very interesting perspective!

Hi, here is something extra. Does i^i^i^... exist? This is the most important thing that I didn't discuss in the video, and I'll try to explain it like a story. Let F(z) = i^z. (picking principal branch*) F(i) = i^i F(F(i)) = i^i^i F(F(F(i))) = i^i^i^i ...doing this infinitely many times and, if it "zooms in on" a single value, then that single value is what we call the infinite imaginary number power tower: i^i^i^i^i^i^... If we look at the complex plane, and apply F to every single point, then most of those points move. Applying F over and over again will make those points move around over and over again, kind of like a choppy video. There are, however, some special points that don't move at all when we apply F to them, and that means they don't move at all no matter how many times we apply F to them! There are infinitely many of these points (countably infinitely many). They are called the fixed points of F. Now there is one special fixed point of F where, when you apply F over and over again to all the points in the complex plane, a sort of little whirlpool appears around this special fixed point. All the points near it are getting sucked into it like a whirlpool. In fact even points far away from it sometimes end up close enough to it that they too get sucked into it. This special fixed point is approximately equal to 0.438 + 0.36i. It's the only fixed point like this. All the others actually do the opposite: they repel points near them away, like a reverse whirlpool. It so happens that if you start at point i, and then apply F over and over again, that it will end up getting close enough to ~ 0.438 + 0.36i that it ends up getting caught in its whirlpool and sucked into it. That is i^i^i^... The math for how to prove all of this requires some complex analysis. * F(z) = e^(z*i*pi/2)