| |
|
What
Is The Mandelbrot Set?
 The
complete Mandelbrot Set
The Mandelbot set
is sometimes referred to as the sum of all Julia sets. So what is
a Julia set? Consider the family of quadratic polynomials f(x)
= x2 - µ, where µ is an imaginary (complex) parameter.
As µ varies, the graph of the Julia set will vary on the complex plane.
Some of these Julia sets will be connected and these connected sets
comprise the Mandelbrot set. So the picture you see (above) is the
plot of all points in the plane where x varies from -1 to 2 and y
varies from -1.2 to 1.2. The Mandelbrot set is the black shape in
the picture. As you explore this plot by zooming into it, you find
many beautiful and surprising pictures.
How
To Use This Mandelbrot Applet
Click
the Draw Set button to start generating the Mandelbrot set. A dialog
box will pop up with the default coordinates/iterations to plot.
The larger the value you select for the maximum iterations, the
more detail (and higher contrast) you will see. Click stop to interrupt
the drawing process. To zoom into an area, click the Zoom In button
and then click on the part of the picture you want enlarged. To
zoom back out, just click the zoom out button.
What
Are Fractals And Who Is Mandelbrot?
Benoit
B. Mandelbrot founded a new branch of mathematics, fractal geometry.
In conventional geometry, an object’s dimension is expressed
in whole numbers; a line, for example, is one-dimensional and plane
has two dimensions. In fractal geometry, objects may have "fractional"
dimensions. For example, a fractal image may have a border that
is infinitely detailed, and thus, a dimension between one and two.
Fractals have the property that they are self-similar over many
scales--that is, a tiny portion of a fractal image may resemble
the entire fractal. Mandelbrot discovered the fractal set bearing
his name in 1980. You will find this self-similarity as you zoom
into the Mandelbrot set and discover "mini" Mandelbrot
sets inside.
Mandelbrot
Applet Source Code
Download
Mandelbrot.zip (9 kb)
|
