currently rendering (lacunaries and automorphisms of the unit disk, oh my)

Setup:
Let
v(q) = \sum_{n=1}^{\infty} (-1)^{n}q^{2^{n}} and
\mathrm{mob}(\theta,q,q_{0}) = e^{i\theta} \frac{q-q_{0}}{1-\bar{q}_{0}q}

I’m currently generating a movie depicting how v(q) changes when we apply the
Mobius transformation to the unit disk \mathbb{D} first and then take v(q). In my
setup q_{0}=-1/\sqrt[4]{3}, and \theta is in the interval [0,2\pi] in steps of
1/1024th the length of the interval. Blake Courter said that it was
pretty easy to do this, and I’m currently at frame 459.

Video is done, here it is:

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