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.

The composite creation of a doctor, a philosopher, a poet, and a sculptor, the word empathy in the modern sense only came into use at the dawn of the twentieth century as a term for the imaginative act of projecting yourself into a work of art, into a world of feeling and experience other than your own. It vesselled in language that peculiar, ineffable way art has of bringing you closer to yourself by taking you out of yourself — its singular power to furnish, Iris Murdoch’s exquisite phrasing, “an occasion for unselfing.” And yet this notion cinches the central paradox of art: Every artist makes what they make with the whole of who they are — with the totality of experiences, beliefs, impressions, obsessions, childhood confusions, heartbreaks, inner conflicts, and contradictions that constellate a self. To be an artist is to put this combinatorial self in the service of furnishing occasions for unselfing in others.

That may be why the lives of artists have such singular allure as case studies and models of turning the confusion, complexity, and uncertainty of life into something beautiful and lasting — something that harmonizes the disquietude and dissonance of living.

Red Hill and White Shell by Georgia O’Keeffe, 1938

In Funny Weather: Art in an Emergency (public library), Olivia Laing — one of the handful of living writers whose mind and prose I enjoy commensurately with the Whitmans and the Woolfs of yore — occasions a rare gift of unselfing through the lives and worlds of painters, poets, filmmakers, novelists, and musicians who have imprinted culture in a profound way while living largely outside the standards and stabilities of society, embodying of James Baldwin’s piecing insight that “a society must assume that it is stable, but the artist must know, and he must let us know, that there is nothing stable under heaven.”

Punctuating these biographical sketches laced with larger questions about art and the human spirit are Laing’s personal essays reflecting, through the lens of her own lived experience, on existential questions of freedom, desire, loneliness, queerness, democracy, rebellion, abandonment, and the myriad vulnerable tendrils of aliveness that make life worth living.

What emerges is a case for art as a truly human endeavor, made by human beings with bodies and identities and beliefs often at odds with the collective imperative; art as “a zone of both enchantment and resistance,” art as sentinel and witness of “how truth is made, diagramming the stages of its construction, or as it may be dissolution,” art as “a direct response to the paucity and hostility of the culture at large,” art as a buoy for loneliness and a fulcrum for empathy.

Summer 1964 by Agnes Martin

Laing writes:

Empathy is not something that happens to us when we read Dickens. It’s work. What art does is provide material with which to think: new registers, new spaces. After that, friend, it’s up to you.

I don’t think art has a duty to be beautiful or uplifting, and some of the work I’m most drawn to refuses to traffic in either of those qualities. What I care about more… are the ways in which it’s concerned with resistance and repair.

A writer — a good writer — cannot write about art without writing about those who make it, about the lives of artists as the crucible of their creative contribution, about the delicate, triumphant art of living as a body in the world and a soul outside standard society. Olivia Laing is an excellent writer. Out of lives as varied as those of Basquiat and Agnes Martin, Derek Jarman and Georgia O’Keeffe, David Bowie and Joseph Cornell, she constructs an orrery of art as a cosmos of human connection and a sensemaking mechanism for living.

In a sentiment to which I relate in my own approach to historical lives, Laing frames her method of inquiry:

I’m going as a scout, hunting for resources and ideas that might be liberating or sustaining now, and in the future. What drives all these essays is a long-standing interest in how a person can be free, and especially in how to find a freedom that is shareable, and not dependent upon the oppression or exclusion of other people.

[…]

We’re so often told that art can’t really change anything. But I think it can. It shapes our ethical landscapes; it opens us to the interior lives of others. It is a training ground for possibility. It makes plain inequalities, and it offers other ways of living.

Throughout these short, scrumptiously insightful and sensitive essays, Laing draws on the lives of artists — the wildly uneven topographies of wildly diverse interior worlds — to contour new landscapes of possibility for life itself, as we each live it, around and through and with art. In the essay about Georgia O’Keeffe — who revolutionized modern art while living alone and impoverished in the middle of the desert, in the middle of the world’s first global war — Laing observes:

How do you make the most of what’s inside you, your talents and desires, when they slam you up against a wall of prejudice, of limiting beliefs about what a woman must be and an artist can do?

[…]

From the beginning, New Mexico represented salvation, though not in the wooden sense of the hill-dominating crosses she so often painted. O’Keeffe’s salvation was earthy, even pagan, comprised of the cold-water pleasure of working unceasingly at what you love, burning anxiety away beneath the desert sun.

Georgia O’Keeffe, Grey Lines with Black, Blue and Yellow, 1923 (Georgia O’Keeffe Museum)

In an essay about another artist — the painter Chantal Joffe, for whom Laing sat — she echoes Jackson Pollock’s’s observation that “every good artist paints what he is,” and writes:

You can’t paint reality: you can only paint your own place in it, the view from your eyes, as manifested by your own hands.

A painting betrays fantasies and feelings, it bestows beauty or takes it away; eventually, it supplants the body in history. A painting is full of desire and love, or greed, or hate. It radiates moods, just like people.

[…]

Paint as fur, as velvet, as brocade, as hair. Paint as a way of entering and becoming someone else. Paint as a device for stopping time.

Art by Basquiat from Life Doesn’t Frighten Me by Maya Angelou

In another essay, Laing offers an exquisite counterpoint to the barbed-wire fencing off of identities that has increasingly made the free reach of human connection — that raw material and final product of all art — dangerous and damnable in a culture bristling with ready indignations and antagonisms:

A writer I was on a panel with said, and I’m paraphrasing here, that it is no longer desirable to write about the lives of other people or experiences one hasn’t had. I didn’t agree. I think writing about other people, making art about other people, is both dangerous and necessary. There are moral lines. There are limits to the known. But there’s a difference between respecting people’s right to tell or not tell their own stories and refusing to look at all.

[…]

It depends whether you believe that we exist primarily as categories of people, who cannot communicate across our differences, or whether you think we have a common life, an obligation to regard and learn about each other.

In a sense, the entire book is a quiet manifesto for unselfing through the art we make and the art we cherish — a subtle and steadfast act of resistance to the attrition of human connection under the cultural forces of self-righteousness and sanctimonious othering, a stance against those fashionable and corrosive forces that so often indict as appropriation the mere act of learning beautiful things from each other.

In another essay — about Ali Smith, the subject to whom Laing feels, or at least reads, the closest — she quotes a kindred sentiment of Smith’s:

Art is one of the prime ways we have of opening ourselves and going beyond ourselves. That’s what art is, it’s the product of the human being in the world and imagination, all coming together. The irrepressibility of the life in the works, regardless of the times, the histories, the life stories, it’s like being given the world, its darks and lights. At which point we can go about the darks and lights with our imagination energised.

Joseph Cornell, Untitled (Celestial Navigation), 1958. (U.S. Department of State / Joseph and Robert Cornell Memorial Foundation)

Among the subjects of a subset of essays Laing aptly categorizes as “love letters” is John Berger, whose lovely notion of “hospitality” radiates from Laing’s own work — a notion she defines as “a capacity to enlarge and open, a corrective to the overwhelming political imperative, in ascendance once again this decade, to wall off, separate and reject.” She reflects on being stopped up short by Berger’s embodiment of such hospitality when she saw him speak at the British Library at the end of his long, intellectually generous life:

It struck me then how rare it is to see a writer on stage actually thinking, and how glib and polished most speakers are. For Berger, thought was work, as taxing and rewarding as physical labour, a bringing of something real into the world. You have to strive and sweat; the act is urgent but might also fail.

He talked that evening about hospitality. It was such a Bergerish notion. Hospitality: the friendly and generous reception and entertainment of guests, visitors or strangers, a word that shares its origin with hospital, a place to treat sick or injured people. This impetus towards kindness and care for the sick and strange, the vulnerable and dispossessed is everywhere in Berger’s work, the sprawling orchard of words he planted and tended over the decades.

[…]

Art he saw as a communal and vital possession, to be written about with sensual exactness… Capitalism, he wrote in Ways of Seeing, survives by forcing the majority to define their own interests as narrowly as possible. It was narrowness he set himself against, the toxic impulse to wall in or wall off. Be generous to the strange, be open to difference, cross-pollinate freely. He put his faith in the people, the whole host of us.

With My Back to the World by Agnes Martin, 1997

In a superb 2015 essay titled “The Future of Loneliness” — an essay that bloomed into a book a year later, the splendid and unclassifiable book that first enchanted me with Laing’s writing and the mind from which it springs — she considers how technology is mediating our already uneasy relationship to loneliness, and how art redeems the simulacra of belonging with which social media entrap us in this Stockholm syndrome of self-regard. In a passage of chillingly intimate resonance to all of us alive in the age of screens and selfies and the vacant, addictive affirmation of people we have never dined with tapping heart- and thumb-shaped icons on cold LED screens, she writes:

Loneliness centres around the act of being seen. When a person is lonely, they long to be witnessed, accepted, desired, at the same time as becoming intensely wary of exposure. According to research carried out over the past decade at the University of Chicago, the feeling of loneliness triggers what psychologists term hypervigilance for social threat. In this state, which is entered into unknowingly, the individual becomes hyperalert to rejection, becoming inclined to perceive their social interactions as tinged with hostility or scorn. The result of this shift in perception is a vicious circle of withdrawal, in which the lonely person becomes increasingly suspicious, intensifying their sense of isolation.

This is where online engagement seems to exercise its special charm. Hidden behind a computer screen, the lonely person has control. They can search for company without the danger of being revealed or found wanting. They can reach out or they can hide; they can lurk and they can show themselves, safe from the humiliation of face-to-face rejection. The screen acts as a kind of protective membrane, a scrim that permits invisibility and also transformation. You can filter your image, concealing unattractive elements, and you can emerge enhanced: an online avatar designed to attract likes. But now a problem arises, for the contact this produces is not quite the same thing as intimacy. Curating a perfected self might win followers or Facebook friends, but it will not necessarily cure loneliness, since the cure for loneliness is not being looked at, but being seen and accepted as a whole person: ugly, unhappy and awkward as well as radiant and selfie-ready.

Illustration by Mimmo Paladino for a rare edition of James Joyce’s Ulysses

Having met with Ryan Trecartin — “a baby-faced thirty-four-year-old” of whom I had never heard (saying more about my odd nineteenth-century life than about his art) but whose early-twenty-first-century films about the lurid and discomposing thrill of digital culture prompted The New Yorker to describe him as “the most consequential artist to have emerged since the nineteen-eighties” — Laing reflects:

My own understanding of loneliness relied on a belief in solid, separate selves that he saw as hopelessly outmoded. In his world view, everyone was perpetually slipping into each other, passing through perpetual cycles of transformation; no longer separate, but interspersed. Perhaps he was right. We aren’t as solid as we once thought. We’re embodied but we’re also networks, expanding out into empty space, living on inside machines and in other people’s heads, memories and data streams as well as flesh. We’re being watched and we do not have control. We long for contact and it makes us afraid. But as long as we’re still capable of feeling and expressing vulnerability, intimacy stands a chance.

Vulnerability — which Laing unfussily terms “the necessary condition of love” — is indeed the bellowing undertone of these essays: vulnerability as frisson and function of art, of life itself, of the atavistic impulse for transmuting living into meaning that we call art.

Complement the thoroughly symphonic Funny Weather with Paul Klee on creativity and why an artist is like a tree, Kafka on why we make art, Egon Schiele on why visionary artists tend to come from the minority, and Virginia Woolf’s garden epiphany about what it means to be an artist — which remains, for me, the single most beautiful and penetrating thing ever written on the subject — then revisit Laing on life, loss, and the wisdom of rivers.

“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.

Vintage Science Face Masks Benefiting the Nature Conservancy (New Designs Added)

The Snail with the Right Heart: A True Story