The Marginalian
The Marginalian

The Courage of Vulnerability: Teenage Frida Kahlo’s Moving Letters to Her First Love

By Maria Popova

The Courage of Vulnerability: Teenage Frida Kahlo’s Moving Letters to Her First Love

One of the 35 girls among the 2,000 students at Mexico’s National Preparatory School, Frida Kahlo (July 6, 1907–July 13, 1954) was fifteen when she met Alejandro Gómez Arias. Both were passionate and erudite, both were members of the anarchist student group known as Los Cachuchas for the pointed cloth caps they wore in defiance of the era’s restrictive dress code, both became each other’s first love. Alejandro was on the bus with Frida that fateful late-summer day shortly after her eighteenth birthday when a tram collision killed several other passengers and left her so severely injured — her pelvis fractured, her stomach and uterus punctured by a rail, her spine broken in three places and her leg in eleven — that the doctors at the Red Cross Hospital did not think she could be saved. It was Alejandro’s unrelenting insistence that made them try. Against all odds, Frida lived — but her life was irrevocably changed. How she coped with what she had to live through in turn changed the history of art.

Her letters to Alejandro, collected in the altogether stirring volume Frida Kahlo: Love Letters (public library) edited by Suzanne Barbezat, offer a rare glimpse of her becoming — as an artist, as a lover, as a person who lived with extraordinary vulnerability, extraordinary courage, and the precocious awareness that the conversation between the two is the measure of a life.

From the outset, her letters command and caress at the same time. “Write to me often and long, the longer the better,” she urges him in one. “On Saturday I’ll bring your sweater, your books and a lot of violets,” she tells him in another. She takes love as seriously as it ought to be taken but also knows it dies without play: “Sorry about constantly repeating the word ‘love’ five times in a row, but it’s just that I’m very silly.” She signs herself “your pretty girl (monkey face),” “your girl, buddy, woman or whatever you like,” “your sister (girlfriend, buddy, wife).” (It starts so early, that trembling gamble of the heart by which a person tries to discern what they mean to another.) Over and over, she offers glimpses into her uncommon inner world. In a letter penned the summer she turned seventeen, after some arrangements for how they can see each other — Frida’s parents disapproved of the relationship — she writes:

Now I’m going to read Salambo until half past 10, it’s 8 o’clock now, and then the Bible in three volumes and, finally, think for a while about huge scientific problems and then go to bed, and sleep until half past 7 in the morning, eh? Until tomorrow, may we have a good night and may we both think that great friends must love each other very, very much, much, much, much, much, mucho . . . with “m” for music or for “mundo.”

A month later, she offers that lovely unasked assurance that makes a fragile young love feel safe and solid:

My Alex, since I won’t see you for two days and I miss you so much, I’m writing you this so that you will start to believe something that you don’t believe, but which is very true.

And then, beneath a drawing, she adds:

Please forgive me for not writing any more but I started to draw the doll at 9 and it took me an astronomical three quarters of an hour to draw and another half hour to write, so it’s about 10 now and you know that makes me sleepy like the hens, but I’ll keep writing this letter in my dreams and you know that I would write enough to fill at least a thousand pages.

I love you very much.

Your pretty girl (monkey face)

On Christmas Day, she tells him:

My Alex: I loved you since I first saw you. What do you say to that? Since we probably won’t see each other for several days, I’m going to beg you not to forget your little woman, eh?

[…]

You must like easy things… I would like to be even easier, a tiny little thing that you could just carry in your pocket always, always… Alex, write to me often and even if it’s not true, tell me that you care for me a lot and that you can’t live without me…

Your girl, buddy, woman or whatever you like
Frieda

Punctuating the teenage ardor is the stuff of life — she tells him about taking classes in shorthand and typing so as not to waste money on paying the telegraph operator, tells him about applying for a job at the Education Library for four pesos an hour, tells him about her material and domestic struggles, but always places him above all else. When he gets sick, she writes to him:

Right now the only thing I want is for you to get better and all the rest is in 5th and 6th place, because in 1st to 4th place is that you get better and that you love me… Get better very, very soon and think about me a little bit, that’s what your sister (girlfriend, buddy, wife) wants.

She couldn’t have known, in comforting him through his minor ailment, that only a few months later her own embodiment would be pushed to the brink of mortality. Twenty-five days after the accident, bedridden at the hospital where her mother had only visited her twice and her father once, she writes in a letter adorned with a drawing of skull and bones:

Alex of my life: You know better than anyone how sad I have been in this filthy hospital… Everyone tells me not to despair; but they don’t know what it is for me to be bedridden for three months, which is what I need to be, after having been a first-class stray cat all my life, but what’s there to do, since la pelona didn’t carry me away. Don’t you think?… The day I see you Alex, I’m going to kiss you, there’s no help for it; now I see more than ever how I love you with all my soul and I won’t trade you for anyone; you see how suffering something is always worthwhile.

On the eve of her discharge, she writes:

Here or there, I’ll be waiting for you. I’m counting the hours as I wait for you wherever, here or at home, because seeing you, the months in bed will pass much faster… Life begins tomorrow…! — I adore you —

But rather than revival, she entered a long convalescence, confined to bed and savaged by pain in every region of her body as both of her parents fell seriously ill. Six weeks into her confinement, just after her mother had a seizure, she writes to Alejandro:

I want you to come see me because I’m in over my head and I can’t help but hold on, because it would be worse if I despaired, don’t you think? I want you to come and talk to me like before, to forget everything and to come see me for the love of your holy mother and to tell me that you love me even if it’s not true, ok? (The pen doesn’t write very well with so many tears.)

Alejandro remained by her side for more than a year into her convalescence, then left for Europe in the early spring of 1927. In her passionate dispatches, she never minimized her pain, but she never let it dominate her stubborn will for life.

Self Portrait with Velvet Dress, 1926.

Four months into their separation, having just completed one of her tenderest self-portraits, she writes:

My Alex: I still can’t tell you I’m doing better, but nevertheless I feel much happier than before, I have so much hope of getting better by the time you return that you shouldn’t be sad on my account for a single moment. I almost never lose hope now… There is no reason for you to suffer for me, everything I tell you in my letters is because I’m such a “cry-baby” and at the end just a young girl, but it is not that much, it is fine to suffer a little, don’t you think, my Alex?… You are coming back, what more could I ask for? You can’t imagine how marvelous it is to wait for you with the same serenity as the portrait… Write to me a little bit more, your letters really heal me.

Two weeks later, amidst worries about having enough money for another X-ray, she writes:

You can’t imagine with what pleasure I would give all my life just to kiss you. I think this time I have really suffered, so I must deserve it.

[…]

Your Frieda
(I adore you)

Seven months into Alejandro’s absence, she names the terror of abandonment trembling in every lover’s heart even in the closest proximity, for between two people there is always an ocean in which to meet or drown:

Life is ahead of us… In Coyoacán the nights are amazing… and the sea, a symbol in my portrait, synthesizes life, my life.

You haven’t forgotten me?

It would almost be unfair, don’t you think?

She had first voiced this fear a season earlier, writing to him at the peak of summer:

Alex: I’m going to confess one thing: there are moments that I think you’re forgetting me, but you aren’t, right? You couldn’t fall in love with the Mona Lisa.

But he did. Alejandro broke off the relationship shortly after returning to Mexico that autumn. Frida may have intuited it, but she was not prepared, the way we never really are even for the blows we feel coming. Barely twenty, her body shattered and her heart broken, she found herself reeling with that most difficult, most eternal question: Where does love go when it goes?

It went where it always goes — into the totality of her person. We make everything we make with everything we are, everything we have touched that has touched us back in that tender and terrifying contact with life we call experience.

Portrait of Alejandro Gómez Arias, 1928.

Several months later, Frida completed a portrait of Alejandro looking plaintive, almost fragile, and inscribed it at the top:

Alex, with affection I painted your portrait, that he is one of my comrades forever, Frida Kahlo, 30 years later.

Frida did not live another thirty years. But this young love that had shaped her life, possibly saved it, pulsates beneath every painting she ever painted to tell the centuries what it is like to be alive, with all the pain and passion of it — an inextinguishable reminder that every love we have ever loved, every loss we have ever suffered, becomes part of us, part of what we have to give; for, in the end, how we love, how we give, and how we suffer is just about the sum of who we are.

BP

Any Common Desolation

By Maria Popova

Any Common Desolation

The morning after a relationship of depth and significance long bending under the weight of its own complexity had finally broken with an exhausted thud, I opened the kiln to discover a month’s worth of pottery shattered — two pieces had exploded, the shrapnel ruining the rest. All that centering, all that glazing, all the hours of pressing letterforms into the wet clay — all of it in shiny shards. And meanwhile spring was breaking outside and a little girl in bright blue rain boots was jumping in a puddle, smashing the reflections of the clouds with savage joy.

And I thought, this is all there is: breaking, breaking apart, breaking open.

Breaking alive.

Card from An Almanac of Birds: 100 Divinations for Uncertain Days, also available as a stand-alone print and as greeting cards.

It is not an easy assignment, being alive. Coming awake from the stupor of near-living that lulls us through our days, awake to the knowledge that on the other side of the neighborhood ICE trucks are handcuffing people and on the other side of the planet children are dying in gunfire, while outside the first birds of spring are singing and everywhere people are falling in love and in some faraway mountain village a shepherd is singing under a thousand stars. And somehow, somehow, all of it has to cohere into a single world in which we, in all our incohesion, must live this single life.

Ellen Bass reckons with all of this in her splendid poem “Any Common Desolation,” originally published in The Academy of American Poets’ poem-a-day newsletter and later included in James Crews’s lifeline of an anthology How to Love the World: Poems of Gratitude and Hope (public library), shared here with Ellen’s blessing.

ANY COMMON DESOLATION
by Ellen Bass

can be enough to make you look up
at the yellowed leaves of the apple tree, the few
that survived the rains and frost, shot
with late afternoon sun. They glow a deep
orange-gold against a blue so sheer, a single bird
would rip it like silk. You may have to break
your heart, but it isn’t nothing
to know even one moment alive. The sound
of an oar in an oarlock or a ruminant
animal tearing grass. The smell of grated ginger.
The ruby neon of the liquor store sign.
Warm socks. You remember your mother,
her precision a ceremony, as she gathered
the white cotton, slipped it over your toes,
drew up the heel, turned the cuff. A breath
can uncoil as you walk across your own muddy yard,
the big dipper pouring night down over you, and everything
you dread, all you can’t bear, dissolves
and, like a needle slipped into your vein —
that sudden rush of the world.

Complement with Henry James on how to stop waiting and start living and Hermann Hesse on how to be more alive, then revisit Ellen’s magnificent poem “How to Apologize.” And if you are looking to break your poetry open, I couldn’t recommend her Living Room Craft Talks more heartily.

BP

The Art of Befriending Time and Change: Debbie Millman’s Illustrated Love Letter to Gardening as a Portal to Self-discovery

By Maria Popova

The Art of Befriending Time and Change: Debbie Millman’s Illustrated Love Letter to Gardening as a Portal to Self-discovery

You may or may not find the meaning of life while pacing a flower bed, but each time you plunge your bare hands into the hummus of the Earth and run your fingers through the roots of something that hungers for the sun, you are resisting the dying of the light and saying “yes” to life.

Gardening may or may not make you a great writer, but it will lavish you with metaphors, those fulcrums of meaning without which all writing — all thinking — would be merely catalog copy for a still life.

You may or may not be able to stop a war by planting a garden, but each time you kneel to press a seed into the ground and bow to look at the ants kissing a peony abloom, you are calling ceasefire on the war within; you are learning to tend to fragility, to cultivate a quiet stubborn resilience, to surrender to forces larger than your will; you are learning to trust time, which is our best means of trusting life. “The gardener,” Derek Jarman wrote in his profound journal of gardening his way through grief, “digs in another time, without past or future, beginning or end… the Amen beyond the prayer.”

This is why Debbie Millman (yes) begins her tenderly illustrated Love Letter to a Garden (public library) at the very beginning, at that first atom of time chipped from the rib of eternity — the singularity that seeded everything.

A seed, she observes, is a kind of singularity — a tiny beginning compacting an entire existence. And so, in consonance with the great naturalist John Muir’s observation that “when we try to pick out anything by itself, we find it hitched to everything else in the universe,” it becomes impossible to contemplate this one thing without contemplating the nature and meaning of existence itself.

Page after painted page, Debbie’s lifelong longing for a garden is slowly revealed as her process of becoming herself, beginning with the portal of wonder that opened the moment her grandmother told her the seeds in the apple she was eating could grow a tree.

Seeds and flowers come to punctuate the story of her life — chapters ending, chapters beginning, the maelstroms of uncertainty, the discomposure of loss, the discomposure of love. They appear at auspicious moments, illustrating the vital difference between signs and omens:

Walking by a few days later, she halts mid-stride upon seeing the peonies blooming once more — only to realize that another mourner had placed a posy of plastic flowers where the real ones had thrived. In the artifice, connection; in the simulacrum, a prayerful bow before the deepest reality we share — time and change, which is another way of saying love and loss.

Half a lifetime later, living in a brownstone of her own, Debbie nurses herself back from heartbreak by making a small hopeful flower garden with a birdbath and tending to it daily with blind devotion.

She falls in love again, marries her soul mate, moves to California for a season and begins growing vegetables.

She navigates the terror and uncertainty of the pandemic by watching the smallest things grow.

And when the world finally regains its precarious balance, she travels its jungles and gardens, orchards and forests, to kneel on the woolly moss of Ireland, to bow before Japan’s sacred lotus, to savor Morocco’s Sanguine oranges and Tuscany’s Pesca Regina di Londa peaches, to run her hands over the elephantine trunks of Cambodia’s banyan trees and her fingers along the fibonacci spines of Mexico’s agave.

Over and over, she returns to her own garden for consolation and calibration. She learns patience. She learns perspective. Watching things come alive after a long germination, she begins to befriend time — the time it takes for a heart to heal, for a world to heal, for an ending to end so that a beginning may begin. Watching things die despite her best efforts, she confronts her lifelong fear of doing anything she isn’t good at — that is, she faces the abyss between the ego and the universe, the will and the world, the abyss in which we live.

What emerges from her Love Letter to a Garden (public library) is a tender reminder that we are here to plant a garden in the abyss, and to trust time.

BP

The Magic of Moss and What It Teaches Us About the Art of Attentiveness to Life at All Scales

By Maria Popova

“Attention without feeling,” Mary Oliver observed in her magnificent memoir of love and loss, “is only a report.” In Gathering Moss: A Natural and Cultural History of Mosses (public library) — an extraordinary celebration of smallness and the grandeur of life, as humble yet surprisingly magical as its subject — botanist Robin Wall Kimmerer extends an uncommon and infectious invitation to drink in the vibrancy of life at all scales and attend to our world with befitting vibrancy of feeling.

One of the world’s foremost bryologists, Kimmerer is a scientist blessed with the rare privilege of belonging to a long lineage of storytellers — her family comes from the Bear Clan of the Potawatomi. There is a special commonality between her heritage and her scientific training — a profound respect for all life forms, whatever their size — coupled with a special talent for rendering that respect contagious, which places her prose in the same taxon as Mary Oliver and Annie Dillard and Thoreau. Indeed, if Thoreau was a poet and philosopher who became a de facto naturalist by the sheer force of poetic observation, despite having no formal training in science, Kimmerer is a formally trained scientist whose powers of poetic observation and contemplative reflection render her a de facto poet and philosopher. (So bewitching is her book, in fact, that it inspired Elizabeth Gilbert’s beautiful novel The Signature of All Things, which is how I first became aware of Kimmerer’s mossy masterwork.)

Scale by Maria Popova

Mosses, to be sure, are scientifically impressive beyond measure — the amphibians of vegetation, they were among the first plants to emerge from the ocean and conquer the land; they number some 22,000 species, whose tremendous range of size parallels the height disparity between a blueberry bush and a redwood; they inhabit nearly every ecosystem on earth and grow in places as diverse as the branch of an oak and the back of a beetle. But beyond their scientific notoriety, mosses possess a kind of lyrical splendor that Kimmerer unravels with enchanting elegance — splendor that has to do with what these tiny organisms teach us about the art of seeing.

She uses the experience of flying — an experience so common we’ve come to take its miraculousness for granted — to illustrate our all too human solipsism:

Between takeoff and landing, we are each in suspended animation, a pause between chapters of our lives. When we stare out the window into the sun’s glare, the landscape is only a flat projection with mountain ranges reduced to wrinkles in the continental skin. Oblivious to our passage overhead, other stories are unfolding beneath us. Blackberries ripen in the August sun; a woman packs a suitcase and hesitates at her doorway; a letter is opened and the most surprising photograph slides from between the pages. But we are moving too fast and we are too far away; all the stories escape us, except our own.

Illustration by Peter Sís from The Pilot and the Little Prince

We, of course, need not rise to the skies in order to fall into the chronic patterns of our myopia and miss most of what is going on around us — we do this even in the familiar microcosm of a city block. Kimmerer considers how our growing powers of technologically aided observation have contributed to our diminished attentiveness:

We poor myopic humans, with neither the raptor’s gift of long-distance acuity, nor the talents of a housefly for panoramic vision. However, with our big brains, we are at least aware of the limits of our vision. With a degree of humility rare in our species, we acknowledge there is much we can’t see, and so contrive remarkable ways to observe the world. Infrared satellite imagery, optical telescopes, and the Hubble space telescope bring vastness within our visual sphere. Electron microscopes let us wander the remote universe of our own cells. But at the middle scale, that of the unaided eye, our senses seem to be strangely dulled. With sophisticated technology, we strive to see what is beyond us, but are often blind to the myriad sparkling facets that lie so close at hand. We think we’re seeing when we’ve only scratched the surface. Our acuity at this middle scale seems diminished, not by any failing of the eyes, but by the willingness of the mind. Has the power of our devices led us to distrust our unaided eyes? Or have we become dismissive of what takes no technology but only time and patience to perceive? Attentiveness alone can rival the most powerful magnifying lens.

But the rewards of attentiveness can’t be forced into manifesting — rather, they are surrendered to. In a sentiment that calls to mind Rebecca Solnit’s spectacular essay on how we find ourselves by getting lost, Kimmerer writes:

A Cheyenne elder of my acquaintance once told me that the best way to find something is not to go looking for it. This is a hard concept for a scientist. But he said to watch out of the corner of your eye, open to possibility, and what you seek will be revealed. The revelation of suddenly seeing what I was blind to only moments before is a sublime experience for me. I can revisit those moments and still feel the surge of expansion. The boundaries between my world and the world of another being get pushed back with sudden clarity an experience both humbling and joyful.

[…]

Mosses and other small beings issue an invitation to dwell for a time right at the limits of ordinary perception. All it requires of us is attentiveness. Look in a certain way and a whole new world can be revealed.

[…]

Learning to see mosses is more like listening than looking. A cursory glance will not do it. Starting to hear a faraway voice or catch a nuance in the quiet subtext of a conversation requires attentiveness, a filtering of all the noise, to catch the music. Mosses are not elevator music; they are the intertwined threads of a Beethoven quartet.

Echoing Richard Feynman’s iconic monologue on knowledge and mystery, Kimmerer adds:

Knowing the fractal geometry of an individual snowflake makes the winter landscape even more of a marvel. Knowing the mosses enriches our knowing of the world.

Moss and air plant sculpture by Art We Heart

This knowing, at its most intimate, is a function of naming — for words are how we come to know meanings. Kimmerer considers this delicate dialogue between a thing’s essence and its name:

Having words for these forms makes the differences between them so much more obvious. With words at your disposal, you can see more clearly. Finding the words is another step in learning to see.

[…]

Having the words also creates an intimacy with the plant that speaks of careful observation.

[…]

Intimacy gives us a different way of seeing, when visual acuity is not enough.

The remarkable diversity of moss varieties known and named only adds to the potentiality for intimacy with the world at all scales. But among this vast multiplicity of mosses is one particular species inhabiting the small caves carved by glaciers into the lakeshore, which alone embodies immense wisdom about the mystery and meaning of life. Kimmerer writes:

Schistostega pennata, the Goblins’ Gold, is unlike any other moss. It is a paragon of minimalism, simple in means, rich in ends. So simple you might not recognize it as a moss at all. The more typical mosses on the bank outside spread themselves to meet the sun. Such robust leaves and shoots, though tiny, require a substantial amount of solar energy to build and maintain. They are costly in solar currency. Some mosses need full sun to survive, others favor the diffuse light of clouds, while Schistostega lives on the clouds’ silver lining alone.

Goblins’ Gold (Photograph: Matt Goff)

This singular species subsists solely on the light reflections emanating from the lake’s surface, which provide one-tenth of one percent of the solar energy that direct sunlight does. And yet in this unlikely habitat, Schistostega has emerged as a most miraculous jewel of life:

The shimmering presence of Schistostega is created entirely by the weft of nearly invisible threads crisscrossing the surface of the moist soil. It glows in the dark, or rather it glitters in the half light of places which scarcely feel the sun.

Each filament is a strand of individual cells strung together like beads shimmering on a string. The walls of each cell are angled, forming interior facets like a cut diamond. It is these facets which cause Schistostega to sparkle like the tiny lights of a far-away city. These beautifully angled walls capture traces of light and focus it inward, where a single large chloroplast awaits the gathering beam of light. Packed with chlorophyll ad membranes of exquisite complexity, the chloroplast converts the light energy into a stream of flowing electrons. This is the electricity of photosynthesis, turning sun into sugar, spinning straw into gold.

But more than a biological marvel, Schistostega presents a parable of patience and its bountiful rewards — an allegory for meeting the world not with grandiose entitlement but with boundless generosity of spirit; for taking whatever it has to offer and giving back an infinity more. Kimmerer writes:

Rain on the outside, fire on the inside. I feel a kinship with this being whose cold light is so different from my own. It asks very little from the world and yet glitters in response.

[…]

Timing is everything. Just for a moment, in the pause before the earth rotates again into night, the cave is flooded with light. The near-nothingness of Schistostega erupts in a shower of sparkles, like green glitter spilled on the rug at Christmas… And then, within minutes, it’s gone. All its needs are met in an ephemeral moment at the end of the day when the sun aligns with the mouth of the cave… Each shoot is shaped like a feather, flat and delicate. The soft blue green fronds stand up like a glad of translucent ferns, tracking the path of the sun. It is so little. And yet it is enough.

This tiny moss is a master of “the patient gleaming of light” — and what is the greatest feat of the human spirit, the measure of a life well lived, if not a “patient gleaming of light”? Annie Dillard knew this when she wrote: “I cannot cause light; the most I can do is try to put myself in the path of its beam.” And Carl Jung knew it when he insisted that “the sole purpose of human existence is to kindle a light in the darkness of mere being.” The humble, generous Schistostega illuminates the darkness of mere being into blazing awe at the miracle of life itself — a reminder that our existence on this unremarkable rock orbiting an unremarkable star is a glorious cosmic accident, the acute awareness of which calls to mind poet Mark Strand’s memorable words: “It’s such a lucky accident, having been born, that we’re almost obliged to pay attention.”

To pay attention, indeed, is the ultimate celebration of this accidental miracle of life. Kimmerer captures this with exuberant elegance:

The combination of circumstances which allows it to exist at all are so implausible that the Schistostega is rendered much more precious than gold. Goblins’ or otherwise. Not only does its presence depend on the coincidence of the cave’s angle to the sun, but if the hills on the western shore were any higher the sun would set before reaching the cave… Its life and ours exist only because of a myriad of synchronicities that bring us to this particular place at this particular moment. In return for such a gift, the only sane response is to glitter in reply.

Gathering Moss is a glittering read in its entirety. Complement it with Annie Dillard on the art of seeing and the two ways of looking.

BP

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

By Maria Popova

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, from 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.

BP

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