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Topology is sometimes referred to as "Geometry on a rubber sheet". It concerns itself with properties that don't change when an object is stretched, bent or otherwise distorted, just provided it's not torn or glued. (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:

We explore these ideas in the workshop: Maths In A Twist

Topology can also be used to analyse Juggling Tricks. For example, the Half Shower is topologically identical to the Shower, even though they look different, feel different, sound different, and are enormously different in difficulty.

Actually, there are two branches of topology, because we have a choice to make, whether we consider twists to be relevant or not. Topologically, a "knot" is simply a circle if we ignore the twists, but when we study how proteins fold, and how robot arms move, twists are important.
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.


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