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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 descirbes 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.


We Are Water Protectors: An Illustrated Celebration of Nature, Native Heritage, and the Courage to Stand Up for Earth

An inspired signal from that sacred place where the spirit of wakeful action meets the bone of ancient wisdom.

We Are Water Protectors: An Illustrated Celebration of Nature, Native Heritage, and the Courage to Stand Up for Earth

“Every story is a story of water,” Native American poet Natalie Diaz wrote in her stunning ode to her heritage, the language of the Earth, and the erasures of history.

We ourselves are a story of water — biologically and culturally, in our most elemental materiality and our mightiest metaphors.

From author Carole Lindstrom, member of the Turtle Mountain Band of Ojibwe, and artist Michaela Goade, member of the Central Council of the Tlingit a Haida Indian Tribes of Alaska, comes We Are Water Protectors (public library) — a lyrical illustrated celebration of cultural heritage and the courage to stand up for nature.

Inspired by the landmark locus of courage and resistance at Standing Rock — the 2016 movement that magnetized people from more than five hundred indigenous nations and thousands of allies to take a stance against the Dakota Access Pipeline, against its concrete assault on a particular piece of land and against its general symbolism as ominous emblem of extractionism — the book invites young people to cast themselves as agents of change and stewards of the natural world.

In the author’s afterword, Lindstrom explains that in Ojibwe culture, women are considered the protectors of the water and men of the fire. In her tradition, there is a prophecy that paints two possible roads from the present to the future: One is the natural path, embracing the sacred relationship between human beings and the rest of the natural world long before biologists and ecologists discovered that “we are human only in contact, and conviviality, with what is not human”; the other is a path of unnatural acceleration, propelled by greed and mindless technological frenzy.

In the prophecy, this second path is strewn with black snakes — a symbolic image ominously reflected in the actuality of the oil pipelines that cross-hatch Native lands with their grim message of turning nature from a source of life-wide vitality and reverence to a resource for human need and greed.

Goade — who grew up in the coastal rainforests of Alaska, with an embodied awareness of the intricate relationship between water and life, and is the first Native artist to earn the Caldecott Medal, the Nobel Prize of children’s book illustration — amplifies the story’s message with her vibrant artwork drawing on motifs from Native folklore and mythology.

Radiating from the spirit of the story is a reflection of the touching message to the next generations, with which Rachel Carson said her farewell to life after awakening the ecological conscience of the American mainstream:

Yours is a grave and sobering responsibility, but it is also a shining opportunity. You go out into a world where mankind is challenged, as it has never been challenged before, to prove its maturity and its mastery — not of nature, but of itself.

Complement We Are Water Protectors with the great Scottish mountaineer and poet Nan Shepherd on the might and mystery of water and a stunning animated poem about our connection to Earth and to each other, then revisit another soulful illustrated celebration of nature in The Blue Hour.

Illustrations courtesy of Roaring Brook Press; photographs by Maria Popova


A Scientist’s Advice on Healing: A Soulful Animated Poem About Getting to the Other Side of Heartbreak

“Try to accept this fat red hurt is your starting point.”

A Scientist’s Advice on Healing: A Soulful Animated Poem About Getting to the Other Side of Heartbreak

“Love your heart. For this is the prize,” Toni Morrison wrote in an exquisite passage from Beloved as she considered the body as an instrument of sanity, joy, and self-respect a century after William James asserted in his groundbreaking work on how our bodies affect our feelings that “a purely disembodied human emotion is a nonentity,” lending the fledgling credibility of a young science to Walt Whitman’s poetic insistence that “the body includes and is the meaning, the main concern and includes and is the soul.”

There is such fertile ground for sensemaking in this space between biology and metaphor that we have always used our bodies as sensemaking instruments for the soul. But no part of the body has taken on more metaphorical meaning than the vital organ depicted in millennia of literature and song as the seat of love.

The Human Heart. One of French artist Paul Sougy’s mid-century scientific diagrams of life. (Available as a print and as a face mask.)

When we speak of the heart breaking, we are speaking metaphorically, and yet anyone who has lived through heartbreak — that is, anyone who has lived at all — knows intimately the awful way in which the psychological condition of loss takes on the quality of physical pain. It is hardly surprising, then, that the body and the soul heal in consanguinity — the heart-as-metaphor heals the same way the heart-as-organ does.

That is what English poet Christy Ducker explores with uncommon sensitivity and lyric splendor in “A Scientist’s Advice on Healing.” A fine poet and a fine scholar who earned her Ph.D. while composing poems about the Victorian lighthouse keeper Grace Darling, Ducker embodies the animating spirit of The Universe in Verse and stands as a testament to Ursula K. Le Guin’s lovely insistence that “science describes accurately from outside, poetry describes accurately from inside, [and] both celebrate what they describe.”

In this enchanting animated poem, Ducker joins visions with artist Kate Sweeney to deliver a soulful prescription partway between science and metaphor, between organ and instrument, as palliating to the physiology of illness as it is to the psychology of heartbreak:

by Christy Ducker

Try to accept
this fat red hurt
is your starting point,
in the way a pen must be put to paper
     in one particular spot,

then move

the globby flap
of blame
     and past
          the mono-sulk
               of pain.

Change the subject,
before it’s too late.
Sketch out
what health
you do possess,
what signal-cascades,
what flotilla of cells
circumnavigate you,

then draw yourself back
     together again,
in a language
     of your own.

Your body’s talk
is loose as lymph —
it’ll have you open out
     as a tree,
or sneak up on pain
     as assassin,
     or wolf.

Encourage this
for healing won’t come at you
Embrace the lack of heroics —
this isn’t Hollywood,
it’s you,
in a plot
that may
or may not resolve.

The poem appears in Messenger (UK edition) — a slim collection of Ducker’s poems exploring “how we wound and how we heal,” drawing on the science of immunology in a collaboration with York’s Center for Chronic Disease, and featuring visual poetics by Sweeney, who also animated poet Linda France’s magnificent “Murmuration.”

Couple with “Antidotes to Fear of Death” — astronomer and poet Rebecca Elson’s stunning cosmic salve for our creaturely tremblings of heart — then revisit Epictetus’s 2,000-year-old Stoic strategy for surviving heartbreak.


Road to Survival: Empowering Wisdom the Forgotten Book That Shaped the Modern Environmental Movement

“If we ourselves do not govern our destiny, firmly and courageously, no one is going to do it for us.”

Road to Survival: Empowering Wisdom the Forgotten Book That Shaped the Modern Environmental Movement

A century after the trailblazing conservationist John Muir observed that “when we try to pick out anything by itself, we find it hitched to everything else in the universe,” and half a century before Maya Angelou urged us in her cosmic clarion call to see that “we, this people, on this minuscule and kithless globe” must correct our course before we destroy ourselves and our kith-hitched globe, the ecologist and conservationist William Vogt (May 15, 1902–July 11, 1968) composed a masterwork of admonition and actionable vision that shaped the ethos of the modern environmental movement and emboldened the generation of pioneers who set it into motion.

During his long recovery from paralytic polio as a child, William had fallen in love with the natural world, escaping his confinement through books about the wilderness and its wondrous creatures. A century before the Oxford Children’s Dictionary discarded dozens of words related to nature as irrelevant to the imagination and its prosthesis in language, the small bedridden boy grew especially enchanted by the feathered creatures of free flight that he met on the page.

Art by Jackie Morris from The Lost Words by Robert Macfarlane

Literature remained his portal into life as he set out on a literary career in New York. But nature beckoned, the birds beckoned.

At thirty, Vogt abruptly left the city for Latin America, traveling on a Peruvian commission to study bird populations on the guano islands. In the three years he spent there, laboring in the salty air on the hot barren rocks, he arrived at an empirical proof of Muir’s insight — Vogt discovered that when changes in ocean currents diminished the population of plankton and anchovies, millions of birds fled the islands in search of food, leaving their defenseless chicks to die in “pitiful, collapsed, downy clumps” that broke his heart.

He saw in this heartbreak a miniature of the whole — the intricate interlacing of creaturely destinies on a planet of finite resources, growing populations, and infinite interconnectedness of needs.

Art from If You Come to Earth by Sophie Blackall

Vogt spent the next decade and a half developing these ideas by devouring countless books and scientific papers, talking with scientists and sea captains, farmers and diplomats, presidents and sheepherders in Patagonia, engineers and trappers in Manitoba. In 1948 — the year the great nature writer Henry Beston cast his poetic hope that “perhaps there is still time to take a stand for the Kingdom of Life [which] needs defenders” and that “perhaps, mighty as its enemies may be, allies will come who are even mightier” — Vogt distilled his learnings in Road to Survival (public library), a mighty manifesto that roused a generation of allies to stand for the Kingdom of Life.

At the crux of his inquiry into the key motive forces of our civilization’s course is Vogt’s insistence that while much is broken with the exploitive consumerism behind the world’s government and industry, much more depends on and is mendable by the concerted collective action of ordinary people. In a humbling testament to the slow work of cultural change and civilizational course-correction, it would take nearly a century for his ideas to come alive and actionable in such pillars of ecological responsibility as the Paris Agreement, the Green New Deal, and Greta Thunberg’s global school climate strike movement.

William Vogt

In an era when humanity regarded the rest of nature as a world parallel to our own — a world at best visited in pleasant excursions, at worst reduced to extractable resources to serve human needs — Vogt insists that “ecological health is one of the indispensables” in the “flowering of human happiness and well-being.” A century after Ernst Haeckel coined the term ecology and a decade before Rachel Carson made it a household word, Vogt draws on his island discovery to contour the totality of Earth’s ecological interdependence:

Drastic measures are inescapable. Above everything else, we must reorganize our thinking. If we are to escape the crash we must abandon all thought of living unto ourselves. We form an earth-company, and the lot of the Indiana farmer can no longer be isolated from that of the Bantu… An eroding hillside in Mexico or Yugoslavia affects the living standard and probability of survival of the American people… Today’s white bread may force a break in the levees, and flood New Orleans next spring. This year’s wheat from Australia’s eroding slopes may flare into a Japanese war three decades hence.

Art from If You Come to Earth by Sophie Blackall

Rising from the wakeful pages is the assurance, at once empowering and disquieting with its weight of responsibility, that “we the people” is not some agglomeration of abstract others but the sum total of each and every one of us as individual and concrete agents of change.

In a chapter aptly titled “History of Our Future,” Vogt writes:

When I write “we” I do not mean the other fellow. I mean every person who reads a newspaper printed on pulp from vanishing forests. I mean every man and woman who eats a meal drawn from steadily shrinking lands. Everyone who flushes a toilet, and thereby pollutes a river, wastes fertile organic matter and helps to lower a water table. Everyone who puts on a wool garment derived from overgrazed ranges that have been cut by the little hoofs and gullied by the rains, sending runoff and topsoil into the rivers downstream, flooding cities hundreds of miles away.

Writing shortly before he became National Director of Planned Parenthood — a position he would hold for more than a decade in parallel with his conservation work — Vogt adds:

Especially do I mean men and women in overpopulated countries who produce excessive numbers of children who, unhappily, cannot escape their fate as hostages to the forces of misery and disaster that lower upon the horizon of our future.


The direction of these curves and the misery they write across the earth are not likely to be changed in the proximate future. Their direction is fixed for some decades. Great masses of people have a preponderantly young population; as they come into the breeding age we must, despite all possible efforts short of generalized slaughter, expect human numbers to increase for a time. The drag imposed by ignorance, selfishness, nationalism, custom, etc., is certain to retard, by some decades, any effective or substantial improvement of resource management.

In an admonition of astonishing prescience, Vogt insists that changing the trend requires revising our past ideals to calibrate them to an evolving ecological reality:

The freebooting, rugged individualist, whose vigor, imagination, and courage contributed so much of good to the building of our country (along with the bad), we must now recognize, where his activities destroy resources, as the Enemy of the People he has become… Above all, we must learn to know — to feel to the core of our beings — our dependence upon the earth and the riches with which it sustains us. We can no longer believe valid our assumption that we live in independence.

Art by Aurélia Fronty from Wangari Maathai: The Woman Who Planted Millions of Trees by Franck Prévot

Writing shortly before Rachel Carson penned her prescient open letter to America, insisting that “the real wealth of the Nation lies in the resources of the earth — soil, water, forests, minerals, and wildlife,” Vogt calls for long-term thinking and broad-range feeling:

We must — all of us, men, women, and children — reorient ourselves with relation to the world in which we live… We must come to understand our past, our history, in terms of the soil and water and forests and grasses that have made it what it is. We must see the years to come in the frame that makes space and time one.

In a world he considered already overcrowded and resource-strained by the two billion people inhabiting it — a population that would triple within a generation — Vogt adds:

As we are crowded together… on the shrinking surface of the globe, we have set in motion historical forces that are directed by our total environment.

Art by Ping Zhu from The Snail with the Right Heart

Half a century after Alfred Russel Wallace issued his prophetic prescription for civilizational course-correction — an admonition that fell on deaf ears at the outset of the twentieth century and would remain a point of willful deafness in the climate denial of the twenty-first — Vogt makes an impassioned and empowering appeal to the only substantive force of change in the grand scheme of any society and civilization:

If we ourselves do not govern our destiny, firmly and courageously, no one is going to do it for us. To regain ecological freedom for our civilization will be a heavy task. It will frequently require arduous and uncomfortable measures. It will cost considerable sums of money. Democratic governments are not likely to set forth on such a steep and rocky path unless people lead the way.


So that the people shall not delude themselves, find further frustration through quack nostrums, fight their way into blind alleys, it is imperative that this world-wide dilemma be made known to all mankind. The human race is caught in a situation as concrete as a pair of shoes two sizes too small. We must understand that, and stop blaming economic systems, the weather, bad luck, or callous saints. This is the beginning of wisdom, and the first step on the long road back.

For a contemporary counterpart to Vogt’s Road to Survival, see the excellent anthology of policy, poetry, and science All We Can Save: Truth, Courage, and Solutions for the Climate Crisis, shining a kaleidoscope of actionable perspective by women leading the way on the road Vogt envisioned, then revisit Eve Ensler’s extraordinary letter of apology to Mother Earth.


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