A topologist can’t tell the difference between a coffee cup and a donut, or so the saying goes. That’s because topology is the study of geometrical objects without considering things like length, angles, or shape. A coffee cup can be transformed into a donut by squashing, stretching and twisting, so the two objects are topologically equivalent. However, a ball can’t be turned into a donut without tearing or cutting, so they are topologically different.
It makes for some clever party tricks, but does topology have any practical applications? If you’ve ever looked at a map of the London Underground, you’ve seen topology in action. This iconic representation of the Tube ignores the distances and physical locations of the various stations, but preserves the links between them. The resulting map is much clearer than the unwieldy real life mess.
Topology also comes in handy elsewhere. Cosmologists use a lot of topology when they are studying the structure of our universe. The exact shape of our universe has very important implications for how it began, how it behaves today, and how it might end. Researchers believe the universe could be in the shape of a sphere, a saddle, or even a horn.
Back on Earth, engineers put topology to work helping them design robots. The movements that robots can make can be thought of as shapes in n-dimensional space, where n is the number of joints the robot has. By exploring the shapes with topology, roboticists can get a better idea of how their robots will move.
Topology is often associated with highly theoretical maths, but these examples show it does have some practical uses.