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Geometry of Grief

Reflections on Mathematics, Loss, and Life

Geometry of Grief

Reflections on Mathematics, Loss, and Life

In this profound and hopeful book, a mathematician and celebrated teacher shows how mathematics may help all of us—even the math-averse—to understand and cope with grief.
We all know the euphoria of intellectual epiphany—the thrill of sudden understanding. But coupled with that excitement is a sense of loss: a moment of epiphany can never be repeated. In Geometry of Grief, mathematician Michael Frame draws on a career’s worth of insight—including his work with a pioneer of fractal geometry Benoit Mandelbrot—and a gift for rendering the complex accessible as he delves into this twinning of understanding and loss. Grief, Frame reveals, can be a moment of possibility.

Frame investigates grief as a response to an irrevocable change in circumstance. This reframing allows us to see parallels between the loss of a loved one or a career and the loss of the elation of first understanding a tricky concept. From this foundation, Frame builds a geometric model of mental states. An object that is fractal, for example, has symmetry of magnification: magnify a picture of a mountain or a fern leaf—both fractal—and we see echoes of the original shape. Similarly, nested inside great loss are smaller losses. By manipulating this geometry, Frame shows us, we may be able to redirect our thinking in ways that help reduce our pain. Small‐scale losses, in essence, provide laboratories to learn how to meet large-scale losses.

Interweaving original illustrations, clear introductions to advanced topics in geometry, and wisdom gleaned from his own experience with illness and others’ remarkable responses to devastating loss, Frame’s poetic book is a journey through the beautiful complexities of mathematics and life. With both human sympathy and geometrical elegance, it helps us to see how a geometry of grief can open a pathway for bold action.

200 pages | 45 halftones | 5 1/2 x 8 1/2 | © 2021

Biography and Letters

Mathematics and Statistics

Physical Sciences: Physics--Popular Books


"How the fractal nature of grief is both the key to understanding it and the doorway to moving through it is what mathematician Frame explores in his unusual book Geometry of Grief: Reflections on Mathematics, Loss, and Life. 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. . . . Attentiveness to beauty is the instrument of transcendence—that essential facet of Frame’s geometry of grief and readjustment."

Maria Popova | The Marginalian

"Frame has written a poignant and beautiful book. . . . Treat yourself to the wisdom of this sweet, gentle soul."

Steven Strogatz | @stevenstrogatz

"Frame's new book, Geometry of Grief, suggests that thinking about fractals—and thinking geometrically, in general—can help us process life's most difficult moments... Zooming out instead of in, we might see our individual griefs as small versions of other tragedies in the world. Maybe, Frame says, thinking of grief as a fractal can inspire empathy and lead us to channel our sadness into helping others."

Boston Globe

"This brief, intriguing personal meditation is inspired by mathematician Michael Frame’s lifelong love of geometry — including 20 years’ collaboration with fractal geometer Benoit Mandelbrot — and the childhood loss of his aunt, who set him on his career path. He writes: 'Grief informs geometry and geometry informs grief.' How so? His epiphany on first understanding any beautiful mathematical idea is always tinged with sadness, because it is unrepeatable. With quirky illustrations, he integrates the lives of his Mom and Dad."


"The word fractal, from the Latin fractus meaning 'broken glass,' was coined by Benoît Mandelbrot, who brought mathematics closer to nature by showing that iterating a simple geometric pattern can result in complex, rough-edged, and beautiful shapes. In Geometry of Grief, Frame, a former colleague of Mandelbrot’s at Yale, aims to unite maths with our lived experience still further by showing how fractals both inform and can help us cope with the experience of irreversible loss. . . . Ambitious and moving.”

Times Literary Supplement

"Frame has written a wonderful memoir. Combining his passion for mathematics and his mastery of the geometry of fractals, in this text he seeks to educate, encourage, and inspire readers in a personal way. Identifying grief as irreversible, Frame makes surprising connections to foundational mathematical concepts such as continuity and self-similarity. . . . Here, he elucidates the foundational intertwining of science and grief in his own life through deeply and surprisingly personal stories from lived experience. Weaving together references from literature, art, popular culture, and mathematics, Frame clarifies the transcendence found through grief as a universal human experience. The text offers a wealth of resources for curious readers and is a must have for the bookshelf of any mathematics teacher who wants to inspire curiosity in students or relate with empathy to a student suffering loss. This book will be a wonderful addition to any advanced or graduate-level seminar, and preprofessionals in mathematics education will certainly benefit from exploring the book's diverse fields of discourse. Highly recommended."


"Frame writes: ‘Times folds up. So many ghosts crowd into my head. Parents, grandparents, aunts and uncles, dear friends, students… And far too many cats.’ Live long enough oneself and one realizes that half or more of one’s friends and relatives have departed the planet, ‘summoned,’ as the poet Robert Southey had it, ‘on the grand tour of the universe,’ before one. One lives with it, saddened yet grateful oneself still to be in the game. Yet some holes never successfully fill up."


“With poignancy and audacity, Frame builds an unexpected bridge between mathematical beauty and human sorrow, illuminating both.”

Francis Su, author of "Mathematics for Human Flourishing"

“I expected to enjoy the experience of thinking in fresh ways with Frame about grief—and encountering his love of cats, really of all nature, made manifest on the page. What blew me away were the exciting new connections among love, grief, beauty, and resilience that flowered in my mind as I read. Immersed in Frame’s world of geometry, including fractals, and its applications to real-world emotions, I sometimes felt afloat in a mysterious, and always inviting, dream. It’s a beautiful place for all of us to spend time.”

Barbara J. King, author of "How Animals Grieve"

“With concision and compassion, Frame shows how a mathematical mind makes sense of a grieving heart. The result is a peculiar, wise, and beautiful book.”

Ben Orlin, author of "Math with Bad Drawings" and "Change Is the Only Constant"

“A unique, meaningful, and moving work that connects the irreversibility of loss that comes with grief and the irreversibility of first deeply understanding something—particularly something mathematical.”

Susan Jane Colley, Oberlin College, editor of "American Mathematical Monthly"

“Captured perfectly an experience that I've long tried to put into words but couldn't—the nostalgic sense that comes immediately after a discovery. It's astonishing how so many important ideas come to us in an almost dream-like state, fully formed and with a hazy beauty that somehow falls away when we more fully work out to implications of that idea and put it into words and onto paper; the whole process makes the idea more real but somehow less charming."

Christina Stankey, medical student and former student of Michael Frame

“Frame believes everyone can fall in love with math if it’s presented with empathy and humor and clarity and context. He portrays math as math-lovers know it: a beautiful garden, a place of curiosity and delight, a tribute to human creativity and the wonders of nature.”

Steven Strogatz, from the foreword to "Fractal Worlds"

Table of Contents

1          Geometry
2          Grief
3          Beauty
4          Story
5          Fractal
6          Beyond
Appendix: More Math


Late in my eleventh year, Ruthie got sick. Hodgkin’s lymphoma, survivable now but not so much in the early 1960s. She was treated with the chemotherapy drug Mustargen, I believe, but lived only a few more months in some misery and died early in my twelfth year. I visited Ruthie when she was sick, but I couldn’t do much. I stood beside her bed, rested my little hand on her forearm and tried to talk with her. But I couldn’t think of anything to say.

At home after these visits, Mom hugged me, stroked my hair. I knew I should have talked more with Ruthie. She had done so much for me, and she needed me now. She needed me to talk with her because I was her favorite. Later I understood that Mom was working through her own grief. She knew the situation far better than I did, knew this disease would win and Ruthie would lose. Dad began to talk with me about his sister’s illness. He was straightforward: Ruthie was going to die. I appreciated his honesty. No nonsense about Ruthie going away, or—worse—going to live with the angels. Her life would end, and soon. “This isn’t fair. There’s so much more for Ruthie and me to do. She promised we’d get a telescope to look at the planets. I’ve saved my allowance for six months already. This just isn’t fair.”

“Son, life isn’t fair. Ruthie isn’t sick because she did anything bad. She just got sick. Sometimes good things happen, sometimes bad things happen. All we can do is try to make a few more good things happen and a few less bad things happen. But a lot of things that happen to us, we can’t do anything about.”

“Dad, that’s really scary.”

“Yes, son, it really is.”

That night I thought of a plan. I’d work very, very hard. Study all the time, no more hide-and-seek or listening to silly stories. I’d finish high school years early, go to college, then graduate school and medical school, become a medical researcher, find a cure for Hodgkin’s lymphoma, administer it to Ruthie, and save her. In one version of the fantasy, I flew in a helicopter from my university laboratory to Ruthie’s hospital. I was so pleased with my plan...

But Ruthie died. Dad was at the hospital with her, holding her hand, when she died. When he came home, his expression told me all I needed to know...

Dad didn’t want us kids to go to the funeral. Mom and Dad went while we stayed with Mom’s parents, Burl and Lydia Arrowood. I found a bag of balloons in Gramp’s workshop. Gramp was a jeweler and repaired clocks and watches. Because he used a gas torch to melt some alloys, his workshop had a gas jet. I filled a balloon with gas, tied it off, walked into the front yard away from the trees and let the balloon fly. The symbolic content of this was melancholy: it represented all the experiments that Ruthie and I had planned to do, that now were lost forever. It represented the closing of a door.

Later that year, I read a supplementary problem in my algebra text. For much of the weekend I tried all sorts of tricks. Eventually I found a solution, but it was clunky, mechanical, inelegant. It worked, but I knew it wasn’t the solution the author intended.

After math class on Monday, I asked my teacher. She smiled, said she was happy I tried the problem, then wrote the simple, beautiful solution.

At that moment, my world folded in on itself, disappeared, and I knew what I thought was a different flavor of grief. The solution used only tricks I knew but applied them in a clever way that hadn’t occurred to me. At that moment, I began to suspect I was not bright enough to be a good scientist. Determination and hard work would get me into the tribe of scientists, but would life as a supporting character be enough? Choosing that path carried the real risk that from the end of life, where I am now, a backward look would reveal decades of steady work punctuated by very few moments of modest insight. To be sure, those moments have been amazing. The pleasure of understanding some bits of the architecture of ideas is ample reward. But I wanted to do so much more....

For me, the path I choose—exploring some structures of math—has afforded new perspectives on grief. I believe grieving exhibits some similarities to doing math; we’ll find echoes of each in the other. Wrestling with mathematical questions has helped me to parse my own episodes of grief. That is my subject here...

Grief is a response to an irreversible loss. A corollary: there is no anticipatory grief. 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. We’ll focus on these three aspects of grief: it is irreversible, it carries emotional weight, and it is transcendent. Grief is not the only experience that exhibits these properties...

The initial “aha” moment, when first we understand something, can occur only once. If the thing understood is important to us and hints at deeper mysteries, we can grieve the loss of this moment, which, once we experience it, is lost forever. Beauty seen in a mirror reflects grief. This is the key to the connection I’ll make between geometry and grief....
  • 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...
  • Moreover, grief is self-similar: the grief of losing a parent contains many “smaller” griefs. No more conversations, no more comparing memories of times good or bad, no more quiet walks together. Each of these is a scaled version of the response to the loss of a parent, a smaller copy that can act as a laboratory to find effective projections. Projecting outward, grief can point to actions that can help others. My most optimistic thought is that some of the energy of grief can be redirected in this way. Small steps or large steps, but steps all the same. 
This book is a love song to my late parents, to friends we’ve lost and to cats we’ve lost. And the book is a love song to geometry, the brightest point in my mind. In old age, my understanding of geometry disappears more with each passing year and adds complex fractures to my breaking heart.

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