Barnsley's Fern

The Attractor of the Iterated Function System given by the set of ``fern functions''

$$f_1(x,y) = \left[\begin{array}{cc}0.85 & 0.04\\ -0.04 & 0.85\end{array}\rig... ...c}x\\ y\end{array}\right]+\left[\begin{array}{c}0.00\\ 1.60\end{array}\right]$$ (1)

$$f_2(x,y) = \left[\begin{array}{cc}-0.15 & 0.28\\ 0.26 & 0.24\end{array}\rig... ...c}x\\ y\end{array}\right]+\left[\begin{array}{c}0.00\\ 0.44\end{array}\right]$$ (2)

$$f_3(x,y) = \left[\begin{array}{cc}0.20 & -0.26\\ 0.23 & 0.22\end{array}\rig... ...c}x\\ y\end{array}\right]+\left[\begin{array}{c}0.00\\ 1.60\end{array}\right]$$ (3)

$$f_4(x,y) = \left[\begin{array}{cc}0.00 & 0.00\\ 0.00 & 0.16\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]$$ (4)

(Barnsley 1993, p. 86; Wagon 1991). These Affine Transformations are contractions. The tip of the fern (which resembles the black spleenwort variety of fern) is the fixed point of $f_1$, and the tips of the lowest two branches are the images of the main tip under $f_2$ and $f_3$ (Wagon 1991).

See also Dynamical System, Fractal, Iterated Function System

References

Barnsley, M. Fractals Everywhere, 2nd ed. Boston, MA: Academic Press, pp. 86, 90, 102 and Plate 2, 1993.

Gleick, J. Chaos: Making a New Science. New York: Penguin Books, p. 238, 1988.

Wagon, S. ``Biasing the Chaos Game: Barnsley's Fern.'' §5.3 in Mathematica in Action. New York: W. H. Freeman, pp. 156-163, 1991.