The Mandelbrot set
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 .
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
iterating 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).
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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gopher://life.anu.edu.au:70/I9/.WWW/complex_systems/mandel1.gif .
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
iterating 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).
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_.)