P.S. you can “graphically” demonstrate the non-commutativity of 3-dimensional rotation to a layperson (including children) in a few minutes by turning around an

P.S. you can “graphically” demonstrate the non-commutativity of 3-dimensional rotation to a layperson (including children) in a few minutes by turning around an arbitrary 3-dimensional object in your hands. P.P.S. The hypervolume of an n-dimensional ball of unit radius only tends to zero as n tends to infinity if you use an n-dimensional hypercube with unit side length as your unit for hypervolume. If you use a standard simplex as your unit for hypervolume, then the volume of the ball tends to infinity. If you use the volume of the ball as your unit, then the volume of the ball tends to 1 by definition. Using a unit-length n-cube as your unit hypervolume is a convenient but arbitrary cultural convention. One main reason this ratio is ”counterintuitive” is because we conventionally use a unit-side-length square or cube but a unit-radius disk or ball, and the extraneous factors of 2 caused by the discrepancy cause confusion in low dimensions; people weirdly think of disks as being bigger than squares. If we used a unit-diameter disk/ball instead, we would see how the ratio of hypervolume of ball/cube goes strictly down as dimension goes up: dimension 1: 1 dimension 2: π/4 ≈ 0.79 dimension 3: π/6 ≈ 0.52 dimension 4: π²/32 ≈ 0.31 dimension 5: π²/60 ≈ 0.16 dimension 6: π³/384 ≈ 0.08 dimension 7: π³/840 ≈ 0.04 ...