The Coffee Cup and the Donut

Topology · Axiom Academy

INTRO A Topologist Can't Tell the Difference Discover the famous topology joke about coffee cups and donuts Welcome to topology, where shapes are more flexible than you ever imagined! Before we dive into the mathematics, let's start with one of the most famous jokes in all of mathematics: Wait... what? How can anyone confuse a coffee cup with a donut? They look completely different! Let's explore this bizarre statement and discover the beautiful idea behind it. In topology, shapes can be stretched, bent, and deformed as if they were made of clay. The only rule: no tearing and no gluing. Watch as we smoothly transform a coffee cup into a donut. Notice that throughout the entire transformation, something very important stays the same... Did you notice what stayed the same? The hole! Both the coffee cup (with its handle) and the donut have exactly one hole. Topological Invariant: Number of Holes The number of holes (called the genus ) doesn't change under continuous deformations. Let's look at other shapes and count their holes. Objects with the same number of holes are topologically equivalent to each other!

This is the written version of the interactive lesson above. See the full Topology course.