A mathematician sees reassuring logic in the twists and turns of life

When the leaves fall this fall, 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 little pieces that look like the whole shape,” says Michael Frame, a retired mathematician from Yale University.

Fractals are all around us – in the feathery leaves of a fern or in our own branching blood vessels. Their geometry has applications in fields such as 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 deal with life’s most difficult times.

This is just one of a handful of books published this fall that make the case for viewing the world through the lens of mathematics. “Thinking Better: The Art of Shortcuts in Mathematics and in Life” by Marcus du Sautoy explores how each of us can use the tools of mathematics to solve everyday problems and make predictions. A similar idea is behind David Sumpter’s “The Ten Equations That Rule the World: and How You Can Use Them, too”.

Frame’s book is less a manual than a meditation on his decades of studying and teaching fractal geometry and the insights they gave him on visualizing life’s unexpected turns.

Many of us imagine our lives as having a trajectory or a path. But our view of this path is incomplete, says Frame. To illustrate his point, he holds out his hand with his palm facing down. Imagine a light shining from above, he said. In the table below, you would see the shadow of five fingers spread apart. But if he turned his hand so that his thumb was pointing towards the ceiling, the shadow would change. The spaces between the fingers would narrow until the shadow became a line. The picture on the table does not reveal the whole story.

Likewise, Frame says, the way we understand the stories of our own lives is limited because we can’t see all of the factors at once.

When a loved one passes away, we can perceive a huge hole in the trajectory of our life. But from another perspective, this gap may not seem so big. For example, Frame says her father loved to tell stories about his time in the military. After her father died, Frame decided to continue telling her father’s stories and ask other veterans to share their own stories.

From this perspective, part of her father is still alive, Frame explains: “By focusing on different aspects of your life, you can reduce the apparent size of the gap and therefore reduce the pain of loss a bit.”

He also finds comfort in viewing grief itself as a fractal. “Each loss is made up of a bunch of smaller losses,” he says. Her father’s death, for example, contains the loss of walking together in the woods and sitting on the deck watching the fireflies. Changing his perspective on those small losses – rotating the shape, seeing what connections remain and the angles at which the gaps are smaller – has helped him learn to deal with the biggest loss, Frame says.

By zooming out rather than in, we might see our individual sorrows as little versions of other tragedies in the world. Perhaps, Frame says, viewing grief as a fractal can inspire empathy and cause us to channel our sadness to help others.

Frame points out that even studying mathematics is damaging. “The first time you understand geometric proof,” he says, “you changed the way you see a small part of the world. This complicated idea suddenly fits together in a beautiful way. But that thrill can only happen once, he says. “The exhilaration you feel when you first understand something – it will never come back. Like death, this loss is irreversible.

In fact, he says, “every time you understand something important, it comes with a little sorrow. And whenever you love someone, there is the potential for heartache.

Frame says, however, that there are limits to what mathematics can help us understand. He loves fractals because their complex shapes can be defined by simple equations. But, he admits, not everything is like a fractal – for example, human behavior. “Sometimes,” he said, “complicated things have complicated explanations.”

Elizabeth preston is a science journalist from the Boston area.

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