When the leaves fall this autumn, the bare trees will reveal their fractal nature: The pattern of twigs on a branch resembles the larger pattern of branches on the tree. “The shape is made of smaller bits that look like the whole shape,” says Michael Frame, a retired Yale University mathematician.
Fractals are all around us — in a fern’s feathery leaves or our own branching blood vessels. Their geometry has applications in fields including economics, medicine, and physics. Frame also sees them as a useful metaphor for visualizing and understanding grief.
Frame’s new book, “Geometry of Grief: Reflections on Mathematics, Loss, and Life,” suggests that thinking about fractals — and thinking geometrically, in general — can help us process life’s most difficult moments.
It’s just one of a few books being released this fall that make the case for viewing the world through the lens of mathematics. Marcus du Sautoy’s “Thinking Better: The Art of the Shortcut in Math and Life” explores how any of us can use the tools of math to solve everyday problems and make predictions. A similar idea drives David Sumpter’s “The Ten Equations That Rule the World: And How You Can Use Them Too.”
Frame’s book is less of a how-to than a meditation on his decades studying and teaching fractal geometry and the insights they gave him about visualizing unexpected turns in life.
Many of us imagine our lives as having a trajectory or a path. But our view of that path is incomplete, Frame says. To illustrate his point, he holds out his hand with the palm facing down. Imagine a light shining from above, he says. On the table below, you’d see the shadow of five spread fingers. But if he rotated his hand so his thumb pointed to the ceiling, the shadow would change. The spaces between the fingers would shrink until the shadow became one line. The picture on the table doesn’t reveal the whole story.
Similarly, Frame says, the way we understand the stories of our own lives is limited because we can’t see every factor at once.
When a loved one dies, we may perceive a huge hole in our life’s trajectory. But from another angle, that gap may not look so big. For example, Frame says his father loved to tell stories about his time in the military. When his father died, Frame decided to continue telling his father’s stories and asking other veterans to share tales of their own.
From that angle, part of his dad is still alive, Frame says: “By focusing on different aspects of your life, you can reduce the apparent size of the gap, and therefore make the pain of loss a bit smaller.”
He also finds comfort in thinking about grief itself as a fractal. “Each loss is made up of a bunch of smaller losses,” he says. The death of his father, for example, contains the loss of walking in the woods together and of sitting on the deck to watch fireflies. Changing his perspective on those smaller losses — rotating the shape, seeing the connections that remain and the angles from which the gaps are smaller — helped him learn to cope with the greater loss, Frame says.
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.
Frame points out that even the study of mathematics carries grief. “The first time you understand a geometrical proof,” he says, “you’ve changed the way you see some small part of the world. This complicated idea suddenly fits together in a beautiful way.” But that thrill can happen only once, he says. “The elation that you feel when you first understand something — that will never come back.” Like a death, that loss is irreversible.
In fact, he says, “every time you understand something important, there’s built in with that a little bit of grief. And every time you love somebody, there’s built in with that a potential for grief.”
Frame says, though, that there are limits to what math can help us understand. He loves fractals because their complex shapes can be defined by simple equations. But, he acknowledges, not everything is like a fractal — for example, human behavior. “Sometimes,” he says, “complicated things have complicated explanations.”
Elizabeth Preston is a science journalist in the Boston area.