Barnsley's Fern

Definition Barnsley’s fern is the attractor of the iterated function system made of the four maps in the table. Each map sends a point $(x, y)$ to the point with the new coordinates shown.

It is usually drawn with the chaos game. Start at the point $(0, 0)$. At each step choose one of the four maps at random, with the probabilities in the last column and independently of all earlier choices, apply it to the current point, and plot the result. All four maps are contractions, with ratios of about $0.16$, $0.85$, $0.34$ and $0.38$, so by Hutchinson’s theorem exactly one nonempty, closed and bounded set $A$ satisfies

and that set is the fern. The fixed point of each map lies on the attractor, because applying that map again and again to a point of the attractor stays on the attractor and approaches the fixed point, which the closed attractor must then contain. The point $(0, 0)$ is the fixed point of $f_1$, so every plotted point lies on the fern. With probability $1$ the plotted points come arbitrarily close to every point of it. The probabilities only affect how quickly and how evenly its parts fill in. The fern is a geometric model that resembles a plant, not a description of how a living fern grows, which involves cells, light, water and genes that appear nowhere in these maps. Michael Barnsley presented it in his 1988 book Fractals Everywhere.

Definition An affine transformation of the plane is a map that sends $(x, y)$ to the point $(x', y')$ with

for fixed numbers $a$, $b$, $c$, $d$, $s$ and $t$. The numbers $a$, $b$, $c$, $d$ form its linear part, which keeps the origin in place, and $(s, t)$ is the shift applied afterward. When $ad - bc \ne 0$ the map sends straight lines to straight lines, parallel lines to parallel lines and squares to parallelograms, and it multiplies every area by $\lvert ad - bc \rvert$. A negative value of $ad - bc$ means the map also flips the picture over, like a mirror. When $ad - bc = 0$ it flattens the whole plane onto a line or a single point. The map is a similarity, which stretches or shrinks every segment by the same factor, exactly when $a = d$ and $b = -c$, or $a = -d$ and $b = c$.

Definition A set is self-affine when it is the attractor of an iterated function system whose maps are affine transformations and contractions, so that it is made of finitely many affine copies of itself. It is self-similar when all of those maps are similarities, each shrinking every segment by the same factor and possibly turning or flipping. Every self-similar set is self-affine, but not every self-affine set is self-similar. Barnsley’s fern is self-affine, since it is the union of its four affine copies $f_1(A)$, $f_2(A)$, $f_3(A)$ and $f_4(A)$. Its four maps are not all similarities, so the fern is not described here as made of exact scaled copies of itself. The simple dimension formula for sets built from similar copies, all shrunk by the same ratio, does not apply to it, and this page gives no value for the fern’s dimension.

Definition In the chaos game for an iterated function system $f_1, \ldots, f_k$, the map probabilities are numbers $p_1, \ldots, p_k$ with every $p_i > 0$ and

At each step the map $f_i$ is chosen with probability $p_i$, independently of the earlier steps. For the fern they are $0.01$, $0.85$, $0.07$ and $0.07$. Whichever positive values are used, with probability $1$ the plotted points come arbitrarily close to every point of the same attractor. The probabilities decide how often each map is used, since over a long run the fraction of steps that use $f_i$ approaches $p_i$, and so they decide how densely the points cover each part of the attractor. If some $p_i$ is $0$, that map is never used, and the points fall on the attractor of the remaining maps instead, which can be a different set.