Iterated function systems
Barnsley fern
Michael Barnsley introduced this fern in 1988 in his book Fractals Everywhere. Four affine contractions with carefully chosen probabilities produce the silhouette of the black spleenwort, Asplenium adiantum-nigrum.
The four maps
| Map | a | b | c | d | e | f | p | Effect |
|---|---|---|---|---|---|---|---|---|
| f1 | 0 | 0 | 0 | 0.16 | 0 | 0 | 0.01 | Stem |
| f2 | 0.85 | 0.04 | −0.04 | 0.85 | 0 | 1.60 | 0.85 | Main frond |
| f3 | 0.20 | −0.26 | 0.23 | 0.22 | 0 | 1.60 | 0.07 | Left side branch |
| f4 | −0.15 | 0.28 | 0.26 | 0.24 | 0 | 0.44 | 0.07 | Right side branch |
Chaos game
Pick any starting point. Repeatedly: pick a map at random according to pi, apply it, plot the new point. After a short burn-in, the plotted points are on the IFS attractor, the fern itself. Hutchinson’s theorem guarantees the existence and uniqueness of this attractor.
References
- Barnsley, M., Fractals Everywhere, Academic Press, 1988.
- Hutchinson, J., “Fractals and self similarity,” Indiana Univ. Math. J., 1981.
- IFS Playground