Q6a : What is the Mandelbrot set?
A6a: The Mandelbrot set is the set of all complex c such that iterating z -> z^2 + c does not go to infinity (starting with z = 0).
Other images and resources are:
Q6b : How is the Mandelbrot set actually computed?
A6b: The basic algorithm is: For each pixel c, start with z = 0. Repeat z = z^2 + c up to N times, exiting if the magnitude of z gets large. If you finish the loop, the point is probably inside the Mandelbrot set. If you exit, the point is outside and can be colored according to how many iterations were completed. You can exit if |z| > 2, since if z gets this big it will go to infinity. The maximum number of iterations, N, can be selected as desired, for instance 100. Larger N will give sharper detail but take longer.
Frode Gill has some information about generating the Mandelbrot Set at http://www.krs.hia.no/~fgill/mandel.html.
Q6c : Why do you start with z = 0?
A6c: Zero is the critical point of z = z^2 + c, that is, a point where d/dz (z^2 + c) = 0. If you replace z^2 + c with a different function, the starting value will have to be modified. E.g. for z -> z^2 + z, the critical point is given by 2z + 1 = 0, so start with z = -0.5. In some cases, there may be multiple critical values, so they all should be tested.
Critical points are important because by a result of Fatou: every attracting cycle for a polynomial or rational function attracts at least one critical point. Thus, testing the critical point shows if there is any stable attractive cycle. See also:
Note that you can precompute the first Mandelbrot iteration by starting with z = c instead of z = 0, since 0^2 + c = c.
Q6d: What are the bounds of the Mandelbrot set? When does it diverge?
A6d: The Mandelbrot set lies within |c| <= 2.
If |z| exceeds 2, the z sequence diverges.
Proof: if |z| > 2, then |z^2 + c| >= |z^2| - |c| > 2|z| - |c|.
If |z| >= |c|, then 2|z| - |c| > |z|.
So, if |z| > 2 and |z| >= c, then |z^2 + c| > |z|,
so the sequence is increasing. (It takes a bit more work to prove it is
unbounded and diverges.) Also, note that |z| = c, so if |c| > 2,
the sequence diverges.
Q6e : How can I speed up Mandelbrot set generation?
A6e: See the information on speed below (see "Fractint"). Also see:
Q6f: What is the area of the Mandelbrot set?
A6f: Ewing and Schober computed an area estimate using 240,000 terms of the Laurent series. The result is 1.7274... However, the Laurent series converges very slowly, so this is a poor estimate. A project to measure the area via counting pixels on a very dense grid shows an area around 1.5066. (Contact rpm%mrob.uucp@spdcc.com for more information.) Hill and Fisher used distance estimation techniques to rigorously bound the area and found the area is between 1.503 and 1.5701. Jay Hill's latest results using Root Solving and Component Series Evaluation shows the area is at least 1.506302 and less than 1.5613027. See Fractal Horizons edited by Cliff Pickover and Hill's home page for details about his work.
References:
Q6g: What can you say about the structure of the Mandelbrot set?
A6g: Most of what you could want to know is in Branner's article in Chaos and Fractals: The Mathematics Behind the Computer Graphics.
Note that the Mandelbrot set in general is not strictly self-similar; the tiny copies of the Mandelbrot set are all slightly different, mainly because of the thin threads connecting them to the main body of the Mandelbrot set. However, the Mandelbrot set is quasi-self-similar. However, the Mandelbrot set is self-similar under magnification in neighborhoods of Misiurewicz points (e.g. -.1011 + .9563i). The Mandelbrot set is conjectured to be self-similar around generalized Feigenbaum points (e.g. -1.401155 or -.1528 + 1.0397i), in the sense of converging to a limit set.
References:
The "external angles" of the Mandelbrot set (see Douady and Hubbard or brief sketch in "Beauty of Fractals") induce a Fibonacci partition onto it.
The boundary of the Mandelbrot set and the Julia set of a generic c in M have Hausdorff dimension 2 and have topological dimension 1. The proof is based on the study of the bifurcation of parabolic periodic points. (Since the boundary has empty interior, the topological dimension is less than 2, and thus is 1.)
Reference:
Q6h: Is the Mandelbrot set connected?
A6h: The Mandelbrot set is simply connected. This follows from a theorem of Douady and Hubbard that there is a conformal isomorphism from the complement of the Mandelbrot set to the complement of the unit disk. (In other words, all equipotential curves are simple closed curves.) It is conjectured that the Mandelbrot set is locally connected, and thus pathwise connected, but this is currently unproved.
Connectedness definitions:
Connected: X is connected if there are no proper closed subsets A and B
of X such that A union B = X, but A intersect B is empty. I.e. X is connected
if it is a single piece.
Simply connected: X is simply connected if it is connected and every closed curve in X can be deformed in X to some constant closed curve. I.e. X is simply connected if it has no holes.
Locally connected: X is locally connected if for every point p in X, for every open set U containing p, there is an open set V containing p and contained in the connected component of p in U. I.e. X is locally connected if every connected component of every open subset is open in X. Arcwise (or path) connected: X is arcwise connected if every two points in X are joined by an arc in X.
(The definitions are from Encyclopedic Dictionary of Mathematics.)
Reference:
Douady, A. and Hubbard, J., "Comptes Rendus" (Paris) 294, pp.123-126,
1982.
Q6i: What is the Mandelbrot Encyclopedia?
A6i: The Mandelbrot Encyclopedia is a web page by Robert Munafo <rpm%mrob.uucp@spdcc.com> about the Mandelbrot Set. It is available via WWW at <http://home.earthlink.net/~mrob/muency.html>.
Q6j: What is the dimension of the Mandelbrot Set?
A6j: The Mandelbrot Set has a dimension of 2. The Mandelbrot Set contains and is contained in a disk. A disk has a dimension of 2, thus so does the Mandelbrot Set.
The Koch snowflake (Hausdorff dimension 1.2619...) does not satisfy this condition because it is a thin boundary curve, thus containing no disk. If you add the region inside the curve then it does have dimension of 2.
The boundary of the Mandelbrot set and the Julia set of a generic c in M have Hausdorff dimension 2 and have topological dimension 1. The proof is based on the study of the bifurcation of parabolic periodic points. (Since the boundary has empty interior, the topological dimension is less than 2, and thus is 1.) See reference above
Q6k: What are the seahorse and the elephant valleys?
A6k: The Mandelbrot set being the most famous fractal, its various regions are well known and many of them have popular names evoking graphic details found by zooming into them. The seahorse valley is the limit border of the main cardioid at the negative side of the x axis (near to x=-0.75, y=0.0). You can see here convoluted and complex buds looking more or less like seahorses. The elephant valley is near the symetry plane on the positive side of the x axis (x=0.25, y=0.0). Spirals protuding from the border evoke trunks of elephants. By zooming in these regions many interesting structures can be seen.
A nice guide (by Paul Derbyshire) to explore the various regions of the Mandelbrot set can be found at :