FRACTAL, page 8 of 36, hit 002'560

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:

Frank Rousell's hyperindex of clickable/retrievable Mandelbrot images

http://www.cnam.fr/fractals/mandel.html

Neal Kettler's Interactive Mandelbrot

http://www.vis.colostate.edu/~user1209/fractals/explorer/

Panagiotis J. Christias' Mandelbrot Explorer

http://www.softlab.ntua.gr/mandel/mandel.html

2D & 3D Mandelbrot fractal explorer (set up by Robert Keller)

http://reality.sgi.com/employees/rck/hydra/

Mandelbrot viewer written in Java (by Simon Arthur)

http://www.mindspring.com/~chroma/mandelbrot.html

Mandelbrot Questions & Answers (without any scary details) by Paul

Derbyshire

http://chat.carleton.ca/~pderbysh/mandlfaq.html

Quick Guide to the Mandelbrot Set (includes a tourist map) by Paul

Derbyshire

http://chat.carleton.ca/~pderbysh/manguide.html

Beginner's guide to the Mandelbrot Set by Eric Carr

http://www.cs.odu.edu/~carr/mandelbr.html

Java program to view the Mandelbrot Set by Ken Shirriff

http://www.sunlabs.com/~shirriff/java/

Mu-Ency The Encyclopedia of the Mandelbrot Set by Robert Munafo

http://home.earthlink.net/~mrob/muency.html

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:

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 02 + 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:

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 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.

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 via

World Wide Web (in Postscript format)

http://inls.ucsd.edu/y/Complex/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. 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:

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:

ftp://math.sunysb.edu/preprints/ims91-7.ps.Z

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 (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.