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Seeking an Aurora: A Wondrous Illustrated Celebration of Earth’s Most Otherworldly Spectacle of Light and Color

Transcendence and tenderness in the lacuna of awe between the creaturely and the cosmic.

Seeking an Aurora: A Wondrous Illustrated Celebration of Earth’s Most Otherworldly Spectacle of Light and Color

In 1621, already questioning his life in the priesthood — the era’s safest and most reputable career for the educated — the 29-year-old Pierre Gassendi, a mathematical prodigy since childhood, traveled to the Arctic circle as he began diverting his passionate erudition toward Aristotelian philosophy and astronomy. There, under the polar skies, he witnessed an otherworldly spectacle on Earth — our planet’s most intimate and dramatic contact with its home star, a chromatic swirl of the ephemeral and the eternal unloosed as solar winds blow millions of charged particles from the Sun across the orrery of the Solar System and into Earth’s atmosphere, where our magnetic fields carry them toward the poles. As they collide with the particles of different atmospheric gasses, they ionize and discharge energy as photons of different colors — red, blue, green, and violent — painting the nocturne with the waking dream of a pastel-technicolor dawn.

Awestruck with the natural poetry and the mythic feeling-tone of the luminous spectacle, Gassendi named what he saw Aurora borealis — after Aurora, the Roman goddess of dawn, and borealis, the Latin word for “northern.” Eventually, as explorers braved the icy oceanic expanses to visit the polar regions of the Southern hemisphere over the following centuries, they adapted Gassendi’s etymology to name the Antarctic version of the luminous display Aurora australis, after the Latin word for “southern.”

From the land of Aurora australis comes Seeking an Aurora (public library) — a work of transcendence and tenderness by New Zealand author-artist duo Elizabeth Pulford and Anne Bannock, whose spare poetic prose and soulful paintings interleave to enlush an inner landscape of wonder, suspended between the creaturely and the cosmic.

Late one night, a father awakens his child — a child of ambiguous gender and ethnicity, a touching effort to approximate the universal in the human — to slip out of the house together, past the soundly sleeping mother and the baby in the crib, and out into the winter nocturne on a quest of wonder.

They walk with brisk excitation across the open field and through the skeletal trees as the warm humanity of their breath puffs into the cold night air, into the silence they share with the other breathing creatures that make this planet a world.

Outside everything was still.
Even the dogs were quiet, and the cows looked like prehistoric creatures, their noses streaming smoke.

The adventure unfolds from the narrative vantage point of the child, who turns around to look back at the house with its “warm, buttery light spilling from the kitchen window,” back at the two sets of “footprints in the silvery frost,” then up at the sky, “a ship of shivering stars.”

As the pair ascend the steep hill toward their lookout, the cows and the dogs recede into the distance, leaving only the stars, the Moon, and the swell of anticipation.

And then, suddenly, the aurora appears, its “wide wings of light” sweeping across the sky to widen the child’s eyes with wonder.

Dancing light, glowing and glimmering,
shimmering and shining.
Colored ribbons swirling and twirling,
lighting up the sky on the still, dark night.

Father and child are silent under the soft technicolor sky — an awed silence that evokes the works of the poet Diane Ackerman, who wrote long ago in her stunning Cosmic Pastoral of feeling “stricken by the ricochet wonder of it all: the plain everythingness of everything, in cahoots with the everythingness of everything else.”

On the walk home, back to the house with the warm buttery light, the father shares everything he knows about the aurora — a secret everythingness revealed on the last page of the book, in a brief science primer of an afterword, sweetly titled “Everything Dad Knew about the Aurora.”

Couple Seeking an Aurora with the inspiring picture-book biography of trailblazing astronomer Maria Mitchell, then revisit a literary titan’s account of the other great cosmic spectacle visible from Earth — Virginia Woolf’s arresting meditation on the total solar eclipse.

Illustrations courtesy of Blue Dot Kids Press; photographs by Maria Popova


Immunity, Interdependence, and the Shared Root of Our Safety and Our Sanity: Eula Biss on the Science and Social Dynamics of Health as Communal Trust

“We are protected not so much by our own skin, but by what is beyond it. The boundaries between our bodies begin to dissolve here… Immunity… is a common trust as much as it is a private account.”

Immunity, Interdependence, and the Shared Root of Our Safety and Our Sanity: Eula Biss on the Science and Social Dynamics of Health as Communal Trust

Months after Rachel Carson’s Silent Spring awakened humanity to the delicate interdependence of nature, Dr. King awakened humanity to our delicate dependence on each other. “We are caught in an inescapable network of mutuality [and] whatever affects one directly, affects all indirectly,” he wrote from his cell at the Birmingham City Jail.

When Robert Hooke looked at a piece of cork through an early handcrafted leather-and-gold microscope in 1665, he named the strange irregular “pores” of its honeycomb-like tissue structure cells, after the small adjacent spaces in which monks spend their voluntary solitary confinement. It would take another two centuries for scientists to discover that cells are the basic biological units of life, that they are in constant osmotic communication with one another, and that they replicate themselves to become new cells, each a whispered word from the language in which life talks to the future.

Cork structure from Robert Hooke’s Micrographia, 1665. (Available as a print.)

Biological and social, our interdependence is a defining feature not only of our civilization, not only of our species and all living species, but of life itself — life the physiological process and life the psychosocial phenomenon. “Every atom belonging to me as good belongs to you,” Walt Whitman exulted in the golden age of chemistry — the new science he saw as “the elevating, beautiful, study… which involves the essences of creation.” Meanwhile, the development of cell theory was revolutionizing biology, making of this philosophical field as old as Aristotle an even newer science that illuminated the essence of life. Cells became to biology what atoms were to chemistry. Biology ushered in the revelation that every cell belonging to me as good — as healthy, as vital, as fit for replication — belongs to you.

That delicate interdependence of life and lives, with its tangled roots in biology and cultural history, is what Eula Biss explores in On Immunity: An Inoculation (public library) — a book of penetrating and poetic insight, drawn with that rare scholarship capable of correcting the warped cultural hindsight we call history; a book of staggering foresight, conceived in the wake of the H1N1 flu pandemic, yet speaking with astonishing prescience to the complex epidemiological realities and social dynamics of the COVID-19 pandemic unfolding more than five years after its publication.

For Biss — the daughter of a medical scientist and a poet — even her own biological inheritance as a universal donor with type O negative blood becomes a potent metaphor for the mechanism of vaccination, a lens through which to view the permeable membrane between the biological and social realities of immunity. With an eye to the blood banks that collect her donations to save other lives, she writes:

If we imagine the action of a vaccine not just in terms of how it affects a single body, but also in terms of how it affects the collective body of a community, it is fair to think of vaccination as a kind of banking of immunity. Contributions to this bank are donations to those who cannot or will not be protected by their own immunity. This is the principle of herd immunity, and it is through herd immunity that mass vaccination becomes far more effective than individual vaccination.

It is a rather unfortunate term for an unassailable scientific principle — we humans, especially in this culture of rugged individualism nursed on the Emersonian ideal of self-reliance, bristle at thinking of ourselves as members of a herd. In our long history of thinking with animals, herd animals have been the butt of our derogatory metaphors for mindless conformity.

Art by Olivier Tallec from Louis I, King of the Sheep
Art by Olivier Tallec from Louis I, King of the Sheep

And yet inside the unfortunate linguistic container, an unfaltering biological reality resides: On large enough a scale, even a fairy ineffective vaccine that fails to produce immunity in some individuals will slow down the spread of infection in the community; as the virus fails to replicate itself in more and more new hosts, the vaccine will eventually halt it altogether. In consequence, even such a mediocre vaccine will protect all members of the community, even those for whom inoculation has not worked as intended on the individual level. This is why it is more dangerous to be the vaccinated animal amid a largely unvaccinated herd than the other way around. Biss writes:

The unvaccinated person is protected by the bodies around her, bodies through which disease is not circulating. But a vaccinated person surrounded by bodies that host disease is left vulnerable to vaccine failure or fading immunity. We are protected not so much by our own skin, but by what is beyond it. The boundaries between our bodies begin to dissolve here. Donations of blood and organs move between us, exiting one body and entering another, and so too with immunity, which is a common trust as much as it is a private account. Those of us who draw on collective immunity owe our health to our neighbors.

With an eye to the origin of herd immunity theory — a theory developed in the 1840s by a doctor treating smallpox, which has taken manyfold more human lives than any other infectious disease in the history of our species and which has since been eradicated — Biss proposes an alternative, both more poetic and more precise, to the imperfect term that so perfectly describes the biosocial reality:

Herd immunity, an observable phenomenon, now seems implausible only if we think of our bodies as inherently disconnected from other bodies. Which, of course, we do.

The very expression herd immunity suggests that we are cattle, waiting, perhaps, to be sent to slaughter. And it invites an unfortunate association with the term herd mentality, a stampede toward stupidity. The herd, we assume, is foolish. Those of us who eschew the herd mentality tend to prefer a frontier mentality in which we imagine our bodies as isolated homesteads that we tend either well or badly. The health of the homestead next to ours does not affect us, this thinking suggests, so long as ours is well tended.

If we were to exchange the metaphor of the herd for a hive, perhaps the concept of shared immunity might be more appealing. Honeybees are matriarchal, environmental do-gooders who also happen to be entirely interdependent. The health of any individual bee, as we know from the recent epidemic of colony collapse, depends on the health of the hive.

Diagram of bee anatomy by French artist Paul Sougy, 1962. (Available as a print.)

Biss quotes a succinct summation by her father, a doctor:

Vaccination works by enlisting a majority in the protection of a minority.

No one person has done more to undermine this vital mutuality of protection than Andrew Wakefield — the British gastroenterologist who, in the 1990s, infected the hive mind with his causal claims linking vaccines and autism. Preying on the understandable human impulse toward concretizing blame for amorphous and ambiguous problems, the theory went viral before multiple subsequent studies debunked his results, before it was exposed that Wakefield was paid for his research by a lawyer readying a lawsuit against a vaccine maker, before the General Medical Council of the United Kingdom concluded its investigation with the verdict that Wakefield had been “irresponsible and dishonest” in conducting and publishing his work.

Despite the scientific and ethical denunciation of Wakefield’s study, its ideological meme had already spread beyond retrieval. (Richard Dawkins coined the word meme in 1976 by borrowing from biology — a word that came alive anew a quarter century later in the context of “viral” content on the internet, which has its own roots in epidemiology.) A quarter century later, echoes of Wakefield’s disproven falsehoods bellow with formidable vocality. That group of voices is often referred to as the anti-vaccination movement, but I find the term movement extremely ill-suited — such groupthink is not in movement but static, frozen in time and frozen with fear, petrified in the cultural amber of a time before the Age of Reason and lashed about by the same errors of magical thinking, willful blindness, and confusion of causation and correlation that made our medieval ancestors take comets for indisputable omens of future events and left-handedness for indisputable evidence of possession by the Devil.

Art from The Comet Book, 1587. (Available as a print).

Biss is more generous in her own assessment of anti-vaccination:

Those who went on to use Wakefield’s inconclusive work to support the notion that vaccines cause autism are not guilty of ignorance or science denial so much as they are guilty of using weak science as it has always been used — to lend false credibility to an idea that we want to believe for other reasons.

Writing shortly after the birth of the Occupy movement — the self-described “99%” launching “an ongoing global protest of capitalism” — she considers a friend’s half-joke, half-koan about vaccination as a matter of “occupy immune system,” and reflects on the basic moral syllogism of anti-vaccination as a political stance claiming to protest the capitalist forces behind modern medicine:

Immunity is a public space. And it can be occupied by those who choose not to carry immunity. For some… a refusal to vaccinate falls under a broader resistance to capitalism. But refusing immunity as a form of civil disobedience bears an unsettling resemblance to the very structure the Occupy movement seeks to disrupt — a privileged 1 percent are sheltered from risk while they draw resources from the other 99 percent.


We are justified in feeling threatened by the unlimited expansion of industry, and we are justified in fearing that our interests are secondary to corporate interests. But refusal of vaccination undermines a system that is not actually typical of capitalism. It is a system in which both the burdens and the benefits are shared across the entire population. Vaccination allows us to use the products of capitalism for purposes that are counter to the pressures of capital.

Emissary by Maria Popova

In a lovely antidote to the tragic human tendency toward cynicism — that touchingly misguided and ineffective effort at self-protection, that particularly virulent strain of cowardice to which our culture has grown increasingly hospitable as it has grown increasingly impatient with the slow and vulnerable work of nuance — Biss adds:

That so many of us find it entirely plausible that a vast network of researchers and health officials and doctors worldwide would willfully harm children for money is evidence of what capitalism is really taking from us. Capitalism has already impoverished the working people who generate wealth for others. And capitalism has already impoverished us culturally, robbing unmarketable art of its value. But when we begin to see the pressures of capitalism as innate laws of human motivation, when we begin to believe that everyone is owned, then we are truly impoverished.

Complement On Immunity — a redemptive and salutary read in its entirety — with Virginia Woolf on illness as a portal to self-understanding and Bessel van der Kolk on the science of how our minds and our bodies converge in healing, then revisit Adrienne Rich on resisting capitalism through the arts of the possible.


Wintering: Resilience, the Wisdom of Sadness, and How the Science of Trees Illuminates the Art of Self-Renewal Through Difficult Times

“Wintering… is the courage to stare down the worst parts of our experience and to commit to healing them the best we can. Wintering is a moment of intuition, our true needs felt keenly as a knife.”

Wintering: Resilience, the Wisdom of Sadness, and How the Science of Trees Illuminates the Art of Self-Renewal Through Difficult Times

Rilke reverenced winter as the season for tending to the inner garden of the soul: “Suddenly to be healed again and aware that the very ground of my being — my mind and spirit — was given time and space in which to go on growing,” he wrote to a grief-stricken young woman who had reached out to him for consolation. “In the depths of winter, I finally learned that within me there lay an invincible summer,” Albert Camus wrote a generation later in his stunning essays about travel, which are really meditations on homecoming to our strength. Camus was soon to become the second-youngest Nobel laureate of all time and soon to die in a car crash with an unused train ticket to the same destination in his pocket. We are not invincible. But in how we garden the winters of the soul, we find the summer of our strength and the bloom of our fragile aliveness.

That is what Katherine May explores in Wintering: The Power of Rest and Retreat in Difficult Times (public library) — a gorgeous book, a generous book, a layered book of uncommon sensitivity and substance, drawn from May’s own experience of living through a deep and disquieting winter of life. She writes:

[Since childhood] we are taught to ignore sadness, to stuff it down into our satchels and pretend it isn’t there. As adults, we often have to learn to hear the clarity of its call. That is wintering. It is the active acceptance of sadness. It is the practice of allowing ourselves to feel it as a need. It is the courage to stare down the worst parts of our experience and to commit to healing them the best we can. Wintering is a moment of intuition, our true needs felt keenly as a knife.

Art by Valerio Vidali from The Shadow Elephant by Nadine Robert — a subtle children’s book about honoring sadness.

Like happiness — which, as George Eliot well knew, is a skill we incrementally master as we grow older — sadness, May reminds us, is also a skill: There are self-punishing ways to be sad, and self-salving ways to be sad. In skillful wintering, we learn the difference between the two. Rilke, who wintered amply and wisely, knew that great sadnesses clarify us to ourselves — winters of the spirit come in various sizes and cycles, each meaningful, all cumulative in their soul-sculpting beneficence. May writes:

When you start tuning in to winter, you realise that we live through a thousand winters in our lives — some big, some small… Some winters creep up on us so slowly that they have infiltrated every part of our lives before we truly feel them.


To get better at wintering, we need to address our very notion of time. We tend to imagine that our lives are linear, but they are in fact cyclical.

This cyclical nature of the seasons of the spirit is counter to our dominant cultural narrative of self-improvement, with its ethos of linear progression toward states of ever-increasing flourishing. It is counter, too, to the world’s major spiritual traditions, with their ideas of salvation and enlightenment. (Any longtime practitioner of Zen or metta meditation, for instance, knows that while we do reach moments of so-called enlightenment — a gladsome dissolution of the self into an all-pervading lovingkindness — these moments are inevitably punctuated by visitations of our habitual tendencies toward egoic shortness of temper, the self-absorption we call melancholy, and other conditioned modes of unenlightened conduct.) And yet befriending this cyclical rhythm of our inner lives, May observes with life-tested clarity, is the key to wintering — to emerging from the coldest seasons of the soul not only undiminished but revitalized.

Ever/After by Maria Popova. (Available as a print.)

Drawing on the analogy of tress — these most fertile metaphors for our humanity, in which we see ourselves and see quiet wisdom on how to live with ourselves, on how to live with each other, on the root of authenticity, on what it means to be an artist and what it means to be human — she writes:

We are in the habit of imagining our lives to be linear, a long march from birth to death in which we mass our powers, only to surrender them again, all the while slowly losing our youthful beauty. This is a brutal untruth. Life meanders like a path through the woods. We have seasons when we flourish and seasons when the leaves fall from us, revealing our bare bones. Given time, they grow again.

In one of the book’s wonderful portals into the world of science as a means of comprehending our elemental humanity, May considers the astonishing actuality of trees beyond the merely metaphorical:

The dropping of leaves by deciduous trees is called abscission. It occurs on the cusp between autumn and winter, as part of an arc of growth, maturity, and renewal. In spring and summer, leaf cells are full of chlorophyll, a bright green substance that absorbs sunlight, fueling the process that converts carbon dioxide and water into the starch and sugar that allow the tree to grow. But at the end of the summer, as the days grow shorter and the temperature falls, deciduous trees stop making food. In the absence of sunlight, it becomes too costly to maintain the machinery of growth. The chlorophyll begins to break down, revealing other colours that were always present in the leaf, but which were masked by the abundance of green pigment: oranges and yellows, derived from carotene and xanthophyll. Other chemical changes take place to create red anthocyanin pigments. The exact mix is different for each tree, sometimes producing bright yellows, oranges, and browns, and sometimes displaying as reds or purples.

But while this is happening, a layer of cells is weakening between the stem and the branch: this is called the abscission zone. Gradually it severs the leaf from access to water, and the leaf dries and browns and in most cases falls off, either under its own weight or encouraged by wintery rains and winds. Within a few hours, the tree will have released substances to heal the scar the leaf has left, protecting itself from the evaporation of water, infection, or the invasion of parasites.

Sundown Poem by Maria Popova. (Available as a print.)

I have always cherished the bare beauty of winter trees, so fractal and pulmonary against the somber sky — so skeletal, yet so alive. Anyone willing to look closely — and why be alive at all if not to relish the ecstasy of noticing, that crowning glory of our consciousness? — is rewarded with the gasping recognition that the branches are already covered in tiny dormant buds encoding the Braille promise of spring.

May writes:

Most trees produce their buds in high summer, and the autumn leaf fall reveals them, neat and expectant, protected from the cold by thick scales… from the sharp talons of the beech to the hooflike black buds of the ash. Many trees also display catkins in the winter, like the acid-green lambs’ tails of the hazel and the furry grey nubs of the willow. These employ the wind or insects to spread pollen, ready for the new year.

The tree is waiting. It has everything ready. Its fallen leaves are mulching the forest floor, and its roots are drawing up the extra winter moisture, providing a firm anchor against seasonal storms. Its ripe cones and nuts are providing essential food in this scarce time for mice and squirrels, and its bark is hosting hibernating insects and providing a source of nourishment for hungry deer. It is far from dead. It is in fact the life and soul of the wood. It’s just getting on with it quietly. It will not burst into life in the spring. It will just put on a new coat and face the world again.

Art from Trees at Night by Art Young, 1926. (Available as a print.)

Looking back on her own barren-branched seasons of the soul, she reflects:

Here is another truth about wintering: you’ll find wisdom in your winter, and once it’s over, it’s your responsibility to pass it on. And in return, it’s our responsibility to listen to those who have wintered before us. It’s an exchange of gifts in which nobody loses out. This may involve the breaking of a lifelong habit, one passed down carefully through generations: that of looking at other people’s misfortunes and feeling certain that they brought them upon themselves in a way that you never would. This isn’t just an unkind attitude. It does us harm, because it keeps us from learning that disasters do indeed happen and how we can adapt when they do. It stops us from reaching out to those who are suffering. And when our own disaster comes, it forces us into a humiliated retreat, as we try to hunt down mistakes that we never made in the first place or wrongheaded attitudes that we never held. Either that, or we become certain that there must be someone out there we can blame. Watching winter and really listening to its messages, we learn that effect is often disproportionate to cause; that tiny mistakes can lead to huge disasters; that life is often bloody unfair, but it carries on happening with or without our consent. We learn to look more kindly on other people’s crises, because they are so often portents of our own future.

Art by Arthur Rackham for a rare 1917 edition of the Brothers Grimm fairy tales. (Available as a print.)

The whole of Wintering — which explores the biological, psychological, neurochemical, and philosophical subtleties of our state of being in winter the season and winter the metaphor — is a splendid and soul-salving read. Complement it with Thoreau’s transcendentalist strategy for finding inner warmth in the cold of life, Annie Dillard on how winter awakens us to life, Adam Gopnik’s lyrical love letter to the white season, and D.H. Lawrence on trees, solitude, and how we root ourselves when our worlds collapse, then savor more of May’s writing and the personal story from which it springs in her wonderful On Being conversation with Krista Tippett.


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, 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.


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