First, an old idea by Henri Poincaré (1854-1912): when studying a vector field in space, we can sometimes find a small disc that the trajectories hit repeatedly. Studying the points on the disc where the trajectories pass through is often a lot simpler than studying the vector field as a whole. We go from dynamics in continuous time to dynamics in discrete time.
What happens when you multiply shapes? This is part 2 of our episode on multiplying things that aren’t numbers.
Multiplication of numbers is an associative property and we can make sense of “multiplication” between things that aren’t numbers but that’s not considered as associativity.
What shape do you most associate with a standard analog clock? Your reflex answer might be a circle, but a more natural answer is actually a torus. Surprised? Then stick around.
If you needed to tell someone what numbers are and how they work, without using the notion of number in your answer, could you do it?