Tuesday, Oct 2, 2012

First, we review some exercises on the algebra of dimensions.

Fractal dimension can measure roughness. Many objects have varying degrees of roughness, so a single dimension is not an adequate measure.

Multifractals are a way to quantify this variability.

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 non-uniform way, revealing many fractals. Continuing with this example, here are histograms representing the probabilities of the first four generations. Note the highest-probability region has a familiar shape. We hypothesize a power-law scaling for these probabilities, and introduce the coarse Holder exponent as the exponent in that power-law. Now we stratify the square into regions having the same Holder exponent α. Computing the dimensions of these strata is how multifractals are quantified. A plot of dimension as a function of α is the f(α) curve. 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 β(q) curve, from which the f(α) curve can be calculated. 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.