Quanta Magazine

In the mid-1980s, like Walkman cassette players and tie-dyed shirts, the buglike silhouette of the Mandelbrot set was everywhere.

Students plastered it to dorm room walls around the world. Mathematicians received hundreds of letters, eager requests for printouts of the set. (In response, some of them produced catalogs, complete with price lists; others compiled its most striking features into books.) More tech-savvy fans could turn to the August 1985 issue of Scientific American. On its cover, the Mandelbrot set unfolded in fiery tendrils, its border aflame; inside were careful programming instructions, detailing how readers might generate the iconic image for themselves.

By then, those tendrils had also extended their reach far beyond mathematics, into seemingly unrelated corners of everyday life. Within the next few years, the Mandelbrot set would inspire David Hockney’s newest paintings and several musicians’ newest compositions — fuguelike pieces in the style of Bach. It would appear in the pages of John Updike’s fiction, and guide how the literary critic Hugh Kenner analyzed the poetry of Ezra Pound. It would become the subject of psychedelic hallucinations, and of a popular documentary narrated by the sci-fi great Arthur C. Clarke.

The Mandelbrot set is a special shape, with a fractal outline. Use a computer to zoom in on the set’s jagged boundary, and you’ll encounter valleys of seahorses and parades of elephants, spiral galaxies and neuron-like filaments. No matter how deep you explore, you’ll always see near-copies of the original set — an infinite, dizzying cascade of self-similarity.

That self-similarity was a core element of James Gleick’s bestselling book Chaos, which cemented the Mandelbrot set’s place in popular culture. “It held a universe of ideas,” Gleick wrote. “A modern philosophy of art, a justification of the new role of experimentation in mathematics, a way of bringing complex systems before a large public.”

The Mandelbrot set had become a symbol. It represented the need for a new mathematical language, a better way to describe the fractal nature of the world around us. It illustrated how profound intricacy can emerge from the simplest of rules — much like life itself. (“It is therefore a real message of hope,” John Hubbard, one of the first mathematicians to study the set, said in a 1989 video, “that possibly biology can really be understood in the same way that these pictures can be understood.”) In the Mandelbrot set, order and chaos lived in harmony; determinism and free will could be reconciled. One mathematician recalled stumbling across the set as a teenager and seeing it as a metaphor for the complicated boundary between truth and falsehood.