“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 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.

It was always a rapture, a rebellion, a gauntlet against gravity and girlhood — skulking past the teachers, pushing through the boys, and racing across the schoolyard to climb the colossal walnut tree, whose feisty fractal vivacity mocked the bleak Brutalist architecture of my elementary school in Bulgaria.

Then there was my rural-grandmother’s cherry tree, into whose balding crown I would disappear to sulk when my parents discarded me to the country for yet another endless summer. And the copse of horse-chestnut across the street from my city-grandmother’s lightless apartment, whose friendly open-palmed leaves beckoned me to find the first of the spiky green fruit, before they released their shiny brown pebbles of seed onto the cracked sidewalk.

Even as my wrist-bones turned from twigs to branches and adulthood carried me across the Atlantic to lay down my sovereign roots, the impulse never left me, perhaps because the child never leaves us. I climbed — not with the skills and scientific motive force of an arbornaut but with the sylvan transcendence of Blake — oaks in Brooklyn and coastal redwood in Bolinas and Douglas Fir in Olympia and the Tree of Life in New Orleans.

Oak-hopping in New Orleans, September 2020. (Photograph: Milène Lichtwarck.)

Along the way, I came to cherish trees not only as aerial playgrounds, but as wonders of immense poetic, philosophical, and ecological import.

Imagine, then, my delight when a friend handed me a copy of More Women in Trees (public library) by the German photography editor, collector, and curator Jochen Raiss, a follow-up to his improbable hit Women in Trees (public library) — an entry in the ledger of lovely things created by the confluence of chance and choice (which, as Simone de Beauvoir observed with her keen existentialist eye, actually includes our very lives and what makes us who we are.)

It began with a single photograph Raiss found while rummaging through the bin of hodgepodge vintage ephemera at a Frankfurt flea market — a woman, in a tree, happy and carefree.

It delighted him enough to take home and use as a bookmark.

But then, by the marvelous pattern-recognition virtuosity of the human brain, he started finding others during his flea market excursions. He started collecting them. He started carefully cataloguing them in antique wooden crates.

Over the course of a quarter century, he amassed some 140 specimens of the genre, the anthropology of a secret tribe — strange, sweet, subversive photographs of anonymous women engaged in acts of arboreal daring, taken before color film became a commonplace and feminism a conscience.

Some of the photographs were taken when Germany was the roiling epicenter of World War II. Some of the women in them probably hailed Hitler. Some probably died in concentration camps. But for those moments suspended in the branches above the current of their epoch, islanded in space and time, they shared something singular and lovely, united in a sisterhood of sylvan joy.

Their mute, defiant delight seems to be saying, “My grandmother was jailed for wearing trousers but I can win the Nobel Prize in Physics”; seem to be saying, “My mother could not vote but my daughter can be chancellor”; seem to be saying, “I can go as high as I please, damned be gravity and grace, so I can peer at broader horizons.”

Complement the mischievous and marvelous Women in Trees and More Women in Trees with Dylan Thomas’s “Being But Men” — his love poem to trees and the wonder of being human, composed in the same era, an era when “man” included “woman” while erasing women — then revisit artist Art Young’s century-old meditation on human nature in tree silhouettes and Italian visual philosopher Bruno Munari’s existential lesson in how to draw a tree to see yourself.

“When you realize you are mortal you also realize the tremendousness of the future. You fall in love with a Time you will never perceive,” the polymathic poet, painter, novelist, and philosopher Etel Adnan (February 24, 1925–November 14, 2021) wrote at the foot of a mountain she saw as a lens on the meaning of life.

Half a century later and a landmass over, as dawn was silvering the clouds of the Parisian night, she slipped out of the mortal and into the timeless, less than 1000 days shy of having lived 100 years. (No amount of life is enough life, and any amount of life is enough life — as with love.)

Adnan’s uncommon reckoning with mortality and meaning lives on in her final book, Shifting the Silence (public library) — a lucid and luminous stream-of-consciousness outpouring of insight into the nature of existence, an inquiry into what gives meaning to our mortal lives, partway between poetry and philosophy, between requiem and redemption, between Gertrude Stein’s meditation on belonging in a love letter to Paris and Patti Smith’s meditation on dreams in a love letter to time. What emerges is the wakeful work, a life’s work, of naming what is — the ultimate Is beyond the explanations that masquerade as meaning yet containing the ultimate meaning.

Painting by Etel Adnan from Etel Adnan: Light’s New Measure, Guggenheim Museum, 2021. (Photograph: Maria Popova)

Adnan writes:

When we name things simply, with words preceding their meaning, a cosmic narration takes place. Does the discovery of origins remove the dust? The horizon’s shimmering slows down all other perceptions. It reminds me of a childhood of emptiness which seems to have taken me near the beginnings of space and time.

[…]

Word-languages are a trap… They created chaos and made us sink in incoherence… Our words don’t suit prophecies anymore. That power is left to other species: to oak trees, for example, to the tides, which through their restlessness carry a phosphorescence we’re not equipped to hear.

From the fortunate, ramshackle dock of her nine decades — having lived through the splitting of the atom and the Moon landing, through the rise and fall of the Iron Curtain, through a civil war that savaged her homeland and a world war that savaged our civilization, through the heyday Picasso and particle physics and Plath — she observes:

My favorite time is in time’s other side, its other identity, the kind that collapses and sometimes reappears, and sometimes doesn’t. The one that looks like marshmallows, pomegranates, and stranger things, before returning to its kind of abstraction… Today I see eternity everywhere. I had yesterday an empty glass of champagne on the table, and it looked both infinite and eternal.

Visitor to the Guggenheim Museum’s Etel Adnan: Light’s New Measure. (Photograph: Maria Popova)

Writing in the final season of her life, while around her a record heatwave is swarming Paris and wildfires are ravaging the Californian landscapes of her prime and her paintings, Adnan wonders whether this might be the final season of civilization, of the world itself as we know it, wonders whether we can “keep that strange sense of sacredness that we knew, as if by inheritance, in our old days.” She paces the periphery of Paris one timeless step at a time, watches the fog turn the Eiffel Tower into “a faint mark on pure space,” marvels at the magnolia in her garden “thriving in this non-tropical country,” marvels at the first image of an enormous frozen lake newly discovered on Mars and its “pinkish land covered with ice,” savors “the night’s different shades, its infinite richness,” reads a book of poems written by an artificial intelligence and ponders the meaning of reality, the meaning of intelligence. Her mind wanders to the physics of tides, to the Trojan War, to the epoch-making spacecraft that has just landed on the dark side of the Moon, to Picasso’s late erotic etchings of women, to the burning mountain she once lived in and loved with the fire of life. Her wandering mind observes itself:

I am in the midst of whatever I am thinking of. There are fires in California, they have returned. I am burning. Am one of the trees that’s disappearing in the fires. My body black and grey becoming ashes.

And yet there is something else beyond the cinder of the thinking-mind, some vaster consciousness in which the crests and troughs of being and not-being merge into the continuous sine wave of what is, ruffling the oceanic surface of timelessness:

I need to simplify my thinking: to come to the roots of the olive trees I have planted on my island, sit close to them, look at every leaf. Start early in the morning. Then close my eyes and let the morning sun touch my face. Go to the Mediterranean at the street corner, go into its water, its salt, its acid colors, its heat… stop thinking… just be, and for many hours in a row, merge with this vegetal and metallic kind of consciousness which is so overpowering.

Painting by Etel Adnan from Etel Adnan: Light’s New Measure, Guggenheim Museum, 2021. (Photograph: Maria Popova)

A philosopher-friend comes to visit, one of those visits that “lift the sky,” and they talk about “the necessity of an urgent shift of destiny away from the cycle of the eternal return of the same, beyond whatever already is.” A poet-friend dies. “Dear San Francisco, cry for him.” Invoking another friend’s long-ago death that she still carries, and folding into it the incomprehensible awareness of her own mortality — as we invariably do in apprehending another’s — she reflects:

Being, or not being, cannot be dealt with with thinking, but are matters of experience, experienced often in murky waters… Their intensity creates waves that invade us, that leave us stunned. There’s no resolution to somebody’s final absence.

Another friend vanishes into the fog of mental illness, leaving Adnan to contemplate the discomposing dialogue between neurochemistry and identity:

To witness a mind go wild, like the California fires right now, is the hardest thing one can experience. And still, we do. The mind gets so fluid that you can’t stop it with your will, you watch the will’s annihilation. The question arises: are we just a series of chemical reactions? If we were courageous enough we would say yes, we are. But there is something in those chemical reactions that make us reject the acknowledgment of their own nature. We’re body and soul, we say, let’s accept this myth. Plato did it.

Painting by Etel Adnan from Etel Adnan: Light’s New Measure, Guggenheim Museum, 2021. (Photograph: Maria Popova)

Even our ordinary minds, she observes, are too often befuddled by their own mindless activity, the thoughts of which we presume to be the authors — but as any neuroscientist and any longtime meditator can attest, this too is part of the dream of selfhood, the dream by which we flee from the reality that we are each a passing flicker in the consciousness of time and matter.

With an eye to her own experience of “double thinking” — something all of us have experienced in one form or another — she writes:

One thought sliding on another, was startled, didn’t know which one to follow, lost sight of both… Are thoughts bouncing balls? Do we really own them?

She talks to herself, talks to the universe, talks to no one in particular — and then — in a handful of arresting cascades in this stream of consciousness, she talks to you, talks to me, with ravishing intimacy. “I am talking to you because I need you, and to need means to love.” She is talking to us, too, because she has something to impart, the way an oracle does. (Living a century with unrelenting wakefulness to life renders anyone an oracle.)

You know, sunsets are violently beautiful, I would say that they are so by definition, but there are lights, not even colorful in the habitual sense, lights elemental, mercurial, silvery, sulfurous, copper-made, that make us stop, then lose balance, make us open our arms not knowing what else to do, arrest us as if struck by lightning, a soft lightning, a welcome one. I wait for those lights, I know some of you do too, wherever you are, I mean when you are standing by an ocean, alone, within the calmness of your spirit. Be planetary.

To be planetary, she intimates, is to recognize that we are completely together and completely alone all at once, a murmuration of solitudes hurtling through space, out of time:

We’re on a planet sustained by nothing, carried through pure space by a willful star made of fire and in constant ebullition. We’re travelers covering traveling grounds. Going, always going.

Painting by Etel Adnan from Etel Adnan: Light’s New Measure, Guggenheim Museum, 2021. (Photograph: Maria Popova)

The undertone of the book, of Adnan’s farewell message to the living, is the intimation that only in the stillness of silence can we begin to discern where we are going and why:

The universe makes a sound — is a sound. In the core of this sound there’s a silence, a silence that creates that sound, which is not its opposite, but its inseparable soul. And this silence can also be heard.

This silence is the preparation of things to come, but is not free standing. It’s rather the shadow of whatever is, which precedes or follows at will any element that presents itself to this world. Its favorite time is the night.

Half a lifetime after she explored the relationship between dreaming and creativity, Adnan returns to the strange kingdom of sleep and those untrammeled territories of the nocturnal mind beyond thought:

In silence, in the dark, the tides shine, get slippery, their fluidity turns them into a mirage. There’s a persistent hum to the ocean that translates into a back-and-forth movement of our body. Walls disappear and new visual formations invade the imagination. One is not in usual dimensions. Sleep belongs to the past, and the hours too. Luminosity enmeshed with darkness makes us cross over new territories. You move into galaxies in a few seconds, space-time becomes just a game.

Thinking is dimmed when familiar forms of reality disappear. This is not a loss. Long periods of inner silence favor clearings, they let the light in, the flooding, the blinding, the bedazzlement. We need spaces for the reshuffling of new cards, need to be nowhere. Thinking doesn’t always come from preceding thoughts: I suspect it’s always being born, even when it seems related to the past.

Exhibition fragment from the Guggenheim Museum’s Etel Adnan: Light’s New Measure. (Photograph: Maria Popova)

With an eye to Plato’s immortal allegory of the cave, she writes:

Now it’s time to open the cave’s window and leave it open. Let reality fill the space.

Echoing Walt Whitman, who contemplated what makes life worth living after a stroke left him paralyzed, and echoing Mary Shelley, who contemplated what makes life worth living as she envisioned a twenty-first-century world savaged by a deadly pandemic, Adnan adds:

What’s left? This season of heat and wind, this dinner tonight, and these large bands of trembling waves of various shades of green that split my heart with their incredible beauty.

This is Adnan’s parting gift to this world, to us: the life-tested assurance that even when there is too much past and too little future, life is only ever lived day to day, for the living day is all we have — or what Muriel Rukeyser, another visionary of uncommon poetic insight into the nature of being, reverenced as “the living moment… this moment in which we touch life and all the energy of the past and future.”

One such living day, finding herself “at the door of Time’s immensity,” Adnan writes:

The day is blustery, one more day following an infinity of days. And this one on its way out, according to its fate. If everything is alive, this day is too, a life independent from mine, and still interdependent.

Painting by Etel Adnan from Etel Adnan: Light’s New Measure, Guggenheim Museum, 2021. (Photograph: Maria Popova)

On another living day, after rejoicing in having lived to see a human space-probe reach the unseen side the Moon — “I felt the grounds open up under my feet, I felt I reached a landmark of cosmic proportion. I drank beer differently than usual.” — she echoes the civilizational sense that Bach might be our prophet-laureate of aliveness, echoes San Francisco poet Ronald Johnson’s lovely formulation of the elemental poetic truth that “matter delights in music, and became Bach,” echoes philosopher Josef Pieper’s insistence that “music opens a path into the realm of silence” and Aldous Huxley’s insistence that the only thing better able to express the inexpressible than music is silence, and writes:

My hands are getting cold, a musician is playing Bach on a lute on television, and it fits: Bach’s music is the needle of the cosmic balance.

This has taken me into the core of a silence that underlines the universe: underneath the mesh of sounds that never cease there’s a strange phenomena, a counter-reality, the rolling of silent matter. Silence is a flower, it opens up, dilates, extends its texture, can grow, mutate, return on its steps. It can watch other flowers grow and become what they are… Silence is the creation of space… Silence demands the nature of night, even in full day, it demands shadows.

But in all my wanderings I never forgot the light.

Radiating from these pages is at once a welcome and a parting — an invitation to the banquet of life at the deathbed of one particular human who will never again recur as that particular ripple in the consciousness of time but who once lived a long, wide, deep life fully awake to the ephemeral ecstasy of aliveness:

I have invited to my banquet you and your neighbor, and animals too, and stones and mountains, rivers will bring their floods. I will tell you history is made of wars, of ideas, of misery, of glory preceding misery. History is made of everything that has ever happened, the whole trajectory of humans, of dirt and galaxies. You are History, the squirrel is History, the Universe is History. It includes God too.

Painting by Etel Adnan from Etel Adnan: Light’s New Measure, Guggenheim Museum, 2021. (Photograph: Maria Popova)

Complement the portable universe that is Adnan’s Shifting the Silence with poet Lisel Mueller, who lived to the same age as Adnan, on what gives meaning to our ephemeral lives, then revisit Adnan’s stunning painted poem about life, death, and our cosmic redemption, created half a century before she returned her borrowed stardust to the silence of spacetime.

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