IFS Fractals

What is this doing?

An iterated function system is a finite set of contraction maps $f_1,\dots ,f_k$ on the plane. The Hutchinson operator

acts on closed subsets of the plane. Banach's fixed-point theorem applied to the Hausdorff metric guarantees that $F$ has a unique compact fixed point — the attractor of the IFS — and that the iterates $F^n(S)$ converge to it regardless of the starting set $S$.

Each affine map is shown as a colored parallelogram: it is the image of the unit square under $f_i$. The square handle is the origin $(e,f)$; the circle and diamond are the tips of the two basis vectors, so dragging them edits the linear part $\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$ of the map directly.

The IFS used to draw the figure above is the one shown in the function list — the affine maps drawn as coloured parallelograms on the canvas. Each iteration of $F$ takes the current black region and replaces it with the union of its images under the maps. Convergence is geometric: the rate is controlled by the largest contraction ratio of the maps, and after a handful of iterations the visible figure is already indistinguishable from the attractor at pixel resolution.

The attractor is what it is regardless of where you started — try the same IFS with a disk, a stripe, or a doodle, and the iterates settle onto the same shape. This is why the IFS is the natural object to think about, not the starting set: the maps determine the limit, the starting set determines only the early frames.

One subtlety lives in Barnsley's fern. Its stem map has rank one — it collapses two-dimensional area to a one-dimensional line segment. The backward pixel sampling this visualizer uses for every other map would miss a measure-zero set entirely. For rank-deficient maps the worker switches to forward iteration, splatting each currently-set pixel forward through the map, so the stem actually shows up on the grid.