Fractal measure

Although the chaos game guarantees that fractal $\mathcal{F}$ will be filled out provided $\mathcal{F}$ is described by an IFS where each transformation is a contraction and the probabilities associated with each are nonzero, there are some practical considerations.

The first is that on a computer a ``random'' sequence usually means a pseudo-random sequence with a finite (though large) period. Thus, you'd better be sure that the number of points required to fill out $\mathcal{F}$ to the desired resolution is substantially smaller than your pseudo-random number generator's period.

Another consideration becomes obvious when you play with the probabilities associated with the IFS for $\mathcal{F}$. Some choices fill out certain addresses much more quickly than others. Indeed, the choice of probabilities adds some extra structure to $\mathcal{F}$: the density or measure associated with subsets of $\mathcal{F}$.

Barnsley (Fractals Everywhere) describes one implementation of this measure, defined for a grid containing $\mathcal{F}$: for each grid point $p$ in the grid the density is the fraction of total game points generated by the chaos game that land on $p$. This measure combines both the size of $p$'s intersection with $\mathcal{F}$ and how efficiently the chosen probabilities fill in $p\cap \mathcal{F}$.

I have chosen a slightly different approach. For each sub-square of a grid containing a fractal-like object $\mathcal{F}$, I first approximate the size (``area'') of its intersection with $\mathcal{F}$ by counting the number of addresses of $\mathcal{F}$ to an arbitrary depth ($6$ for example). I then divide the number of ``hits'' generated by the chaos game on this sub-square by the ``area'' of its intersection with $\mathcal{F}$ to determine a density. I re-colour $\mathcal{F}$ to indicate the relative density of various sub-squares, the densest coloured white, the sparsest red. The size of the sub-squares is user-selectable (by selecting address length). I think this approach gives a good qualitative description of how efficiently $\mathcal{F}$ is being filled out.