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The Geometry of Grief: A Mathematician on How Fractals Can Help Us Fathom Loss and Reorient to the Ongoingness of Life

“The distance between here and there is the answer to the wrong question.”

The Geometry of Grief: A Mathematician on How Fractals Can Help Us Fathom Loss and Reorient to the Ongoingness of Life

“What exists, exists so that it can be lost and become precious,” Lisel Mueller wrote in her stunning poem about what gives meaning to our mortal lives as she neared, but never quite reached, the triumph of having lived a century — a bittersweet triumph, for to live at all, however long or short, is an unbidden bargain to lose everything you hold precious: every love and every life, including your own. Loss is the price of life — a price we never chose to pay any more than we chose to be born, and yet a price not merely worth paying but beyond questions of worth and why.

One corollary is that, both in the evolutionary sense and in the existential, every loss reveals what we are made of. But every loss also reveals what it is made of, which is more loss: Each loss takes a piece of us — a piece soft and alive — and leaves in its place something cold and heavy; each subsequent loss becomes the magnet that draws out those old leaden pieces, pulls them out from the reliquary of scar tissue where we have been keeping them in order to live, makes them rip through our being afresh. And yet the shrapnel pieces that surface are smaller and softer-edged than when they first entered through the open wound of raw bereavement, smoothed and contracted by the ongoingness of life.

In this sense, grief if fractal, each new instance containing within itself a set of self-similar sub-griefs — miniatures of the same emotional structure, rendered smaller in salience by time and tenacity, those twin inevitabilities of aliveness.

The Mandelbrot set. (Illustration by Wolfgang Beyer.)

How the fractal nature of grief is both the key to understanding it and the doorway to moving through it is what mathematician Michael Frame explores in his unusual book Geometry of Grief: Reflections on Mathematics, Loss, and Life (public library). After twenty years of working with the visionary father of fractals and another twenty years of teaching fractal geometry at Yale, Frame draws on a lifetime of loss and a lifetime of delicate attention to the details of aliveness we call beauty to interleave memoir and mathematics in an uncommon tapestry of thought, twining Borges and quantum mechanics, evolutionary biology and Islamic art, music and multiverse theory.

Because every sound theorem rests upon precise formulation, Frame offers a basic definition:

Grief is a response to an irreversible loss… To generate grief rather than sadness, the thing lost must carry great emotional weight, and it must pull back the veil that covers a transcendent aspect of the world. Breathe out to push the fog away from a brilliant pinpoint of light.

Total eclipse of the sun, observed July 29, 1878, at Creston, Wyoming Territory
Total eclipse of 1878, one of Étienne Léopold Trouvelot’s groundbreaking astronomical drawings. (Available as a print and as a face mask.)

This trifecta of irreversibility, emotional heft, and transcendence anchors Frame’s model of grief and his map for navigating the landscape of loss not as a journey of recovery but as one of readjustment — of reconstituting our model of the world within, which governs our entire experience of the world without. Because the two basic building blocks of our world-model — inner and outer — are attention and narrative, readjustment to life after loss requires deliberate wielding of both. Frame writes:

All moments of our lives are immensely rich, with many — perhaps infinitely many — variables we could notice.

We can view our lives as trajectories, parameterized by time, through story space.

We can never simultaneously view all of the possible variables; rather, we focus on a few variables at a time, restricting our attention to a low-dimensional subspace of story space.

Our trajectories through these subspaces are the stories we tell ourselves about our lives; they are how we make sense of our lives, but always they miss some elements of our experiences.

Irreversible loss appears as a discontinuity, a jump, in our path through story space.

By focusing on certain subspaces, by projecting our trajectories into these spaces, we can reduce the apparent magnitude of the jumps, and consequently find a way to confront the emotional loss and perhaps reduce its impact.

The most gladdening thing about grief parallels the most gladdening thing about science: However meticulous our projections and our models of reality may be, however triumphant in their conquest of knowledge, they are not only perennially incomplete but could be — and, throughout the history of our species, have often been — fundamentally wrong. Science, like life itself, rests upon the abstract art of otherwise — things could be other than what they appear to be, other than what we assume them to be: stranger, more slippery, more possible. Frame writes:

Geometry is a way to organize our models of the world, its shapes and dynamics. But isn’t this all contingent, balanced on a knife’s edge? Could our models have turned out very differently? If the fractal geometry of Mandelbrot had been discovered before the geometry of Euclid, would manufacture be the same? If you think the question is far-fetched, consider the iterated branching of our pulmonary, circulatory, and nervous systems, or the recursive folding of our DNA, or the large surface area and small volume of our lungs and our digestive tract. Evolution has discovered and uses fractal geometry. If people had looked more closely at the geometry of nature, rather than emulating the “celestial perfection” imposed by the church’s interpretation of the works of Euclid and Aristotle, our constructions could be very different now.

Solids from Kepler’s Harmony of the World, exploring the relationship between harmony and geometry. (Available as a print and as a face mask.)

To be fair, the rare few did look and did see different constructions of reality — the Hungarian teenager who, two hundred years ago, subverted Euclid and equipped Einstein with the building blocks of relativity; the sickly German mathematician who, four hundred years ago, subverted the celestial interpretations of the church to give us the revolutionary laws of planetary motion while defending his mother in a witchcraft trial.

Frame writes:

Unless there were only one geometry, only one story — only one world — we should not expect the same categories to grid our views of the universe… Could the world be different than we think? Is it different? Must it be only one thing, or can it be many? If we view the world in one way, does this forever bar us from all others?

Art from An Original Theory or New Hypothesis of the Universe by Thomas Wright, 1750. (Available as a print, as a face mask, and as stationery cards.)

Pointing to a resounding “no” in the many-worlds model of quantum mechanics — a model in which “every observation of every particle splits the universe into branches, one in which each measurement outcome occurs, and communication between these branches is impossible” — he adds:

Once they are recognized, these patterns cannot go unnoticed. They change forever how the image of the world unfolds in our minds, change forever the categories of the models we build.

This recognition-as-model-revision, Frame intimates, is also the way to view and live through grief — an exercise in continual dilation of perspective, so that life can be seen from more and more angles besides the acuteness of loss, noticing more and more of what is there, what remains and what grows in the wake of the lost; an exercise in remembering, again and again, that healing is subtle and unpredictable, unfolding in tiny, quiet, immeasurable increments that eventually add up to profound changes of measurable difference.

Returning to the consolation of fractals — the mathematical language composing chaos theory — Frame writes:

Small changes may not cause large differences, but small changes, invisible because of our inability to measure exactly, can mask our ability to predict whether, when, and where large differences can occur. Chaos is about the breakdown of our ability to forecast for more than a short time.

One of Wilson Bentley’s pioneering 19th-century photographs of snowflakes, one of nature’s fractal masterpieces.

What most readily unblinds us to that vital smallness comprising the grandeur of change and aliveness is a willful attentiveness to beauty — so often the antipode to the brutality of life, so often the portal to aliveness in the face of death, always the supreme testament to pioneering psychologist and philosopher William James’s insight that our experience is what we choose to attend to.

Attentiveness to beauty is the instrument of transcendence — that essential facet of Frame’s geometry of grief and readjustment. In consonance with Willa Cather’s lovely insistence that “unless you can see the beauty all around you everywhere, and enjoy it, you can never comprehend art” — or life — he writes:

Beauty is a bridge between grief and geometry.


Beauty and grief are next-door neighbors, or maybe grief is beauty in a dark mirror… To see beauty is to glimpse something deeper; to grieve is to glimpse a loss whose consequences we will not unpack for years, and maybe never. The beauty of geometry likewise involves great emotional weight, irreversibly alters our perceptions, and is transcendent. For we don’t see all of geometry, only a hint, a shadow of something much deeper.

The Dreaming Horses by Franz Marc, 1913. (Available as a print and as stationery cards.)

In one of the book’s tenderest moments, illustrating this sidewise gleam at the depths, Frame shares a short lyrical essay he composed after his mother’s death, in response to a creative prompt from a student compiling meditations on gravity:

Gravity holds my feet on the ground. Gravity keeps the earth traveling around the sun, the sun dancing around the galaxy, the galaxy threading through the Local Group, and on and on.

Gravity pulls rain out of the sky. And snowflakes. And leaves in autumn. And tears from my eyes when I knew you really are gone. Where did you go?


The distance between here and there is the answer to the wrong question.


I thought gravity pulled my mind into the past, stuck in memories. But now I know I can’t trust memories. Some are invented, all are edited. The whole web of who I am — what I’ve seen and done, what skills I’ve found — is nothing but fog.

Gravity pulls me to the future, bits of me falling off along the way. Each of us disappears into the mist of the possible. In our minds, time is gravity’s other side.

Complement Frame’s Geometry of Grief with Emily Dickinson (who believed that “best witchcraft is geometry”) on the dual spell of love and loss, Hannah Arendt on the antidote to the irreversibility of life, Derek Jarman on gardening as a means of growing though grief, and Nick Cave on loss as a portal to aliveness, then revisit the story of how Benoit Mandelbrot’s discovery of fractals illuminated the hidden order behind chaos.


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.


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

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

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

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

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

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

Howard Nemerov

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

by Howard Nemerov

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

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

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

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


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

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

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

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

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

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

John Herschel (artist unknown)

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

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

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

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

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

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

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


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