An Important Class of Fractals

   One of important classes of fractals is represented by parameterized curves. A continuous curve definition was appeared in J.Hutchinson's paper "Fractals and Self Similarity"
   Definition: Iterated Function System F=IFS(X;S₁,..,S_n) is called a fractal parameterized curve if the following condition is satisfied:

• S_i(x_n)=S_i+1(x₁), for all i=1,..,n-1, where x₁ and x_n is fixed points of according maps S₁ and S_n.

   Hutchinson proved that for any system of contraction, with such condition, there exist a continuous map f from [0,1] segment to F, such that f( [0,1] ) = F.
   The most known fractal curves are: Koch curve, Levy curve, Peano curve and Sierpinsky triangle. Regrettably Dragon set does not satisfy Hutchinson's fractal curve definition. The following Java applet can draw some continuous curves which approximates a curves and sets considered above.

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• Fractal microscope snowflake java applet. is alternative curve drawing applet. Although it does not draw such curves as Sierpinsky Triangle and Dragon set but it allows to modify maps of curve. Due to such freedom you can draw your own curves.
• With my IFS editor applet you can create explore and modify all sets considered above and other self-similar fractals.