At a moment of profound collective grief, we are all looking for tools that will help us to process and grow from the challenges of our time. Few would suspect that such tools might be found in geometry. In his profound and hopeful book, Geometry of Grief: Reflections on Mathematics, Loss, and Life, mathematician and celebrated teacher Michael Frame draws on a career’s worth of mathematical insight—including his work with pioneer of fractal geometry Benoit Mandelbrot—as well as his and others’ personal experiences with loss to develop a mathematically-informed theory of grieving and growth. Complete with original illustrations and surprising moments of humor, Geometry of Grief helps us to see both the beauty of mathematics and the possibilities beyond our pain.

It is also a book that has given rise to an unexpected correspondence between Frame and anthropologist and writer Barbara J. King, whose own books include How Animals Grieve and Animals’ Best Friends: Putting Compassion to Work for Animals in Captivity and in the Wild. Read on for an illuminating exchange between Frame and King.

BJK: Geometry of Grief came out in September, was chosen by Andrew Robinson at Nature magazine as one of his five best science books for the week, led to a Boston Globe profile and your own essay in the magazine PSYCHE, and drew praise from other authors—including me. I was captivated by your writing about geometry as a way to cope with grief, an emotion that inevitably emerges for each one of us at some point. Given your lifelong love of math, and teaching math, did the idea emerge for you gradually—or more as a sudden flash of insight—that aspects of grief can be illuminated and perhaps ameliorated by geometry?  

MF: This is an example of a simple question that does not have a simple answer. Early on I recognized that the sense of surprise, joy even, that accompanies first understanding how ideas fit together in a geometrical proof could not be repeated. You still understand the proof, but the amazement that the approach works can occur only once. Of course, there are many geometric proofs, so the supply of potential “aha” moments is in no danger of being exhausted.  

Still, there is something irreversible about first understanding an idea. And the loss of someone dear to us is irreversible. I’ll never have another new conversation with my aunt Ruthie, I’ll never show her my first book or introduce her to Jean. These and thousands of other things won’t happen. This is part of the irreversibility of death and it can be incandescent.

At least consciously, I saw no connection between the irreversibility of the loss of a loved one and the irreversibility of first understanding a complex idea. Then I read Ethan Canin’s novel A Doubter’s Almanac and began to think there might be a connection between these two types of loss. A very interesting conversation with my friend Joe Calamia made the connection seem plausible, so I thought “let’s see if this makes sense.” Geometry of Grief is my attempt to fill in the bits of this notion.

As for amelioration, I used my experiences with geometry to guide how I approached grief at the loss of a person or a cat. For me, this does help. Will this approach adapt to others?  I do hope it might help some people look at geometry a bit differently. As for approaches to grief, if geometry cannot guide you, maybe you can apply these ideas to something you do love and use that to direct your approach to grief. 

BJK: You write, “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.” Can you say more about the impact on you of the losses you speak of in the book—of loved ones but also of aspects of your own youth and health? Does it bring hope to you that this book is now out in the world?  

MF: Loss of family members is something we all share; how we experience these losses depends on our relations with these family members. I’ll tell my story, but your mileage may vary. I loved Mom and Dad tremendously, always enjoyed time with them, especially the stories they told. When they died, part of my grief was knowing I’d never ever hear any more stories from them. But when I thought some more about stories, I began to recall that stories were more frequent when we were young.  Maybe Mom and Dad wanted us to learn some from their experiences. When a parent tells a story to a little kid, often the kid nods politely and then forgets the story. That’s sort of what happened to me, but not quite. The stories still are in there, but the access path is complicated. Eventually, I began to remember some; my brother and sister reported similar memories, so I am confident these stories are not invented memories. Even though Mom and Dad are gone, the impossibility of new stories from them gave me a laboratory to recover old stories long inaccessible. This loss contains a key to blunting the pain of the loss.

I want to mention one more example, illustrated clearly by a passage from Ann Pancake’s novel Strange as This Weather Has Been. “I learned what it is to grieve your life lost while you’re still living, and I learned that there are few losses harsher than that. It was grief beyond anything I’d imagined. I can still feel sometimes that dry raw socket. The slash, then the body-burning pain.” Sometimes fiction can help us to see so very clearly. For decades and decades, I studied math and developed some skills as a teacher. But eventually memory, especially what concepts gave problems to which students, began to fade. Cognitive plasticity faded, too. I couldn’t understand what question a student was asking, what was the underlying issue. I couldn’t do the job as well as I used to, as well as I wanted to, so I resigned in 2016. And still, I dream I am teaching; wake to recognize the dream and have a white-hot moment of “I’ve made a terrible mistake.” I wanted to be an eighty-year-old, doddering into the classroom with a strip of toilet paper stuck to my shoe, still telling the same dumb jokes halfway through each class. But I’m not able to do that, and sadly I know I’m not able to do that. I grieve for what I could have done with my life, and yet I can find small echoes of what might have been done folded within what has been done.   

As for hope, of course, I have it, but mostly it is based on wishful thinking. This book is an experiment. I don’t know whether these ideas or how I’ve expressed them, can help other people. I do hope they can.

Mostly I hope that this book helps some people think about geometry a bit differently, that a few will apply this approach to dealing with emotion in other settings. I would be so happy to see this geometric approach provide insights into other emotions. ”Geometry of Love,” “Geometry of Loneliness,” or “Geometry of Kindness” would be wonderful to see.           

BJK: We share a love of cats. When I got to the story of Scruffy’s life and death in the book, I had to stop, breathe, and compose myself. Can you say something about how you came to cope with Scruffy’s death, as an example of the type of redirecting- grief work you discuss in the book?

MF: Scruffy had feline leukemia, but lived for five years with us. After I overcame my cat allergies—that took a year and a LOT of antihistamines—Scruffy slept every night curled up on Jean’s shoulder. He sat on the floor in front of our feet, and when he caught our eyes, he’d jump straight up onto our shoulders. He was a sweet kitty, but five years after he moved into our house, he developed a true leukemia and our vet told us he had maybe a week left, maybe less. The irreversibility of imminent death hit me violently. I didn’t want Scruffy to die, couldn’t imagine our days without his sweet kitty self. When we knew it was time, we took Scruffy to our vet. The first shot relaxed Scruffy. He sat on the examination table and purred while we petted him. In a few minutes, Scruffy’s front legs slid out from under him and he closed his eyes. We called the vet in. He gave the other shot, and Scruffy was gone.  

But where did he go? His sweet, playful, tender catness was gone, never to return. I understood the breakdown of neural connections and the increase in entropy, but these don’t help with emotional distress. At this time I hadn’t thought of self-similarity of grief, but I realized that other neighborhood cats exhibited some of Scruffy’s traits. Scruffy was gone, but some of his actions and attitudes remained. Early on, this seemed to amplify how much I missed Scruffy, but eventually, I understood that it also could give me ways to see aspects of him still in the world. Our cat was gone, but some of his characteristics weren’t. And this did help.  

BJK: I have derived great pleasure from our newfound correspondence—all flowing from our mutual interests in books, writing, animals, nature, and of course grief. Because we exchange emails, I know that you grew up in West Virginia, have an abiding interest in Appalachia, and are working with your cousin on a new book. What can you tell us about that project?  

MF: I still can’t believe we’re trading emails. The part of the world you inhabit is so much larger than the part I see. Getting to know you is one of the unexpected benefits of writing Geometry of Grief.  

My cousin Patti Reid and I independently read J. D. Vance’s Hillbilly Elegy, and while we have sympathy for the hardships Vance overcame, the hill folk he describes are not at all those of the southern West Virginia and eastern Kentucky where Patti and I grew up. Vance’s memoir can be read as implying that hill culture is monolithic, but this simply isn’t the case. Initially, we had planned to compile our own stories, to present a view of hill culture different from Vance’s. We believe much of, maybe most of, Appalachia’s problems come from outside sources: aggressive lumber, coal, and natural gas companies tricked thousands of hill families into signing away mineral and timber rights in exchange for land “rights,” only to learn too late that the fine print of the contracts allowed companies to extract resources without concern or compensation for destroyed homes, farmlands, and even cemeteries. Then there’s the whole opioid mess. And don’t forget the negative tropes of hillbillies as mentally deficient, lazy, hyperviolent lunatics. So we decided we needed to read some history of the region, the Mine Wars and the Battle of Blair Mountain, Mother Jones, the UMWA, strip mining, and on and on. Then too we both believe that literature, art, and music can give clear portraits of many aspects of a culture. Our background research includes reading a bunch of novels and stories by Jayne Anne Phillips, Breece Pancake, Ann Pancake, Chris Offutt, Pinckney Benedict, Harriette Arnow, Pearl Buck, Homer Hickam, and others.    

This is a lot of fun, though in a different direction for me. Still, late in life we sometimes do head for home. I can’t do that physically now, so writing stories is the best I can manage. Working with Patti is a delight. We think just differently enough that each of us always sees something else in the other’s writing.  

BJK: I can hardly let you go without asking about fractals. You worked for twenty years with Benoit Mandelbrot and discuss fractals fairly extensively in Geometry of Grief. But I would like to ask about the role of fractals in your personal life. Do you actively look for them in nature? In what ways might we stand to gain, if we, all of us, go looking for them?

MF: When I taught fractal geometry, the first few classes were focused on simple geometrical examples because with these we see more easily that the whole shape is made of pieces, each looking like a shrunken copy of the whole. But the world is not simple geometry: interacting growth and decay processes add random elements into natural fractals. Once we learn to recognize these shapes, examples abound. A tree in winter, branching of the lungs, the roughness of coastlines and river drainage networks, and on and on. Most years I taught, I got emails from roommates of some of my students complaining that whenever they go for a walk with my student, they are bombarded with “There’s a fractal, and there’s another over there.” Once you learn to recognize fractals, you can’t unlearn that.  

By now I’ve seen so many examples of natural fractals that each I see now is more like meeting a relative of an old friend than being introduced to someone new. To recover the initial sense of wonder, I began to look for fractals in abstract spaces, the space of emotions, for example. Each grief contains many smaller griefs: the loss of a parent contains the loss of sharing stories with the parent. Smaller griefs can be laboratories to explore ways to reduce the pain of grief, then extrapolate these approaches to the larger grief.  

How many other emotional states can be studied in this way? What I most hope is that this geometric and fractal study of grief will help people adapt this approach to other circumstances of our lives. The more similarities we see, the more connections with other people, the more porous become the tribal boundaries that appear to separate us so violently now. If this book can bring even the tiniest amount of understanding between people, it will have done more than my wildest wish.

Michael Frame retired in 2016 as adjunct professor of mathematics at Yale University. He is coauthor of Fractal Worlds: Grown, Built, and Imagined and coeditor of Benoit Mandelbrot: A Life in Many Dimensions.

After twenty-eight years of teaching anthropology at the College of William and Mary, Barbara J. King retired early to become a science writer and public speaker. The most recent of her seven books are Personalities on the Plate: The Lives and Minds of Animals We Eat and How Animals Grieve, both published by the University of Chicago Press. King’s work has been featured at Scientific American, Aeon, Undark, SAPIENS, NPR, the BBC, Times Literary Supplement, the World Science Festival, and the annual TED conference in Vancouver. Her TED talk on animal love and grief is available online at https://www.ted.com/speakers/barbara_j_king.

Geometry of Grief: Reflections on Mathematics, Loss, and Life is available now from our website or your favorite bookseller.