Why is a doughnut the same shape as a coffee mug?

To a topologist, a coffee mug and a doughnut are the same shape. Topology is the study of properties that survive stretching, twisting, crumpling and bending, but not tearing, gluing, closing a hole or opening a new one. Two shapes that can be squished into each other like that are called homeomorphic, and the mug and the doughnut are the textbook example, known as the "Topologist's Breakfast".

Here is how the squishing goes. Start with a pliable doughnut, a torus. Press a dimple into one side and keep enlarging it, while the central hole shrinks into the mug's handle. The dimple becomes the place where you pour your coffee. Nothing was cut or glued, and the hole you started with is still there, now as the handle.

That is also why a doughnut is not the same as a sphere. A cube and a sphere are homeomorphic, but a sphere and a doughnut are not: you would have to make or close a hole. The whole subject rests on this idea that some questions depend not on exact shape but on the way things are put together.

Topology began, arguably, with Leonhard Euler. His solution to the Seven Bridges of Königsberg puzzle did not depend on the lengths of the bridges, only on which bridges connect which islands and banks. His polyhedron formula, V − E + F = 2, counts vertices, edges and faces and gives the same answer for every convex polyhedron.

A fun consequence: you cannot comb the hair flat on a hairy ball without a cowlick. It applies to any smooth blob with no holes.