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.

Another corollary to this is that every loss, in both the evolutionary and existential senses, reveals our true nature. However, every loss can also reveal 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. Yet, the shrapnel fragments that are now on the surface of our bodies is smaller and more delicate than they were when they entered the open wound of bereavement. They have been shaped and contract by the continual flow 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.

Mandelbrot Set. (Illustration by Wolfgang Beyer.

Mathematicians can help you understand grief’s fractal nature and how to move through it. Michael FrameHis unusual book Geometry of Grief (public library) explores these themes. Frame spent 20 years working alongside the visionary father fractals. Frame also taught fractal geometry at Yale for another twenty years. Frame uses a lifetime experience of loss, as well as an uncanny ability to pay attention to what we call beauty, to weave together mathematics and memoir in an extraordinary tapestry, including twining Borges and quantum mechanics, genetic biology, Islamic art, and multiverse theories.

Frame is a fundamental definition of sound because each theorem depends on precise formulation.

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. Allow the fog to lift away and let in a bright spot 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. This print is also available as a mask for the face.

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 wrote this:

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

Our lives can be viewed as stories, which are governed by time and space.

It is impossible to simultaneously see all possible variables. Instead, we tend to focus our attention on just a handful of variables, limiting our ability and resources to view a subspace of story space.

Trajectories in these subspaces reflect the stories we tell of ourselves and our lives. However, they do not always include some aspects of our experience.

In our story space journey, irreversible loss is a transition, a leap.

Focusing on subspaces can help us reduce the magnitude of jumps by projecting our trajectories onto these areas. This will allow us to find a way for emotional loss to be confronted and possibly 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 It 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 beThey are more than we imagine them to be. Frame says:

Geometry can be used to help us organize models of the world’s shapes and dynamics. But isn’t this all contingent, balanced on a knife’s edge? Our models could have been very different. What if Mandelbrot’s fractal geometry was discovered prior to Euclid? Would the manufacturing process be identical? You might be skeptical about the absurdity of the question. Consider the iterated branches of our nervous, circulatory and pulmonary systems. Also, the recursive folding and large area of our DNA. The fractal geometry has been discovered by evolution. 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 WorldThis book explores the relation between harmony and geometric. Available as both a printed version and as a mask for the face.

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.


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? Are they different? Can it only be one thing or could it be multiple? Does this mean that we are forever bound to one view of the world?

Thomas Wright (1750): Art from an Original Theory or New Hypothesis about the Universe. This print is also available as a stationery card, a face mask and a printed copy.

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:

They are easily recognized and can be noticed. These patterns change the way the world appears in our heads forever, and they change the types of models that we create.

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 isThere is what remains, and what grows from the loss. This exercise reminds us that healing can be subtle and unpredictable and unfolds in small, quiet and immeasurable steps that add up over time to make a significant difference.

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

Although small differences may not be significant, they can cause big changes. However, our inability of measuring accurately can make it difficult to see how large and subtle changes could affect our ability to forecast when, where, and why. Chaos can be described as the failure to accurately predict events for more than a few minutes.

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 bridges 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. Also, the beauty and transcendence of geometry is a matter of immense emotional weight. For we don’t see all of geometry, only a hint, a shadow of something much deeper.

Dreaming HorsesFranz Marc, 1913. This print is also available as stationery card.

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:

My feet are held down by gravity. Gravity prevents the earth from moving around the sun.

The sky is pulled by gravity. Snowflakes. The leaves of autumn. When I realized you were gone, tears came from my eyes. I don’t know where you went.


There is no distance between here, and there. This is why you are asking the wrong question.


Gravitation thought I was being pulled into the past and stuck with memories. But now I know I can’t trust memories. Many memories can be fabricated, but all of them are also edited. The whole web of who I am — what I’ve seen and done, what skills I’ve found — is nothing but fog.

I am pulled by gravity to the future. Some bits of me are lost along the way. Every one of us is lost in the fog of possibility. 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.

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Over the past decade, I spent hundreds of hours and thousands each month writing. MarginalianThe magazine, which bore for fifteen years the unsettling name Brain Pickings. Thanks to the support of readers, it has been free from ads and still exists. I have no staff, no interns, no assistant — a thoroughly one-woman labor of love that is also my life and my livelihood. Donations are a great way to make your own life better. It makes a huge difference to support this cause.

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