Fractal Geometry of the Mandelbrot Set (Cover Page)

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The Mandelbrot set puts some geometry into the fundamental observation
above. The precise definition is: The Mandelbrot set ** M **
consists of all of those (complex) **c**-values for which the
corresponding orbit of **0** under ** x ^{2} + c**
does not escape to infinity.
From our previous calculations, it follows that

At this point, a natural question is: Why would anyone care about the
fate of the orbit of **0** under ** x ^{2} + c** ?
Why not the orbit of

Before turning to this idea, note that the very definition of the
Mandelbrot set gives us an algorithm for computing it. We simply
consider a square in the complex plane (usually centered at the origin
with sides of length 4). We overlay a grid of equally spaced points in this
square. Each of these points is to be considered a complex **c**-value.
Then, for each such **c**, we ask the computer to check whether the
corresponding orbit of **0** goes to infinity (escapes) or does not go to
infinity (remains bounded). In the former case, we leave the
corresponding **c**-value (pixel) white. In the latter case, we paint
the **c**-value black. Thus the black points in Figure 3
represent the Mandelbrot set.

** Figure 3. The Mandelbrot set**

Two points need to be made. Figure 3 is only an approximation
of the Mandelbrot set. Indeed, it is not possible to determine whether
certain **c**-values lie in the Mandelbrot set. We can only iterate a
finite number of times to determine if a point lies in ** M **.
Certain **c**-values close to the boundary of ** M ** have orbits
that escape only after a very large number of iterations.

A second question is: How do we know that the orbit of **0** under ** x ^{2} + c**
really does escape to infinity? Fortunately, there is an easy
criterion which helps:

** The Escape Criterion:** Suppose **|c|** is less than or equal to 2.
If the orbit of 0
under ** x ^{2} + c** ever lands outside of the circle of radius 2 centered at
the origin, then this orbit definitely tends to infinity.

It may seem that this criterion is not too valuable, as it only works
when **|c|** is less than or equal to 2. However, it is known that the entire Mandelbrot set
lies inside this disk, so these are the only **c**-values we need
consider anyway.

Fractal Geometry of the Mandelbrot Set (Cover Page)

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Prof. Robert L. Devaney (Boston University)