The Mandelbrot Set
The Complex Plane
The Mandelbrot and Julia Sets both are found in the complex plane, the coordinate system in which the x-axis measures the real part of a number and the y-axis measures the imaginary part (i) of a number. A value of 4 + 6i would be plotted with an x-value of 4 and a y-value of 6 (4,6). This is different from the Cartesian plane because in the complex plane each point represents one value (a + bi), whereas in the Cartesian plane each point represents two values (x,y).
Complex Functions
Just as a real function maps a real number to another real number, a complex function maps a complex value to another complex value. The only difference is that the imaginary part of the complex number cannot 'mix' with the real part. Like apples and oranges, these two parts of the same value must always be considered separately. For instance, here is a chart comparing the outcomes of various x-values from the function f(x):
| f(x) = x2 for all real and complex x values |
| x-value |
Real x2 |
Complex x2 |
| a |
b |
f(a) |
f(a + bi) |
| 4 |
0 |
16 |
16 |
| 2 |
2 |
4 |
(2 + 2i)(2 + 2i) = 8i |
| 3 |
-2 |
9 |
(3 - 2i)(3 - 2i) = 5 + 12i |
| 6 |
3 |
36 |
(6 + 3i)(6 + 3i) = 27 + 36i |
| -2 |
1 |
4 |
(-2 + i)(-2 + i) = 3 - 4i |
The Mandelbrot Set
The Mandelbrot Set is similar to the Cantor Set, in that both sets consist of the points that did not escape when iterated through a function. However, the Mandelbrot Set is obtained using the complex function f(z) = z2 + c, where z is the complex independent variable and c is the complex parameter. The actual points in the Mandelbrot Set represent the values that did not escape to infinity (diverge). As you will see in the Java applet, the resulting pattern is quite beautiful.
The Julia Set
The Julia Set uses the function f(z) = z2 + c, while z is the complex independent variable and c is the complex parameter that is constant throughout the set. An interesting fact about the Julia Set is that if you pick a constant parameter that is inside the Mandelbrot Set, the Julia Set will be connected. Conversely, if you pick a constant parameter that is outside of the Mandelbrot Set, the Julia Set will be an unconnected 'dust' of points.
Java Applets Involving Mandelbrot and Julia Sets
(The fun stuff)
