That's not strictly accurate, because you're allowed to cut things, provided you glue them back exactly as they were, but it's pretty close and gives you the right idea.

As a simple example, we can use topology to prove that the number of holes in a surface remains constant no matter how it is distorted, which tells us that unless you rip or glue them, you cannot transform a coffee cup into drinking glass. While this may not seem useful at first sight, there's a lot more to it, and topology is useful in several areas of high technology, especially helping to show that some things are impossible. As with many branches of mathematics, the uses only surfaced decades after the work was originally done.

There are a few simple examples of topological curiosities that can be used as a simple introduction. These include

• Moebius Strips
• Klein Bottles
• CrossCaps

Even the shape made in the air by the cascade can be treated topologically. It's really still a circle. Even though it has a half twist in it to make it a figure-of-eight, the balls still travel around in a circuit. Doing an Outside Throw with one hand serves to "untwist" the twisted circle, giving you back the familiar, uncrossed circuit, and there you have the Half Shower.

Quotation from Tim Berners-Lee