What causes this fractal to curl and unwind?

This should really be a comment but it's way too long:

As the comments indicate, all we really have so far is a qualitative observation. It's quite interesting, but turning it into a precise question is nontrivial, and of course we have to do that before we can get an answer.

So let me take a stab at precisiating the phenomenon we see - or rather, precisiating one particular question about whether a general pattern we seem to observe actually holds. Specifically, I want to ask: "Do all the loops roll clockwise?"

There are a couple things we need to pay attention to:

• We need a precise definition of "loop," and we then need a way to talk about the motion of a loop over time.

• Whatever we whip up should be parameterization-invariant: we're really paying attention to the map $$Gamma:cmapsto{(x,y):exists t(x=sum_{n=0}^infty frac{cos(2^nt+cn)}{2^n}quadmbox{and}quad y=sum_{n=0}^infty frac{sin(2^nt+cn)}{2^n})},$$ or if you prefer "successive horizontal slices" of the surface $${(x,y,z):exists t(x=sum_{n=0}^infty frac{cos(2^nt+zn)}{2^n}quadmbox{and}quad y=sum_{n=0}^infty frac{sin(2^nt+zn)}{2^n})}subseteqmathbb{R}^3.$$

So our definition will probably use the function $$F(t,c)=(sum_{n=0}^infty frac{cos(2^nt+cn)}{2^n},sum_{n=0}^infty frac{sin(2^nt+cn)}{2^n})$$ but it will ultimately "lose information."

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OK, so let's start with the notion of a loop at time $c$. There are a couple things this could mean.

From algebraic topology we have the simple notion of a continuous function $S_1rightarrowGamma(c)$ (or rather, an appropriate equivalence class of such functions). This notion is quite nice; for one thing, it produces a groupoid for each $c$, and so overall a "continuously varying groupoid."

However, that's a bit abstract; more importantly, it ignores details like direction of movement, which we definitely care about here. So while it's a very natural thing to look at, I'm not sure it's quite the right idea here.

Instead, I want to go a bit more concrete:

• A loop at time $c$ is just a pair of numbers $a<b$ such that $F(a,c)=F(b,c)$ but for each $zin (a,b)$ we have $F(z,c)not=F(a,c)$. (Note that this does refer to the parameterization; that will go away later, however. The key feature here is that the parameterization $F$ is "locally injective in $c$," that is for each $t,c$ there is some $epsilon>0$ such that $Fupharpoonright (t-epsilon, t+epsilon)times{c}$ is injective.)

• To talk about the motion of a loop over time, we'll introduce the notion of a bubble. Basically, a bubble is a "continuously varying nontrivial loop" which is maximal in the sense that if we could keep going either forward or backward in $c$ we do so. More precisely, we'll say that a prebubble consists of a tuple $(I,f,g)$ where $I$ is an open interval (which is allowed to extend infinitely in one or both directions) and $f,g$ are injective functions defined on $I$ such that $langle f(c),g(c)rangle$ is a loop for all $cin I$. A prebubble doesn't necessarily "tell the whole story" of a loop's lifetime, and so we further define a bubble to be a prebubble $(I,f,g)$ such that there is no other prebubble $(hat{I},hat{f},hat{g})$ with $Isubsetneq I'$ and $hat{f}upharpoonright I=f, hat{g}upharpoonright I=g$.

Now suppose I have a bubble $beta:=(I,f,g)$. At each time $cin I$ there is a "special point" in this bubble, $p_beta(c)=F(f(c),c)=F(g(c),c)$. And via some polar coordinate tedium the following question can be phrased precisely:

If $beta=(I,f,g)$ is a bubble, need $p_beta(c)$ always be moving clockwise as $c$ increases?

(The tedium is basically that the argument of a point is many-valued. It's easily dealt with, though.)

Note that as desired, at this point everything is parameterization-free - the above works as long as we fix any parameterization which is locally injective in $c$.

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Now that we have a precise question, I'm sure finding the answer will be basically trivial so I'll leave that as an exercise to the reader. :P