Animation vs. Math [video]

It’s all about imaginary numbers, the basic trig functions, Euler’s identity, and the relationships between them. Theses all related ways of talking about rotations. The imaginary numbers are on a number line perpendicular to the real numbers, so a multiplication by i is a 90° rotation, or a quarter turn. We can compute rotations by some other angle θ using the formula cos(θ) + i sin(θ). Euler proved that this is entirely equivalent to e^iθ as long as the angle θ is in radians. A 180° rotation (a half turn) is equal to a rotation by π radians, so cos(π) + i sin(π) = e^iπ. If you start at 1 and rotate by 180° you end up at −1, so e^iπ = −1.

e^iπ multiplies itself by i several times in order to teleport away from the animated stick figure. This works because ie^iπ = i−1 = −i, putting it in the imaginary plane below the reals. You might also notice that both e^iπ and the stick figure are hit by flying minus signs, which flip them horizontally and/or cause them to run in the opposite direction.

In the video, when e^iπ starts to defend itself it transforms into the equivalent infinite sum e^x = Σ (x^n)/n!. This is a much scarier version of the formula, but it just allows you to compute e^x using basic arithmetic. Written as an infinite sum it looks like (x^1)/1! + (x^2)/2! + (x^3)/3! + (x^4)/4! + (x^5)/5! + …. The projectiles that these flying sums fire are just the decimal expansion of successive terms of this sequence.

The animation fires back by building a gun out of the sin(x)/cos(x) function, also known as tan(x). The tangent of π radians is zero, so a hit from this weapon is very effective against e^iπ.

After the stick figure and e^iπ stop fighting, the stick figure asks e^iπ if he can be returned to his own reality, using the analogy of multiplying himself by iπ. e^iπ demonstrates multiplying himself by i⁴ (aka multiplying by 1) to teleport to his own location, but that won’t really help the stick figure. e^iπ does some more complicated math involving an infinite sum which uses the gamma function, but unless this part is beyond me then the author of the video is taking a bit of liberty in the interpretation. Even after all the manipulations of the formula, it still just computes e^iπ, which as we know equals −1. It shouldn’t do anything special, but it’s fun.

e^iπ is seen walking away with the Riemann Zeta function, the golden ratio, and a giant aleph. All of these are fun mathematical concepts which aren’t otherwise involved in the video; the stick figure didn’t happen to discover them in his brief visit.

I found this via my YouTube recommendations today:

Seems to be a reasonable attempt at explaining all the references in that original video --

https://youtu.be/tbqU9APUuYI