The Mandelbrot set is a subset of the
complex plane. The number zero is around the center of the big cardiod,
and all the
numbers in the set have modulus, or absolute value, at most 2. A complex number c is an element of the Mandelbrot set if
the
iterations of 0 in the function x2+c remain bounded. That is, the
number c is in the Mandelbrot set if the sequence of numbers
c, c2+c, (c2+c)2 + c, ((c2+c)2
+ c)2 + c, ... remains bounded.
For example: c = -1: 02+ (-1)= -1, (-1)2+(-1) = 0, 02
+ (-1) = -1, so the sequence of iterations is
0,-1, 0, -1, 0,... , and
the number -1 is an element of the Mandelbrot set.
In the picture above, only the black areas are part of the
set. The other colors indicate how close those regions
are to the Mandelbrot set. The picture is deceptive, though, because it
looks like there are small isolated
islands, but in fact the Mandelbrot set is connected.
Other Examples:
Here we represent the iterations as c = x1, c2 + c =
x2, etc.
| x1 | x2 | x3 | x4 | x5 | x6 | x7 | x8 | In the Mandelbrot set? |
| 1 | 2 | 5 | 26 | 677 | 458330 | 210 billion | 4 x 1023 | no way! |
| 0.5 | 0.75 | 1.065 | 1.63 | 3.15 | 10.44 | 109.6 | 12,005 | no |
| i | -1 + i | -i | -1 + i | -i | -1 + i | -i | -1 + i | yes |
| -2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | yes |
| -0.2 + 0.8i | -0.8+0.48i | 0.21+0.03i | -0.16+0.81i | -0.84+0.54i | 0.21-0.11i | -0.17+0.75i | -0.75+0.54i | yes |
| 0.25 | 0.3125 | 0.34766 | 0.37086 | 0.38754 | 0.40019 | 0.41015 | 0.41822 | yes |
How does the computer generate a picture of the Mandelbrot set? Other microscopic parts of the Mandelbrot set an mpeg (extermely large file size!) Edwards main page
This site © 2002, 2004 Steve Edwards