You know that feeling you get when you have two mirrors facing each other and it gives the illusion of there being an infinite tunnel of rooms? Or if they're at an angle with each other, it makes you feel like you're a part of a strange kaleidoscopic world with many copies of yourself all separated by angled pieces of glass. What many people may not realize is that the idea underlying these illusions can be surprisingly helpful for solving serious problems in math. We've already seen two videos describing the block collision puzzle with its wonderfully surprising answer. A big block comes in from the break, lots of cuts, and the total number of clacks looks like pi. And we want to know why. Here, we see one more perspective explaining what's going on. Where if the connection to PI wasn't surprising enough, we add one more unexpected connection to optics. But we're doing more than just answering the same question twice. This alternate solution gives a much richer understanding of the whole setup, and it makes it easier to answer other questions. And fun side note, it happens to be core to how I coded the accurate simulations of these blocks without requiring absurdly small time steps and huge computation time. The solution from the last video involved a coordinate plane where each point encodes a pair of velocities. Here, we'll do something similar, but the points of our plane are going to encode the pair of positions of both blocks. Again, the idea is that by representing the state of a changing system with individual points in some space, problems and dynamics turn into problems in geometry which hopefully are more solvable. Specifically, let the X coordinate of a 2D plane represent the distance from the wall to the left edge...
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