I am trying to find a more general answer to the question What is the function for a 'fractal sine wave'?, i.e. an intuitive generalized definition for a sequence of curves, where $$ \gamma_n(t) = (x_n(t), y_n(t)),$$ where for $n=0$ $$\gamma_0(t) = (t, \sin(t)),$$ and for $n=1,$ $$\gamma_1(t) = (x_1(t), y_1(t)),$$ where $$x_1(t)=t-\sqrt{2}\frac{\cos(t)}{\sqrt{1+\cos^2(t)}}\frac{\sin(12 t)}{12}\\ y_1(t)=\sin(t)+\sqrt{2}\frac1{\sqrt{1+\cos^2(t)}}\frac{\sin(12 t)}{12},$$
the following graph will most likely illustrate what I am trying to achieve:
What is the next step? How does the next graph like look like and how does it look in general? How does it look as $n \rightarrow \infty$ ?
Note: Maybe, the following criterions could be applicable: $$\frac{d\gamma_n}{dt}(t)|_{t=0} = (0, n+1)\\$$ $$ (x_n(t) -x_{n-1}(t))^2 + (y_n(t) -y_{n-1}(t))^2 =(n+1) \frac{y_{n-1}^2(12t)}{12^2}$$