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The Pattern Inside the Pattern: Fractals, the Hidden Order Beneath Chaos, and the Story of the Refugee Who Revolutionized the Mathematics of Reality

“In the mind’s eye, a fractal is a way of seeing infinity.”

The Pattern Inside the Pattern: Fractals, the Hidden Order Beneath Chaos, and the Story of the Refugee Who Revolutionized the Mathematics of Reality

I have learned that the lines we draw to contain the infinite end up excluding more than they enfold.

I have learned that most things in life are better and more beautiful not linear but fractal. Love especially.

In a testament to Aldous Huxley’s astute insight that “all great truths are obvious truths but not all obvious truths are great truths,” the polymathic mathematician Benoit Mandelbrot (November 20, 1924–October 14, 2010) observed in his most famous and most quietly radical sentence that “clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line.”

An obvious truth a child could tell you.

A great truth that would throw millennia of science into a fitful frenzy, sprung from a mind that dismantled the mansion of mathematics with an outsider’s tools.

The Mandelbrot set. (Illustration by Wolfgang Beyer.)

A self-described “nomad-by-choice” and “pioneer-by-necessity,” Mandelbrot believed that “the rare scholars who are nomads-by–choice are essential to the intellectual welfare of the settled disciplines.” He lived the proof with his discovery of a patterned order underlying a great many apparent irregularities in nature — a sweeping symmetry of nested self-similarities repeated recursively in what may at first read as chaos.

The revolutionary insight he arrived at while studying cotton prices in 1962 became the unremitting vector of revelation a lifetime long and aimed at infinity, beamed with equal power of illumination at everything from the geometry of broccoli florets and tree branches to the behavior of earthquakes and economic markets.

Fractal Flight by Maria Popova. Available as a print.

Mandelbrot needed a word for his discovery — for this staggering new geometry with its dazzling shapes and its dazzling perturbations of the basic intuitions of the human mind, this elegy for order composed in the new mathematical language of chaos. One winter afternoon in his early fifties, leafing through his son’s Latin dictionary, he paused at fractus — the adjective from the verb frangere, “to break.” Having survived his own early life as a Jewish refugee in Europe by metabolizing languages — his native Lithuanian, then French when his family fled to France, then English as he began his life in science — he recognized immediately the word’s echoes in the English fracture and fraction, concepts that resonated with the nature of his jagged self-replicating geometries. Out of the dead language of classical science he sculpted the vocabulary of a new sensemaking model for the living world. The word fractal was born — binominal and bilingual, both adjective and noun, the same in English and in French — and all the universe was new.

In his essay for artist Katie Holten’s lovely anthology of art and science, About Trees (public library) — trees being perhaps the most tangible and most enchanting manifestation of fractals in nature — the poetic science historian James Gleick reflects on Mandelbrot’s titanic legacy:

Mandelbrot created nothing less than a new geometry, to stand side by side with Euclid’s — a geometry to mirror not the ideal forms of thought but the real complexity of nature. He was a mathematician who was never welcomed into the fraternity… and he pretended that was fine with him… In various incarnations he taught physiology and economics. He was a nonphysicist who won the Wolf Prize in physics. The labels didn’t matter. He turns out to have belonged to the select handful of twentieth century scientists who upended, as if by flipping a switch, the way we see the world we live in.

He was the one who let us appreciate chaos in all its glory, the noisy, the wayward and the freakish, form the very small to the very large. He gave the new field of study he invented a fittingly recondite name: “fractal geometry.”

It was Gleick who, in his epoch-making 1980 book Chaos: The Making of a New Science (public library), did for the notion of fractals what Rachel Carson did for the notion of ecology, embedding it in the popular imagination both as a scientific concept and as a sensemaking mechanism for reality, lush with material for metaphors that now live in every copse of culture.

Illustration from Chaos by James Gleick.

He writes of Mandelbrot’s breakthrough:

Over and over again, the world displays a regular irregularity.


In the mind’s eye, a fractal is a way of seeing infinity.

Imagine a triangle, each of its sides one foot long. Now imagine a certain transformation — a particular, well-defined, easily repeated set of rules. Take the middle one-third of each side and attach a new triangle, identical in shape but one-third the size. The result is a star of David. Instead of three one-foot segments, the outline of this shape is now twelve four-inch segments. Instead of three points, there are six.

As you incline toward infinity and repeat this transformation over and over, adhering smaller and smaller triangles onto smaller and smaller sides, the shape becomes more and more detailed, looking more and more like the contour of an intricate perfect snowflake — but one with astonishing and mesmerizing features: a continuous contour that never intersects itself as its length increases with each recursive addition while the area bounded by it remains almost unchanged.

Plate from Wilson Bentley’s pioneering 19th-century photomicroscopy of snowflakes

If the curve were ironed out into a straight Euclidean line, its vector would reach toward the edge of the universe.

It thrills and troubles the mind to bend itself around this concept. Fractals disquieted even mathematicians. But they described a dizzying array of objects and phenomena in the real world, from clouds to capital to cauliflower.

Against Euclid by Maria Popova. Available as a print.

It took an unusual mind shaped by unusual experience — a common experience navigated by uncommon pathways — to arrive at this strange revolution. Gleick writes:

Benoit Mandelbrot is best understood as a refugee. He was born in Warsaw in 1924 to a Lithuanian Jewish family, his father a clothing wholesaler, his mother a dentist. Alert to geopolitical reality, the family moved to Paris in 1936, drawn in part by the presence of Mandelbrot’s uncle, Szolem Mandelbrojt, a mathematician. When the war came, the family stayed just ahead of the Nazis once again, abandoning everything but a few suitcases and joining the stream of refugees who clogged the roads south from Paris. They finally reached the town of Tulle.

For a while Benoit went around as an apprentice toolmaker, dangerously conspicuous by his height and his educated background. It was a time of unforgettable sights and fears, yet later he recalled little personal hardship, remembering instead the times he was befriended in Tulle and elsewhere by schoolteachers, some of them distinguished scholars, themselves stranded by the war. In all, his schooling was irregular and discontinuous. He claimed never to have learned the alphabet or, more significantly, multiplication tables past the fives. Still, he had a gift.

When Paris was liberated, he took and passed the month-long oral and written admissions examination for École Normale and École Polytechnique, despite his lack of preparation. Among other elements, the test had a vestigial examination in drawing, and Mandelbrot discovered a latent facility for copying the Venus de Milo. On the mathematical sections of the test — exercises in formal algebra and integrated analysis — he managed to hide his lack of training with the help of his geometrical intuition. He had realized that, given an analytic problem, he could almost always think of it in terms of some shape in his mind. Given a shape, he could find ways of transforming it, altering its symmetries, making it more harmonious. Often his transformations led directly to a solution of the analogous problem. In physics and chemistry, where he could not apply geometry, he got poor grades. But in mathematics, questions he could never have answered using proper techniques melted away in the face of his manipulations of shapes.

Benoit Mandelbrot as a teenager. (Photograph courtesy of Aliette Mandelbrot.)

At the heart of Mandelbrot’s mathematical revolution, this exquisite plaything of the mind, is the idea of self-similarity — a fractal curve looks exactly the same as you zoom all the way out and all the way in, across all available scales of magnification. Gleick describes the nested recursion of self-similarity as “symmetry across scale,” “pattern inside of a pattern.” In his altogether splendid Chaos, he goes on to elucidate how the Mandelbrot set, considered by many the most complex object in mathematics, became “a kind of public emblem for chaos,” confounding our most elemental ideas about simplicity and complexity, and sculpting from that pliant confusion a whole new model of the world.

Couple with the story of the Hungarian teenager who bent Euclid and equipped Einstein with the building blocks of relativity, then revisit Gleick on time travel and his beautiful reading of and reflection on Elizabeth Bishop’s ode to the nature of knowledge.


Figures of Thought: Krista Tippett Reads Howard Nemerov’s Mathematical-Existential Poem About the Interconnectedness of the Universe

A splendid song of praise for the elemental truth at the heart of all art, science, and nature.

Figures of Thought: Krista Tippett Reads Howard Nemerov’s Mathematical-Existential Poem About the Interconnectedness of the Universe

“A leaf of grass is no less than the journey work of the stars,” Walt Whitman wrote in one of his most beautiful poems in the middle of the nineteenth century, just as humanity was coming awake to the glorious interconnectedness of nature — to the awareness, in the immortal words of the great naturalist John Muir, that “when we try to pick out anything by itself, we find it hitched to everything else in the universe.”

A century later, Albert Einstein recounted his takeaway from the childhood epiphany that made him want to be a scientist: “Something deeply hidden had to be behind things.” Virginia Woolf, in her account of the epiphany in which she understood she was an artist — one of the most beautiful and penetrating passages in all of literature — articulated a kindred sentiment: “Behind the cotton wool is hidden a pattern… the whole world is a work of art… there is no Shakespeare… no Beethoven… no God; we are the words; we are the music; we are the thing itself.”

This interleaved thing-itselfness of existence, hidden in plain sight, is what two-time U.S. Poet Laureate Howard Nemerov (February 29, 1920–July 5, 1991) takes up, two centuries after William Blake saw the universe in a grain of sand, in a spare masterpiece of image and insight, found in his altogether wondrous Collected Poems (public library), winner of both the Pulitzer Prize and the National Book Award.

Howard Nemerov

On Being creator and Becoming Wise author Krista Tippett brought the poem to life at the third annual Universe in Verse, with a lovely prefatory meditation on the role of poetry — ancient, somehow forgotten in our culture, newly rediscovered — as sustenance and salve for the tenderest, truest, most vital parts of our being.

by Howard Nemerov

To lay the logarithmic spiral on
Sea-shell and leaf alike, and see it fit,
To watch the same idea work itself out
In the fighter pilot’s steepening, tightening turn
Onto his target, setting up the kill,
And in the flight of certain wall-eyed bugs
Who cannot see to fly straight into death
But have to cast their sidelong glance at it
And come but cranking to the candle’s flame —

How secret that is, and how privileged
One feels to find the same necessity
Ciphered in forms diverse and otherwise
Without kinship — that is the beautiful
In Nature as in art, not obvious,
Not inaccessible, but just between.

It may diminish some our dry delight
To wonder if everything we are and do
Lies subject to some little law like that;
Hidden in nature, but not deeply so.

For more science-celebrating splendor from The Universe in Verse, savor astrophysicist Janna Levin reading “A Brave and Startling Truth” by Maya Angelou and “Planetarium” by Adrienne Rich; poet Sarah Kay reading from “Song of Myself” by Walt Whitman; Regina Spektor reading “Theories of Everything” by the astronomer and poet Rebecca Elson; Amanda Palmer reading “Hubble Photographs: After Sappho” by Adrienne Rich; and Neil Gaiman’s original tributes-in-verse to women in science, environmental founding mother Rachel Carson, and astronomer Arthur Eddington, who confirmed Einstein’s relativity in the wake of a World War that had lost sight of our shared belonging and common cosmic spring.


A Pioneering Case for the Value of Citizen Science from the Polymathic Astronomer John Herschel

“There is scarcely any well-informed person, who, if he has but the will, has not also the power to add something essential to the general stock of knowledge.”

A Pioneering Case for the Value of Citizen Science from the Polymathic Astronomer John Herschel

“It is always difficult to teach the man of the people that natural phenomena belong as much to him as to scientific people,” the trailblazing astronomer Maria Mitchell wrote as she led the first-ever professional female eclipse expedition in 1878. The sentiment presages the importance of what we today call “citizen science,” radical and countercultural in an era when science was enshrined in the pompous pantheon of the academy, whose gates were shut and padlocked to “the man of the people,” to women, and to all but privileged white men.

Two decades earlier, Mitchell had traveled to Europe as America’s first true scientific celebrity to meet, among other dignitaries of the Old World, one such man — but one of far-reaching vision and kindness, who used his privilege to broaden the spectrum of possibility for the less privileged: the polymathic astronomer John Herschel (March 7, 1792–May 11, 1871), co-founder of the venerable Royal Astronomical Society, son of Uranus discoverer William Herschel, and nephew of Caroline Herschel, the world’s first professional woman astronomer, who had introduced him to astronomy as a boy.

Several years before he coined the word photography, Herschel became the first prominent scientist to argue in a public forum that the lifeblood of science — data collection and the systematic observation of natural phenomena — should be the welcome task of ordinary people from all walks of life, united by a passionate curiosity about how the universe works.

John Herschel (artist unknown)

In 1831, the newly knighted Herschel published A Preliminary Discourse on the Study of Natural Philosophy as part of the fourteenth volume of the bestselling Lardner’s Cabinet Cyclopædia (large chunks of which were composed by Frankenstein author Mary Shelley). Later cited in Lorraine Daston and Elizabeth Lunbeck’s altogether excellent book Histories of Scientific Observation (public library), it was a visionary work, outlining the methods of scientific investigation by clarifying the relationship between theory and observation. But perhaps its most visionary aspect was Herschel’s insistence that observation should be a network triumph belonging to all of humanity — a pioneering case for the value of citizen science. He writes:

To avail ourselves as far as possible of the advantages which a division of labour may afford for the collection of facts, by the industry and activity which the general diffusion of information, in the present age, brings into exercise, is an object of great importance. There is scarcely any well-informed person, who, if he has but the will, has not also the power to add something essential to the general stock of knowledge, if he will only observe regularly and methodically some particular class of facts which may most excite his attention, or which his situation may best enable him to study with effect.

Diversity of snowflake shapes from a 19th-century French science textbook. Available as a print.

Pointing to meteorology and geology as the sciences best poised to benefit from distributed data collection by citizen scientists, Herschel adds:

There is no branch of science whatever in which, at least, if useful and sensible queries were distinctly proposed, an immense mass of valuable information might not be collected from those who, in their various lines of life, at home or abroad, stationary or in travel, would gladly avail themselves of opportunities of being useful.

Herschel goes on to outline the process by which such citizen science would be conducted: “skeleton forms” of survey questions circulated far and wide, asking “distinct and pertinent questions, admitting of short and definite answers,” then transmitted to “a common centre” for processing — a sort of human internet feeding into a paper-stack server. (Lest we forget, Maria Mitchell herself was employed as a “computer” — the term we used to use for the humans who performed the work now performed by machines we have named after them.)

Couple with a wonderful 1957 treatise on the art of observation and why genius lies in the selection of what is worth observing, then revisit Maria Mitchell on how to find your calling.


Pioneering Mathematician G.H. Hardy on How to Find Your Purpose and What Is Most Worth Aspiring for

“If a man has any genuine talent he should be ready to make almost any sacrifice in order to cultivate it to the full.”

Pioneering Mathematician G.H. Hardy on How to Find Your Purpose and What Is Most Worth Aspiring for

“Resign yourself to the lifelong sadness that comes from never ­being satisfied,” Zadie Smith counseled in the tenth of her ten rules of writing — a tenet that applies with equally devastating precision to every realm of creative endeavor, be it poetry or mathematics. Bertrand Russell addressed this Faustian bargain of ambition in his 1950 Nobel Prize acceptance speech about the four desires motivating all human behavior: “Man differs from other animals in one very important respect, and that is that he has some desires which are, so to speak, infinite, which can never be fully gratified, and which would keep him restless even in Paradise. The boa constrictor, when he has had an adequate meal, goes to sleep, and does not wake until he needs another meal. Human beings, for the most part, are not like this.”

Ten years earlier, the English mathematician and number theory pioneer G.H. Hardy (February 7, 1877–December 1, 1947) — an admirer of Russell’s — examined the nature of this elemental human restlessness in his altogether fascinating 1940 book-length essay A Mathematician’s Apology (public library).

G.H. Hardy

In considering the value of mathematics as a field of study and “the proper justification of a mathematician’s life,” Hardy offers a broader meditation on how we find our sense of purpose and arrive at our vocation. Addressing “readers who are full, or have in the past been full, of a proper spirit of ambition,” Hardy writes in an era when every woman was colloquially “man”:

A man who is always asking “Is what I do worth while?” and “Am I the right person to do it?” will always be ineffective himself and a discouragement to others. He must shut his eyes a little and think a little more of his subject and himself than they deserve. This is not too difficult: it is harder not to make his subject and himself ridiculous by shutting his eyes too tightly.


A man who sets out to justify his existence and his activities has to distinguish two different questions. The first is whether the work which he does is worth doing; and the second is why he does it, whatever its value may be. The first question is often very difficult, and the answer very discouraging, but most people will find the second easy enough even then. Their answers, if they are honest, will usually take one or other of two forms; and the second form is a merely a humbler variation of the first, which is the only answer we need consider seriously.

Most people, Hardy argues, answer the first question by pointing to a natural aptitude that led them to a vocation predicated on that particular aptitude — the lawyer became a lawyer because she naturally excels at eloquent counter-argument, the cricketer a cricketer because he has a natural gift for cricket. In what may sound like an ungenerous sentiment but is indeed statistically accurate, Hardy adds:

I am not suggesting that this is a defence which can be made by most people, since most people can do nothing at all well. But it is impregnable when it can be made without absurdity, as it can by a substantial minority: perhaps five or even ten percent of men can do something rather well. It is a tiny minority who can do something really well, and the number of men who can do two things well is negligible. If a man has any genuine talent he should be ready to make almost any sacrifice in order to cultivate it to the full.

Illustration by artist Hugh Lieber from Human Values and Science, Art and Mathematics by mathematician Lillian Lieber

But while talent exists in varying degrees within each field of endeavor, Hardy notes that the fields themselves occupy a hierarchy of value — different activities offer different degrees of benefit to society. And yet most people, he argues, choose their occupation not on the basis of its absolute value but on the basis of their greatest natural aptitude relative to their other abilities. (Not to do so, after all, renders one the faintly smoking chimney in Van Gogh’s famous lament about unrealized talent: “Someone has a great fire in his soul and nobody ever comes to warm themselves at it, and passers-by see nothing but a little smoke at the top of the chimney.”) Hardy writes:

I would rather be a novelist or a painter than a statesman of similar rank; and there are many roads to fame which most of us would reject as actively pernicious. Yet it is seldom that such differences of value will turn the scale in a man’s choice of a career, which will almost always be dictated by the limitations of his natural abilities. Poetry is more valuable than cricket, but [the champion cricketer Don] Bradman [whose test batting average is considered the greatest achievement of any sportsman] would be a fool if he sacrificed his cricket in order to write second-rate minor poetry (and I suppose that it is unlikely that he could do better). If the cricket were a little less supreme, and the poetry better, then the choice might be more difficult… It is fortunate that such dilemmas are so seldom.

Presaging the ominous twenty-first-century trend of talented mathematicians and physicists swallowed by Silicon Valley for lucrative jobs ranging from the uninspired to the downright pernicious, Hardy adds:

If a man is in any sense a real mathematician, then it is a hundred to one that his mathematics will be far better than anything else he can do, and that he would be silly if he surrendered any decent opportunity of exercising his one talent in order to do undistinguished work in other fields. Such a sacrifice could be justified only by economic necessity or age.


Every young mathematician of real talent whom I have known has been faithful to mathematics, and not from lack of ambition but from abundance of it; they have all recognized that there, if anywhere, lay the road to a life of any distinction.

Ambition, he argues, has been the motive force behind nearly everything we value as a civilization — every significant breakthrough in art and science, “all substantial contributions to human happiness.” (George Orwell, too, pointed to personal ambition as the first of the four universal motives of great writers.) But while various ambitions can possess us, ranging from the vain and greedy to the most elevated and idealistic, Hardy points to one as the crowning achievement of the purposeful life:

Ambition is a noble passion which may legitimately take many forms… but the noblest ambition is that of leaving behind something of permanent value.

In the remainder of A Mathematician’s Apology, Hardy goes on to explore the particular aspects of mathematics that make it a pursuit of permanent value. Complement this particular portion with Dostoyevsky on the difference between artistic ambition and the ego, David Foster Wallace on the double-edged sword of ambition, and Georgia O’Keeffe on setting priorities for success.


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