A. As a first mathematical example, we see that by adjusting the probabilities,
we can make different parts of the fractal fill in at different rates. 
Here is an example. The IFS of this
example generates the unit square. 
However, the square fills up in a nonuniform way, revealing many fractals. 


B. Continuing with the example of 7.A., here
are histograms representing the probabilities of the first four generations. 
Note the highestprobability region has a familiar shape. 
This is easy to understand. 
The lower left, lower right, and upper right transformations all have the same
probability, and those three transformations together generate a Sierpinski gasket. 


C. Here is another example,
Example B, with p_{1} = 0.2,
p_{2} = p_{3} = 0.25, and p_{4} = 0.3. 
Now structures more complicated than gaskets will appear. 


D. In the length>0 limit, the coarse dimension becomes a local dimension. 
The placedependence of local dimension motivates the name multifractal. 
Here we investigate the distribution of local dimensions
for Example B. 
The resulting curve is called the f(α) curve. 


E. Here is the general method for generating multifractals with IFS. 
We modify the Moran equation, weighting each term with the probability of the transformation. 
This gives the tau(q) curve, from which the f(α) curve can be calculated. 


F. By changing the probabilities of the transformations, we alter the rate at which
different parts of the shape fills in, and consequently change the f(α) curve. 
Here we illustrate this dependence by several examples. 


Sometimes, from visual inspection of a multifractal we can
gather enough information to sketch its f(α) curve. 
Here are some examples. 


Here we dsicuss the method of moments for plotting f(α) curves. This is most easily understood
in the context of examples: time series moments,
planar data moments, and the special case of
IFS moments.  
G. Here are some examples of f(α) curves derived from
financial data using the method of moments. 

