Barnsley's Fern


The attractor of the iterated function system given by the set of "fern functions"

f_1(x,y)=[0.85 0.04; -0.04 0.85][x; y]+[0.00; 1.60]
f_2(x,y)=[-0.15 0.28; 0.26 0.24][x; y]+[0.00; 0.44]
f_3(x,y)=[0.20 -0.26; 0.23 0.22][x; y]+[0.00; 1.60]
f_4(x,y)=[0.00 0.00; 0.00 0.16][x; y]

(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

Barnsley's Tree, Dynamical System, Fractal, Iterated Function System

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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.Trott, M. Graphica 1: The World of Mathematica Graphics. The Imaginary Made Real: The Images of Michael Trott. Champaign, IL: Wolfram Media, pp. 46, 55, and 87, 1999.Wagon, S. "Biasing the Chaos Game: Barnsley's Fern." §5.3 in Mathematica in Action. New York: W. H. Freeman, pp. 156-163, 1991.

Referenced on Wolfram|Alpha

Barnsley's Fern

Cite this as:

Weisstein, Eric W. "Barnsley's Fern." From MathWorld--A Wolfram Web Resource.

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