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© Copyright 2002, Jim Loy
Here are two pictures (made using Fractint and Paint Shop Pro) of the same Julia set (discovered by Gaston Julia), with different colors in order to bring out different features. Julia sets are related to The Mandelbrot Set, and are also a graph on the complex plane. Julia sets use the same formula as the Mandelbrot set: z=z2+c (or z=z^2+c). For the Mandelbrot set, c is the point being tested, on the complex plane. For the Julia set, c remains constant. The one above has c=0.3+0.6i. We start with a point z. Then we calculate a new z using that formula. If the formula blows up (probably going to infinity) then z is not in the set. Other z's do not blow up. The program colors each point, depending upon the behavior of the sequence of z's. Here are some rather famous Julia sets (produced by adjusting the real and imaginary parts of c):
They are called (left to right) Douady's Rabbit Fractal (or the Dragon Fractal) with c=-0.123+0.745i, the San Marco Fractal with c=-0.75, and the Siegel Disk Fractal with c=-0.391-0.587i. Below, I will begin with our first Julia set (second picture below), and make small adjustments to c:
Here are a few more Julia sets:
Finally, here is one using different colors:
On the left is the famous
Dragon Curve (order 13). It is one continuous
space-filling curve. Perhaps you can see some similarity between it and some of
the Julia sets.