Benoit Mandelbrot, Pantheon Books, 324 pages, $30

How long is Britain's coastline? Ask the same question of a piece of string; it depends on the string. But for an island, the tape measure matters too. As a cartographer zooms in, new inlets and promontories appear. Time and money, if not waves and tides, will eventually call a halt to such diligence. But it is a fact that the shorter the ruler, the longer the coastline.

This is the result of "self-similarity": Small stretches of coastline look much like bigger ones. Without buildings or trees to fix the scale, a photograph of a 1-kilometer bay could easily be confused with that of a much larger one. Such "self-similarity" is everywhere once you start to look.

This book is the autobiography of Benoit Mandelbrot, the mathematician who finally recognized the utility and ubiquity of such seeming anomalies. He coined the standard term for them, "fractals," from the Latin fractus, meaning broken or shattered.

Mandelbrot died in 2010; his wife, Aliette, completed the book. It contains little mathematics, focusing instead on the path of discovery and the people who helped along the way.

It was a winding and random-seeming path. Mandelbrot moved from Warsaw to Paris in 1936 when he was a boy, and then to rural France at the outbreak of war. Along the way he picked up a patchy but inspiring mathematical education thanks to clever relatives, old textbooks, some excellent schoolmasters and a geometric intuition that helped him to pass notoriously difficult exams in France.

Eventually Mandelbrot landed at IBM, where the strands of the story come together, not least because the development of computer graphics allowed fractals to be plotted with ease for the first time. Brimming with examples, conjectures and groundbreaking images, his best-known book, "The Fractal Geometry of Nature," published in 1982, launched an avalanche of colorful T-shirts, mugs and calendars.

Mandelbrot made no secret of his belief that glory lay as much in coming up with a conjecture as in proving it, leading some to dismiss him as a mere "hand-waver," as mathematicians call those who elide a proof's tricky steps. But to his admirers Mandelbrot was a spell-worker who saw connections no one else did and united apparently disparate phenomena.