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Fractal Frequently Asked Questions and Answers
Section - The Mandelbrot set

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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).  
  
An image of the Mandelbrot set is available on the WWW at  
gopher://life.anu.edu.au:70/I9/.WWW/complex_systems/mandel1.gif .  
  
Other images and resources are:  
  
Frank Rousells two hyperindex of clickable/retrievable Mandelbrot images:  
ftp://ftp.cnam.fr/pub/Fractals/mandel/Index.gif Mandelbrot Images  
(Frank Rousell)  
ftp://ftp.cnam.fr/pub/Fractals/mandel/Index2.gif Mandebrot Images #2
(Frank Rousell)  
  
http://www.wpl.erl.gov/misc/mandel.html Interactive Mandelbrot  
(Neal Kettler)   
  
http://www.ntua.gr/mandel/mandel.html Mandelbrot Explorer (interactive)  
(Panagiotis J. Christias)   
  
http://www.ncsa.uiuc.edu/Edu/Fractal/Fractal_Home.html  
Fractal Microscope   
  
http://hermes.cybernetics.net/distfract.html Distributed Fractal Generator  
for SunOS Sparcstations (James Robinson)   
  
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.  
  
Q6c: Why do you start with z=0?  
A6c: Zero is the critical point of 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+c, the 
critical point is given by 2z+1=0, so start with z=-1/2. 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:  
  
1. M. Frame and J. Robertson, A Generalized Mandelbrot Set and the  
Role of Critical Points, _Computers and Graphics_ 16, 1 (1992), pp. 35-40.  
  
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, |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 |z1=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:  
  
1. R. Rojas, A Tutorial on Efficient Computer Graphic Representations of the  
Mandelbrot Set, _Computers and Graphics_ 15, 1 (1991), pp. 91-100.  
  
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 mrob@world.std.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.  
  
References:  
  
1. J. H. Ewing and G. Schober, The Area of the Mandelbrot Set, _Numer.  
Math._ 61 (1992), pp. 59-72.  
  
2. Y. Fisher and J. Hill, Bounding the Area of the Mandelbrot Set,  
_Numerische Mathematik_, . (Submitted for publication). Available by  
ftp: legendre.ucsd.edu:/pub/Research/Fischer/area.ps.Z ..  
  
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. The Mandelbrot set is  
self-similar under magnification in neighborhoods of Misiurewicz points,  
however (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:  
  
1. T. Lei, Similarity between the Mandelbrot set and Julia Sets,  
_Communications in Mathematical Physics_ 134 (1990), pp. 587-617.  
  
2. J. Milnor, Self-Similarity and Hairiness in the Mandelbrot Set, in  
_Computers in Geometry and Topology_, M. Tangora (editor), Dekker,  
New York, pp. 211-257.  
  
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:  
  
1. M. Shishikura, The Hausdorff Dimension of the Boundary of the  
Mandelbrot Set and Julia Sets, The paper is available from anonymous ftp:  
math.sunysb.edu:/preprints/ims91-7.ps.Z [129.49.18.1]..  
  
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_.)  
  

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Last Update March 27 2014 @ 02:11 PM