying to take the theory to far.  
  • He continues to support the next generation of scientists in this area of finance by working with Eugene Fama and Merton Miller.  Mandelbrot does not shepherd his work, he pushes it out and extends its reach.  
  • He misses career opportunities that would be soul crushing and unrecoverable for many, but instead he plots onward.  
  • Mandelbrot does not get stuck on the nature of his contribution to finance, despite its enormous value.  He reads the situation accurately and moves on.

Best Quote(s)

“MY INVOLVEMENT WITH THE BEHAVIOR of financial prices—absolutely unplanned—became a constant of my scientific life.”

Mandelbrot’s The Fractalist, Chapter 22

Mandelbrot’s path to achieving his Keplerian dream is not smooth – it is rough. Encounters with financial prices and their behavior would forever be a part of his life.

“The three states of chance—wild, mild, and slow—can be compared to the three states of matter. Are not solid and gas separated by liquid? Absolutely. In my view, the same is true of chance—the counterpart of liquid being “slow” randomness.”

Mandelbrot’s The Fractalist, Chapter 22

Enumeration of options is always fascinating, and ‘mild, wild and slow’ are wonderful names.

“We’ve done all we can to make sense of these cotton prices. Everything changes, nothing is constant. This is a mess of the worst kind.”

Mandelbrot’s The Fractalist, Chapter 22

There is great freedom in declaring a situation in hopeless. By declaring things a mess, “of the worst kind” the stage is set for Mandelbrot to attempt an explanation. As he benefited from his unique position with Zipf in his PhD, here Mandelbrot benefits from encountering a situation where there is low risk of embarrassment.

“Fractals—or their later elaboration, multifractals—do not claim to predict the future with certainty. But they do create a more realistic picture of market risks than does observation alone.”

Mandelbrot’s The Fractalist, Chapter 22

Telling the future with 100% accuracy isn’t yet possible – but telling the future a little bit better tomorrow than was done today is a great improvement for society. Finance had previously lived in the world of smooth trajectories – when everyone knew that this was inaccurate. Through his study of roughness, Mandelbrot improved the understanding of market behavior.

“But not for a moment did I forget that to remain stable and vertical, a bicycle must move sufficiently fast.”

Mandelbrot’s The Fractalist, Chapter 22

Bicycles are wonderful, but they are safest when moving. Balance is possible with minimal equipment, provided that motion is part of the equation. Velocity and direction are essential to safe operation with a minimalist approach.

“I was acutely aware that my findings would have devastating consequences for the accepted standard theory of speculation.”

Mandelbrot’s The Fractalist, Chapter 22

In Part 1 we watched Mandelbrot’s family deftly maneuver through World War 2 and survive where others misread the political environment and perished. Mandelbrot has recognized that he has done work of great merit, but also sees that the work may not exist in fertile ground. The finance industry’s inertia and resistance to change may prevent this phase of his Keplerian Dream from becoming the early landslide victory that it might have been.

Chapter 23

There are times when reading a book that every page, every sentence and every word come together to create a smile on the face of the reader. In Part 1 we’ve followed young Mandelbrot’s survival in the face of Nazi persecution and read with awe the names of the intellectuals he dealt with as he ‘drifted’ in Part 2. In Part 3 the tailor draws the thread, and in doing so we see the design pull together in satisfying way.

Each page in this chapter is great, each turn of phrase is memorable, and it is difficult to not envision an older Mandelbrot working on this document enjoying the time to recall his younger years.

Best Quote(s)

“This chapter’s title seems to make no sense. How can it possibly reflect reality?”

Benoit Mandelbrot, The Fractalist, Chapter 23

Earlier in the book it became clear that the author was attempting to take his field of roughness and apply it to this text. Here he says so directly, implying that the path that took him to study roughness was itself fractal in nature. Of course.

“Numerous additional fields I visited also differ deeply yet share a key feature that to me matters more than any other: roughness.”

Benoit Mandelbrot, The Fractalist, Chapter 23

If things are not smooth, then they are rough. The study of roughness, known as Fractals, shows common patterns to behavior and events that are not smooth.

“The fact that my life’s most productive season came late kept me in a constant hurry, and I could rarely take it easy.”

Benoit Mandelbrot, The Fractalist, Chapter 23

From a life strategy standpoint – Dr. Mandelbrot never shies from the fact that he was late to gain the fame that he seemed destined from at an early age. He admits that this tension drove him and shaped his life.

“In fact, a common thread of my work is that values far from the norm are the key to the underlying phenomenon.”

Benoit Mandelbrot, The Fractalist, Chapter 23

In reading Charlie Munger’s Almanack (?) he goes to great pains to justify that his good decisions on ‘big days’ led to his fund performance. Mandelbrot points out that this is true of all things.

“Ultimately, my interests and achievements were viewed in Chicago as absurdly broad, and at Harvard as absurdly narrow! Unfortunately, I had to agree that those opinions were not entirely unreasonable.”

Benoit Mandelbrot, The Fractalist, Chapter 23

Throughout his life – or at least in this autobiographical and elderly reflection – Mandelbrot is comfortable with who he is. He does not force himself into scenarios where he cannot prosper. There is a self confidence in these actions that is crucial to his finding and achieving his Keplerian dream.

“I had not a single identifying brand name for my activity. Ten more years went by until I gave up and coined the word “fractal.””

Benoit Mandelbrot, The Fractalist, Chapter 23

Dr. Mandelbrot – because you haven’t been part of someone else’s laboratory, you really should make up a word and then use that to brand your work.

Chapter 24

Mandelbrot’s home at IBM is secure, and with that security he continues to search about for interesting problems that fit his growing fractal toolkit. He remains worried that he has started his great work too late in life, and with that worry he remains determined to seek out and pursue new problems which his field of fractals can solve.

Best Quote(s)

“It forced me to gather all I had achieved and fit it into an hour. This effort started me on my 1975 book.”

Benoit Mandelbrot, The Fractalist, Chapter 24 (Location 3706)

Mandelbrot accepts the lecture in France, and by doing so he creates a deadline. This forces him to organize his notes, which become the foundation of the book that would make his name known worldwide.

Chapter 25

Mandelbrot was able to take the lessons he learned in financial prices and other wide ranging fields of interest and tie it back to his core area of strength – mathematics. The Mandelbrot set is named. By thinking of fractal math as dimensions defined by fractions, rather than whole numbers, he further extended humanity’s ability to name and study this field.

Best Quote(s)

“And I understood from readings and course material that a field might simply die for lack of manageable and interesting unsolved questions.” 3848

Benoit Mandelbrot, Chapter 25, Location 3848

Mandelbrot thought about fields of study as a social construct, which required the intrigue, interest and commitment of individual scientists to grow and prosper.

“Stephen Jay Gould (1941–2002), a lively paleontologist with multiple appointments at Harvard. Quite independently, we had become two very visible champions of discontinuity—he in paleontology and I in the variation of financial prices.” 3825

Benoit Mandelbrot, Chapter 25, Location 3825

Good products have competitors. Products that don’t have a competitor warrant skepticism in their review. Gould’s framing of discontinuity in the fossil record – known as punctuated equilibrium, explained the fractured lineages of the fossil record by explaining them as bursts of fast moving periods of evolution.

Chapter 26

Coining the word ‘fractal’ has been foreshadowed from the early chapters – such as Mandelbrot’s study of Latin as a youth and his education in the US near pioneers of newly named fields like biochemistry. He compares his writing style and objectives with that of his uncle Szolem, and summarizes his approach towards writing.

Best Quote(s)

“Throughout my life, it had been my principle never to compete frontally with anybody.” 4087

Mandelbrot’s The Fractalist, Chapter 26 (Location 4087)

Mandelbrot is delighted to find a group that will be publishing a collection of artwork, and with that he writes out a life rule that is unique, and likely colored by the violence that surrounded him in his youth.

“Many scientific articles are completely flat because they are written for people who do not have to be convinced.” 4189

Mandelbrot’s The Fractalist, Chapter 26 (Location 4189)

This is a fantastic insight – if the audience is captive, and if the author knows their information is unique and valuable, then what incentive do they have to make reading a joy?

“Whether it is opera or Greek drama, one must know how to enter into a subject quickly because one cannot assume that the audience will wait to understand. One has to be able to speak to people in their style, to motivate and even amuse the reader a little.” 4193

Mandelbrot’s The Fractalist, Chapter 26 (Location 4193)

Enter the subject to quickly – and the reader is not prepared to receive the message. Enter it too slowly, and they are bored and walk away. How wonderful to enter a topic in conversation, where words can flow back and forth and we can reach the topic at the pace needed in order to achieve our goals.

“Szolem responded, “Yes, there are about fifteen people in the world who read everything I write. That is enough. I find that very comforting.”” 4018

Mandelbrot’s The Fractalist, Chapter 26 (Location 4087)

The humor of this statement, with Mandelbrot’s uncle proudly boasting of his 15 loyal readers, compared to the generations of people who now know Mandelbrot and his fractals, is a great contrast to the anxiety that Mandelbrot faced his whole life.

Chapter 27

Mandelbrot’s time at Yale is punctuated with the institution’s highest honor. He observes twice, both in his life and in the goals identified by Yale’s math department that there was a strategy for setting out on goals that were ‘odd and unreachable’, that can lead to great things. Yale hired him based on a desire to be ‘different’, not ‘lesser’.

Best Quote(s)

“THE ART OF RECEIVING new offers and fast promotions has always baffled me, but I have been lucky on a few occasions.” 4200

Benoit Mandelbrot, The Fractalist, Chapter 27, Location 4200

Mandelbrot’s name is now well known and distinct, but his life is full of big offers and promotions that go unfilled. There is a fractal nature to his life – rarely smooth, and often rough. How many autobiographies are smoothed out in the re-telling and remembering of a life?

“I had often demonstrated the capacity to formulate big dreams that everyone else held to be odd and unreachable—but that I managed to fulfill.” 4253

Benoit Mandelbrot, The Fractalist, Chapter 27, Location 4253

Mandelbrot’s self aware comment matches his insight about his new academic home, Yale, earlier in the chapter, “… they had decided to replace “lesser” with “different”—in particular, by expanding less abstract topics.”

Chapter 28

With two chapters left to go, Mandelbrot’s reflections on his life and accomplishments make every passage notable. Picking ‘Best Quotes’ feels like an affront to the pearls of wisdom dispensed but not chosen.

Best Quote(s)

“An important turn in my life occurred when I realized that something I had long been stating in footnotes should be put on the marquee.”

Benoit Mandelbrot, The Fractalist, Chapter 28 (Location 4278)

Mandelbrot didn’t set out to study roughness; he instead found a series of problems across seemingly unconnected fields that focused his mathematical skills on an area that was universally present, but not always acknowledged.

“Before my work on roughness, it was either undefined or measured by too many irrelevant quantities. Now it can be measured by one, two, or a few numbers.”

Benoit Mandelbrot, The Fractalist, Chapter 28 (Location 4289)

I’m reading John Urschel’s Mind and Matter with my son; Urschel played in the NFL and afterwards completed a PhD in math at MIT. At times his statements of fact seem boastful, but they are true. How wonderful a statement from Mandelbrot! Is he boasting? Perhaps, but how could he state the truth and not be boastful.

Mandelbrot’s Keplerian dream inspired him to organize a theory of roughness from work across many fields and make a great contribution in mankind’s understanding of the world around us.

Chapter 29

With only the Afterword left, these are the final words written by Mandelbrot. He begins with a weighty reflection about his 86 years of life, and closes with a wink to the reader, that the ‘fractal’ hints that were sprinkled in the book about his own life were intentionally inserted. Like the other chapters in Part 3, there are a number of outstanding quotes that provide insight to his strategy for life and models for explaining the world around him.

Best Quote(s)

My favorite quote from this chapter may be my favorite overall for the whole book; it is usable in real life. If things aren’t smooth – they are probably rough. There is a model for how rough things work – fractals.

“What I’m asserting, very strongly, is that when some real thing is found to be unsmooth, the next mathematical model to try is fractal or multifractal. Since roughness is everywhere, fractals are present everywhere.” Location 4460

Dr. Benoit Mandelbrot, Chapter 29, Location 4460

How best to explain one concept of fractals – self-similarity – with examples;

“A cauliflower shows how an object can be made of many parts, each of which is like a whole, but smaller. Another example of this repeated roughness is the cloud.” Location 4456

Dr. Benoit Mandelbrot, Chapter 29, Location 4456

If fractals are so new as a field of study, how many other similarly obvious fields await the focus of the right mind?

“Fractal geometry is one of those concepts which at first invites disbelief but on second thought becomes so natural that one wonders why it has only recently been developed.” Location 4422

Dr. Benoit Mandelbrot, Chapter 29, Location 4422

If other obvious fields exist for study – if fractals were waiting for the right circumstances to be unlocked, how was it that our story’s hero, who had fled the Nazis in Part 1, was the person to find them?

“One reason is my personality—I don’t seek power or run around asking for favors. A second is circumstances—I was in an industrial laboratory because academia found me unsuitable.” Location 4460

Dr. Benoit Mandelbrot, Chapter 29, Location 4460

And lastly, a wink and a nod to those who have noted that fractals have surely filled the life of Dr. Mandelbrot:

“Does not the distribution of my personal experiences remind one of the central topic of my scientific work—namely, extreme fractal unevenness? All counted, I have known few minutes of boredom.” Location 4480

Dr. Benoit Mandelbrot, Chapter 29, Location 4480


Mandelbrot achieved a great deal, but felt his whole life that there was much more discovery that he could have given the world. The afterword closes out with the topics he had hoped to touch on, but was unable to develop – negative dimensions and lacunarity, and also provides some closing observations on his life.

Best Quote(s)

“Like fractals, life is better understood as a process than as a result.”


Use processes to understand the world.

“A fractal is defined as well by what has been removed as it is by what remains.”

Benoit Mandelbrot

Absence defines as much as presence.

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1 Response to Benoit Mandelbrot’s The Fractalist: Chapter by Chapter, Page by Page Review of an Excellent Autobiography

  1. Pingback: Book Review: Charles T. Munger’s Poor Charlie’s Almanack: Expanded Third Edition; Introduction and First Chapter | Fred Lybrand

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