About the video The Riemann Hypothesis, Explained at 658 secs and on:
What are the meanings of the curves (and their analytic forms) that create the spiral or heart shapes and that keep hitting the zeta zeros several times?
What do these mean to have the curves keep hitting the zeta zeros multiple say N times? What is the meaning of this N? if hitting N times? (or can the N be bounded finite or be infinite?)
I recognize the spiral as being the curve parametrized by $\zeta(\frac{1}{2}+it)$ for $t$ in some finite interval $[0,T]$; the spiral is growing larger by making $T$ larger. The value $\zeta(\frac{1}{2})=1.46035...$ is the leftmost point on the spiral in the video, where it starts. If you look closely, the "critical line" $\mathrm{Re}(s)=\frac{1}{2}$ (the vertical orange line halfway between $0$ and $1$) is the one that turns into the spiral.
In general, the straight lines of the grid moments before are a "before" picture, and the collection of curves moments later are the "after" picture. That is, the curves are the images of $\zeta$ applied to the original straight lines. The dark teal curves are those that can be obtained using the original series definition of $\zeta$, while the orange ones are those that can only be obtained by using analytic continuation to extend the domain of $\zeta$.
(I am unsure but very curious how they smoothly transitioned the lines into curves.)
Every time the spiral hits the origin is a zero of the zeta function. So the multiplicity with which the spiral intersects the origin is the number of zeros $\zeta$ has on the critical line with imaginary part restricted to the interval $[0,T]$. As $T$ grows larger, this number will keep growing, forever.