Mathematics
Homotopy Equivalence and the Euler Characteristic
Quick fact
The Euler characteristic of a sphere (V - E + F) is always 2, no matter how it is subdivided into vertices, edges, and faces—but a donut (torus) has Euler characteristic 0, revealing that it is not homotopy equivalent to a sphere.
Why this is interesting
You can stretch, twist, and squeeze a coffee mug until it becomes a donut—but can you ever turn a coffee mug into a ball? The answer lies in a subtle notion of equivalence that ignores shape and focuses on connectivity.