High Dimensional Geometry Of Everyday Motion (Visualizing Projectiles in 5D)
We use simple laws of motion to determine how a projectile moves as a function of it's initial velocity and time. We then look at the problem in terms of phase space (high dimensional geometry), and describe how the situation can be visualized as a three dimensional hypersurface in five dimensional space. The motion problem can be decomposed into a horizontal and vertical component (each of which can easily be understood via basic differential calculus). The two solutions to this problem (describing horizontal and vertical motion, respectively), can each be visualized as two dimensional surfaces (hyperbolic paraboloids) in three dimensional space. We describe how one can take these two surfaces, "multiply them by planes", and then take their intersection, to obtain the aforementioned dimensional hypersurface in five dimensional space, which describes the general solution to our projectile motion problem. The goal is to observe the interesting high dimensional geometry underlying everyday situations.
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