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Barnsley's Fern

\begin{figure}\begin{center}\BoxedEPSF{barnsleys_fern.epsf}\end{center}\end{figure}

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

$\displaystyle f_1(x,y)$ $\textstyle =$ $\displaystyle \left[\begin{array}{cc}0.85 & 0.04\\  -0.04 & 0.85\end{array}\rig...
...c}x\\  y\end{array}\right]+\left[\begin{array}{c}0.00\\  1.60\end{array}\right]$ (1)
$\displaystyle f_2(x,y)$ $\textstyle =$ $\displaystyle \left[\begin{array}{cc}-0.15 & 0.28\\  0.26 & 0.24\end{array}\rig...
...c}x\\  y\end{array}\right]+\left[\begin{array}{c}0.00\\  0.44\end{array}\right]$ (2)
$\displaystyle f_3(x,y)$ $\textstyle =$ $\displaystyle \left[\begin{array}{cc}0.20 & -0.26\\  0.23 & 0.22\end{array}\rig...
...c}x\\  y\end{array}\right]+\left[\begin{array}{c}0.00\\  1.60\end{array}\right]$ (3)
$\displaystyle f_4(x,y)$ $\textstyle =$ $\displaystyle \left[\begin{array}{cc}0.00 & 0.00\\  0.00 & 0.16\end{array}\right]\left[\begin{array}{c}x\\  y\end{array}\right]$ (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.




© 1996-9 Eric W. Weisstein
1999-05-26