Imagine waking up in a Salvador Dalí painting. You reach over to hit the snooze button on your alarm clock, only to discover that it oozed into a puddle during the night. The sun rising outside your window illuminates an elephant tottering down the street, its legs impossibly skinny stilts a hundred feet tall.

In this fantastic reality, everything is slowly but continuously distorting itself. Your coffee mug morphs into a doughnut, as if made of putty that’s been pinched and pulled. Breakfast is a confusing experience, as everyday items lose their identities.

This may sound like a bad trip. We all know that things in the real world tend to have fixed shapes; the letter L is different from the letter M. But in the funhouse world of topology, a field of mathematics that is slowly reshaping how we think about the world, the usual rules don’t apply. Its practitioners believe that an L is essentially the same as an M, or a C, or a Z. To topologists, objects that can be gently bent, twisted and stretched into each other are, in a sense, fundamentally identical.

For centuries, this Play-Doh way of looking at things held little practical value, but that’s starting to change. Topology is guiding how we make sense of big data today. It has helped physicists discover new materials that conduct electricity unlike anything else on Earth — and new physics hidden inside those materials. It has even inspired Microsoft’s efforts to develop a machine that sounds like science fiction: a quantum computer that promises to solve problems beyond the reach of today’s devices.

“Topology was not applied in a serious way to any serious important problems for a long time,” says Gunnar Carlsson, a Stanford University mathematician and expert in topology. “But it has become a force to be reckoned with in the 21st century.”

**Seven Bridges Road
**

The story of topology begins almost 300 years ago, when one of the smartest mathematicians on the planet heard about a puzzle. The denizens of a faraway European city wanted to know: Could you stroll through their town, Königsberg, and cross each of its seven bridges exactly once?

That mathematician, Leonhard Euler, was an intellectual giant who created much of the mathematical notation we use today and produced arguably the most beautiful equation of all time. He didn’t think much of the Königsberg Bridge Problem at first, calling it “banal” in a letter. Thankfully, the Swiss genius decided to give it his attention, anyway. Euler’s contemporaries couldn’t find such a path — but they could not prove one didn’t exist, either.