Dr. Karl Schaffer, of MathDance shows us (as best you can on an audio podcast!) two body movements involving interesting higher level mathematics:

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The Portuguese Waiter Trick: You can rotate your hand, palm upwards, a full 720°, returning to your starting position!

This reflects an amazing mathematical fact: that if you perform any full rotation twice you reach a configuration that can be morphed steadily back to the original state, without any rotations at all! In the waiter trick, this morphing back to the original state is cleverly being carried out even as the rotations are being undertaken. 

(There is a very beautiful way to understand this, which is a little outside the scope of our podcast; but essentially, the rotations of the sphere themselves form a special topological space; within that space, a loop is a series of rotations that takes you back to your starting point. The essential fact is that every “doubled loop”— every loop made by going around another loop twice— is contractible, can be shrunk to a point. In other words, if we perform two full rotations, we can morph our chain of rotations back to having done nothing.

As another illustration,  hold one end of a belt fixed and rotate the other end 720°. The twists all along the belt are like a chain of rotations (a path through our space of rotations). Can you find a way to remove the twist  without rotating either end? This is sometimes called the Dirac Belt Trick, after the famous physicist, Paul Dirac.