Continuous function fractals

The idea is to have iterative fractals of continuous functions (see also IFS or iterated function system). In the limit, it's hard to tell whether such a fractal would be continuous by the end of things, but at the beginning it looks possible. Below is an example of a non-continuous function (Koch curve):

Base (iteration 0):

               _____________________


Generator (iteration 1):
                            
                           
                        /\
                       /  \
               _______/    \________


Iteration 2:
                            
                        
                      __/\__
                      \    /    
              ___/\___/    \___/\___

Okay, so it's pretty ugly. I hope the general idea is clear however. My idea is to take some continuous ("nice") function and iterate it in a similar manner; examples of "nice" functions might include sin, a binomial with rotational symmetry about the origin, or other functions with such rotational symmetry.

Here's one function I'm thinking of (sin(pi*(x+1))):

Base (iteration 0):

----------------------------------------------------------------------------------------------------

Generator (iteration 1):

                    ,,oO```Oo,,
                 ,oO`         `Oo,
               ,/`               `\,
             ,/`                   `\,
           ,/`                       `\,
          /`                           `\
        ,/                               \,
       /`                                 `\
     ,/                                     \,
    /`                                       `\
  ,/                                           \,
,/`                                             `\,
                                                  `\                                           /`
                                                    \,                                       ,/
                                                     `\                                     /`
                                                       \,                                 ,/
                                                        `\                               /`
                                                          \,                           ,/
                                                           `\,                       ,/`
                                                             `\,                   ,/`
                                                               `\,               ,/`
                                                                 `Oo,         ,oO`
                                                                    ``Oo,,,oO``

Iteration 2:

     ,.ooOO''OOo.,,
    /'`         `''OOo.,
   /                  `'\,
  |`                     `\,
  |,                       \
   \                       '|
   |,                      ,|
    \                      /
     \                    |'
      \,                  /
       `\               ,/                                     ,,.ooOO''````'\,
         \,            /`                             ,,.ooOO'``              |
----------`\---------,/----------------------,,.ooOO'``----------------------/`--------\,-----------
                     |              ,,.ooOO'``                             ,/           `\
                     `\.,,,,..ooOO'``                                     /`              \,
                                                                        ,/                 `\
                                                                       ,|                    \
                                                                       /                      \
                                                                      |`                      `|
                                                                      |,                       \
                                                                       \                       `|
                                                                        \,                     ,|
                                                                         `\.,                  /
                                                                           `'OOo..,         ,./
                                                                                ``'OOo..oOOO'`

The generalized form of iterations (as I see them) are as follows:

Variables:

t (for Theta. The equations below are generally considered parametric. The range for t will probably be -q to q where q is an "interesting" solution of the generative function.)
x (the infamous variable from algebra. I'll be using x for graphing as well.)
y (the companion variable for x. Ditto the graphing bit.)
p (not quite sure where I'm getting the label from. This is the number of subsections to divide the prior iteration into when iterating.)
Functions (grouped according to iteration):

Base (iteration 0):

f0(t)=t

g0(t)=0

Generator (iteration 1):

f1(t)=t

g1(t)=(pick an "interesting" function)

Iterations (iteration a):

Let F(t) be the derivative of fa-1(t).

Let G(t) be the derivative of ga-1(t).

Let s be the integral from -q/(p^(a-1)) to q/(p^(a-1)) of sqrt(F(t)+G(t)).

fa(t)=fa-1(t)-sqrt(G(t)/(1+G(t)))*ga-1((p^(a-1))*s*t)/((p^(a-1))*s)

ga(t)=ga-1(t)+sqrt(1/(1+G(t)))*ga-1((p^(a-1))*s*t)/((p^(a-1))*s)

The difficulty I keep having is the integral ("s" above). That is a basic length integral for parametric equations instead of standalone equations. Specifically, sin(x) is nasty in this integral. In graphing the fractal, let x=f_a(t), and y=g_a(t).

It appears that sin(x) would be quite a fascinating function to use here, but for that length integral. x^3-x is also interesting, but also difficult.