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URL: http://www.mta.ca/~mctaylor/sci.fractals-faq/
Copyright: Copyright 1997-1998 by Michael C. Taylor and Jean-Pierre Louvet
Maintainer: Michael C. Taylor and Jean-Pierre Louvet
sci.fractals FAQ (Frequently Asked Questions)
_________________________________________________________________
_Volume_ 5 _Number_ 3
_Date_ March 8, 1998
_________________________________________________________________
_Copyright_ 1997-1998 by Michael C. Taylor and Jean-Pierre Louvet. All
Rights Reserved.
Introduction
This FAQ is posted monthly to sci.fractals, a Usenet newsgroup about
fractals; mathematics and software. This document is aimed at being a
reference about fractals, including answers to commonly asked
questions, archive listings of fractal software, images, and papers
that can be accessed via the Internet using FTP, gopher, or
World-Wide-Web (WWW), and a bibliography for further readings.
The FAQ does not give a textbook approach to learning about fractals,
but a summary of information from which you can learn more about and
explore fractals.
This FAQ is posted monthly to the Usenet newsgroups: sci.fractals
("Objects of non-integral dimension and other chaos"), sci.answers,
and news.answers. Like most FAQs it can be obtained freely with a WWW
browser (such as Mosaic or Netscape), or by anonymous FTP from
ftp://rtfm.mit.edu/pub/usenet/news.answers/sci/fractals-faq (USA). It
is also available from
ftp://ftp.Germany.EU.net/pub/newsarchive/news.answers/sci/fractals-faq
.gz (Europe),
http://graffiti.cribx1.u-bordeaux.fr/MAPBX/louvet/sci.fractals-faq/faq
.html (France) and http://www.mta.ca/~mctaylor/sci.fractals-faq/
(Canada).
Those without FTP or WWW access can obtain the FAQ via email, by
sending a message to mail-server@rtfm.mit.edu with the _message_:
send usenet/news.answers/sci/fractals-faq
_________________________________________________________________
Suggestions, Comments, Mistakes
Please send suggestions and corrections about the sci.fractals FAQ to
fractal-faq@mta.ca. Without your contributions, the FAQ for
sci.fractals will not grow in its wealth. _"For the readers, by the
readers."_ Rather than calling me a fool behind my back, if you find a
mistake, whether spelling or factual, please send me a note. That way
readers of future versions of the FAQ will not be misled. Also if you
have problems with the appearance of the hypertext version. There
should not be any Netscape only markup tags contained in the hypertext
verion, but I have not followed strict HTML 3.2 specifications. If the
appearance is "incorrect" let me know what problems you experience.
Why the different name?
The old Fractal FAQ about fractals _has not been updated for over two
years_ and has not been posted by Dr. Ermel Stepp, in as long. So this
is a new FAQ based on the previous FAQ's information and the readers
of primarily sci.fractals with contributions from the FRAC-L and
Fractal-Art mailing lists. Thus it is now called the _sci.fractals
FAQ_.
______________________________________________________________________
Table of contents
The questions which are answered include:
Q0: I am new to the 'Net. What should I know about being online?
Q1: I want to learn about fractals. What should I read first? New
Q2: What is a fractal? What are some examples of fractals?
Q3a: What is chaos?
Q3b: Are fractals and chaos synonymous?
Q3c: Are there references to fractals used as financial models?
Q4a: What is fractal dimension? How is it calculated?
Q4b: What is topological dimension?
Q5: What is a strange attractor?
Q6a: What is the Mandelbrot set?
Q6b: How is the Mandelbrot set actually computed?
Q6c: Why do you start with z = 0?
Q6d: What are the bounds of the Mandelbrot set? When does it diverge?
Q6e: How can I speed up Mandelbrot set generation?
Q6f: What is the area of the Mandelbrot set?
Q6g: What can you say about the structure of the Mandelbrot set?
Q6h: Is the Mandelbrot set connected?
Q6i: What is the Mandelbrot Encyclopedia?
Q6j: What is the dimension of the Mandelbrot Set?
Q6k: What are the seahorse and the elephant valleys?
Q6l: What is the relation between Pi and the Mandelbrot Set?
Q7a: What is the difference between the Mandelbrot set and a Julia
set?
Q7b: What is the connection between the Mandelbrot set and Julia sets?
Q7c: How is a Julia set actually computed?
Q7d: What are some Julia set facts?
Q8a: How does complex arithmetic work?
Q8b: How does quaternion arithmetic work?
Q9: What is the logistic equation?
Q10: What is Feigenbaum's constant?
Q11a: What is an iterated function system (IFS)?
Q11b: What is the state of fractal compression?
Q12a: How can you make a chaotic oscillator?
Q12b: What are laboratory demonstrations of chaos?
Q13: What are L-systems?
Q14: What are sources of fractal music?
Q15: How are fractal mountains generated?
Q16: What are plasma clouds?
Q17a: Where are the popular periodically-forced Lyapunov fractals
described?
Q17b: What are Lyapunov exponents?
Q17c: How can Lyapunov exponents be calculated?
Q18: Where can I get fractal T-shirts and posters?
Q19: How can I take photos of fractals?
Q20a: What are the rendering methods commonly used for 256-colour
fractals?
Q20b: How does rendering differ for true-colour fractals??
Q21: How can 3-D fractals be generated?
Q22a: What is Fractint?
Q22b: How does Fractint achieve its speed?
Q23: Where can I obtain software packages to generate fractals? New
Q24a: How does anonymous ftp work?
Q24b: What if I can't use ftp to access files?
Q25a: Where are fractal pictures archived? New
Q25b: How do I view fractal pictures from
alt.binaries.pictures.fractals?
Q26: Where can I obtain fractal papers?
Q27: How can I join fractal mailing lists? New
Q28: What is complexity?
Q29a: What are some general references on fractals and chaos?
Q29b: What are some relevant journals?
Q29c: What are some other Internet references?
Q30: What is a multifractal?
Q31a: What is aliasing? New
Q31b: What does aliasing have to do with fractals? New
Q31c: How Do I "Anti-Alias" Fractals? New
Q32: Ideas for science fair projects? New
Q33: Are there any special notices?
Q34: Who has contributed to the Fractal FAQ? New
Q35: Copyright? New
____________________________________________________
Subject: USENET and Netiquette
_Q0_: I am new to sci.fratals. What should I know about being online?
_A0_: There are a couple of common mistakes people make, posting ads,
posting large binaries (images or programs), and posting off-topic.
_Do Not Post Images to sci.fractals._ If you follow this rule people
will be your friend. There is a special place for you to post your
images, _alt.binaries.pictures.fractals_. The other group
(alt.fractals.pictures) is considered obsolete and may not be carried
to as many people as _alt.binaries.pictures.fractals._ In fact there
is/was a CancelBot which will delete any binary posts it finds in
sci.fractals (and most other non-binaries newsgroup) so nearly no one
will see it.
_Post only about fractals_, this includes fractal mathematics,
software to generate fractals, where to get fractal posters and
T-shirts, and fractals as art. Do not bother posting about news events
not directly related to fractals, or about which OS / hardware /
language is better. These lead to flame wars.
_Do not post advertisements._ I should not have to mention this, but
people get excited. If you have some _fractal_ software (or posters)
available as shareware or shrink-wrap commercial, post your _brief_
announcement _once_. After than, you should limit yourself to notices
of upgrades and responding _via e-mail_ to people looking for fractal
software.
If you are new to Usenet and/or being online, read the guidelines and
Frequently Asked Questions (FAQ) in news.announce.newusers. They are
available from:
Welcome to news.newusers.questions
ftp://rtfm.mit.edu/pub/usenet/news.answers/news-newusers-intro
ftp://garbo.uwasa.fi/pc/doc-net/usenews.zip
A Primer on How to Work With the Usenet Community
ftp://rtfm.mit.edu/pub/usenet/news.answers/usenet/primer/part1
Frequently Asked Questions about Usenet
ftp://rtfm.mit.edu/pub/usenet/news.answers/usenet/faq/part1
Rules for posting to Usenet
ftp://rtfm.mit.edu/pub/usenet/news.answers/usenet/posting-rules
/part1
Emily Postnews Answers Your Questions on Netiquette
ftp://rtfm.mit.edu/pub/usenet/news.answers/usenet/emily-postnew
s/part1
Hints on writing style for Usenet
ftp://rtfm.mit.edu/pub/usenet/news.answers/usenet/writing-style
/part1
What is Usenet?
ftp://rtfm.mit.edu/pub/usenet/news.answers/usenet/what-is/part1
Subject: Learning about fractals
_Q1_: I want to learn about fractals. What should I read/view first?
_A1_: _Chaos: Making a New Science_, by James Gleick, is a good book
to get a general overview and history that does not require an
extensive math background. _Fractals Everywhere,_ by Michael Barnsley,
and _Measure Topology and Fractal Geometry_, by G. A. Edgar, are
textbooks that describe mathematically what fractals are and how to
generate them, but they requires a college level mathematics
background. _Chaos, Fractals, and Dynamics_, by R. L. Devaney, is also
a good start. There is a longer book list at the end of the FAQ (see
"What are some general references?").
Also, there are networked resources available, such as :
Exploring Fractals and Chaos
http://www.lib.rmit.edu.au/fractals/exploring.html
Fractal Microscope
http://www.ncsa.uiuc.edu/Edu/Fractal/Fractal_Home.html
Dynamical Systems and Technology Project: a introduction for
high-school students
http://math.bu.edu/DYSYS/dysys.html
An Introduction to Fractals (Written by Paul Bourke)
http://www.mhri.edu.au/~pdb/fractals/fracintro/
Fractals and Scale (by David G. Green)
http://life.csu.edu.au/complex/tutorials/tutorial3.html
What are fractals? (by Neal Kettler)
http://www.vis.colostate.edu/~user1209/fractals/fracinfo.html
Fract-ED a fractal tutorial for beginners, targeted for high
school/tech school students.
http://www.ealnet.com/ealsoft/fracted.htm
Mandelbrot Questions & Answers (without any scary details) by Paul
Derbyshire
http://chat.carleton.ca/~pderbysh/mandlfaq.html
Godric's fractal gallery. A brief introduction to Fractals clear and
well illustrated explanations
http://www.gozen.demon.co.uk/godric/fracgall.html
Lystad Fractal Info complex numbers and fractals
http://www.iglobal.net/lystad/lystad-fractal-info.html
Fractal eXtreme: fractal theory theoritical informations
http://www.cygnus-software.com/theory/theory.htm
Frode Gill Fractal pages mathematical and programming data about
classical fractals and quaternions
http://www.krs.hia.no/~fgill/fractal.html
Fractals: a history
http://graffiti.cribx1.u-bordeaux.fr/MAPBX/louvet/history.html
Basic informations about fractals
http://graffiti.cribx1.u-bordeaux.fr/MAPBX/louvet/jpl1a.html
Fantastic Fractals a very comprehensive site with tutorials for
beginners and more advanced readers, workshops etc.
http://library.advanced.org/12740/cgi-bin/welcome.cgi
Chaos, Fractals, Dimension: mathematics in the age of the computer by
Glenn Elert. A huge (>100 pages double-spaced) essay on
chaos, fractals, and non-linear dynamics. It requires a
moderate math background, though is not aimed at the
mathematician.
http://www.columbia.edu/~gae4/chaos/
Mathsnet this site has several pages devoted to fractals and complex
numbers.
http://www.anglia.co.uk/education/mathsnet/
Fractals in Your Future by Ronald Lewis
http://www.eureka.ca/resources/fiyf/fiyf.html
Subject: What is a fractal?
_Q2_: What is a fractal? What are some examples of fractals?
_A2_: A fractal is a rough or fragmented geometric shape that can be
subdivided in parts, each of which is (at least approximately) a
reduced-size copy of the whole. Fractals are _generally_ self-similar
and independent of scale.
There are many mathematical structures that are fractals; e.g.
Sierpinski triangle, Koch snowflake, Peano curve, Mandelbrot set, and
Lorenz attractor. Fractals also describe many real-world objects, such
as clouds, mountains, turbulence, coastlines, roots, branches of
trees, blood vesels, and lungs of animals, that do not correspond to
simple geometric shapes.
Benoit B. Mandelbrot gives a mathematical definition of a fractal as a
set of which the Hausdorff Besicovich dimension strictly exceeds the
topological dimension. However, he is not satisfied with this
definition as it excludes sets one would consider fractals.
According to Mandelbrot, who invented the word: "I coined _fractal_
from the Latin adjective _fractus_. The corresponding Latin verb
_frangere_ means "to break:" to create irregular fragments. It is
therefore sensible - and how appropriate for our needs! - that, in
addition to "fragmented" (as in _fraction_ or _refraction_), _fractus_
should also mean "irregular," both meanings being preserved in
_fragment_." (The Fractal Geometry of Nature, page 4.)
Subject: Chaos
_Q3a_: What is chaos?
_A3a_: Chaos is apparently unpredictable behavior arising in a
deterministic system because of great sensitivity to initial
conditions. Chaos arises in a dynamical system if two arbitrarily
close starting points diverge exponentially, so that their future
behavior is eventually unpredictable.
Weather is considered chaotic since arbitrarily small variations in
initial conditions can result in radically different weather later.
This may limit the possibilities of long-term weather forecasting.
(The canonical example is the possibility of a butterfly's sneeze
affecting the weather enough to cause a hurricane weeks later.)
Devaney defines a function as chaotic if it has sensitive dependence
on initial conditions, it is topologically transitive, and periodic
points are dense. In other words, it is unpredictable, indecomposable,
and yet contains regularity.
Allgood and Yorke define chaos as a trajectory that is exponentially
unstable and neither periodic or asymptotically periodic. That is, it
oscillates irregularly without settling down.
sci.fractals may not be the best place for chaos/non-linear dynamics
questions, sci.nonlinear newsgroup should be much better.
_Q3b_: Are fractals and chaos synonymous?
_A3b_: No. Many people do confuse the two domains because books or
papers about chaos speak of the two concepts or are illustrated with
fractals.
_Fractals_ and _deterministic chaos_ are mathematical tools to
modelise different kinds of natural phenomena or objects. _The
keywords in chaos_ are impredictability, sensitivity to initial
conditions in spite of the deterministic set of equations describing
the phenomenon.
On the other hand, _the keywords to fractals are self-similarity,
invariance of scale_. Many fractals are in no way chaotic (Sirpinski
triangle, Koch curve...).
However, starting from very differents point of view, the two domains
have many things in common : many chaotic phenomena exhibit fractals
structures (in their strange attractors for example... fractal
structure is also obvious in chaotics phenomena due to successive
bifurcations ; see for example the logistic equation Q9 )
The following resources may be helpful to understand chaos:
sci.nonlinear FAQ (UK)
http://www.fen.bris.ac.uk/engmaths/research/nonlinear/faq.html
sci.nonlinear FAQ (US)
http://amath.colorado.edu/appm/faculty/jdm/faq.html
Exploring Chaos and Fractals
http://www.lib.rmit.edu.au/fractals/exploring.html
Chaos and Complexity Homepage (M. Bourdour)
http://www.cc.duth.gr/~mboudour/nonlin.html
The Institute for Nonlinear Science
http://inls.ucsd.edu/
_Q3c_: Are there references to fractals used as financial models?
_A3c_: Most references are related to chaos being used as a model for
financial forecasting.
One reference that is about fractal models is, Fractal Market Analysis
- Applying Chaos Theory to Investment & Economics by Edgar Peters.
Some recommended Chaos-related texts in financial forecasting.
Medio: Chaotic Dynmics - Theory and Applications to Economics
Cambridge University Press, 1993, ISBN 0-521-48461-8
Vaga: Profiting from Chaos - Using Chaos Theory for Market Timing,
Stock Selection and Option Valuation
McGraw-Hill Inc, 1994, ISBN 0-07-066786-1
Subject: Fractal dimension
_Q4a_ : What is fractal dimension? How is it calculated?
_A4a_: A common type of fractal dimension is the Hausdorff-Besicovich
Dimension, but there are several different ways of computing fractal
dimension.
Roughly, fractal dimension can be calculated by taking the limit of
the quotient of the log change in object size and the log change in
measurement scale, as the measurement scale approaches zero. The
differences come in what is exactly meant by "object size" and what is
meant by "measurement scale" and how to get an average number out of
many different parts of a geometrical object. Fractal dimensions
quantify the static _geometry_ of an object.
For example, consider a straight line. Now blow up the line by a
factor of two. The line is now twice as long as before. Log 2 / Log 2
= 1, corresponding to dimension 1. Consider a square. Now blow up the
square by a factor of two. The square is now 4 times as large as
before (i.e. 4 original squares can be placed on the original square).
Log 4 / log 2 = 2, corresponding to dimension 2 for the square.
Consider a snowflake curve formed by repeatedly replacing ___ with
_/\_, where each of the 4 new lines is 1/3 the length of the old line.
Blowing up the snowflake curve by a factor of 3 results in a snowflake
curve 4 times as large (one of the old snowflake curves can be placed
on each of the 4 segments _/\_). Log 4 / log 3 = 1.261... Since the
dimension 1.261 is larger than the dimension 1 of the lines making up
the curve, the snowflake curve is a fractal.
For more information on fractal dimension and scale, via the WWW
Fractals and Scale (by David G. Green)
http://life.csu.edu.au/complex/tutorials/tutorial3.html
Fractal dimension references:
1. J. P. Eckmann and D. Ruelle, _Reviews of Modern Physics_ 57, 3
(1985), pp. 617-656.
2. K. J. Falconer, _The Geometry of Fractal Sets_, Cambridge Univ.
Press, 1985.
3. T. S. Parker and L. O. Chua, _Practical Numerical Algorithms for
Chaotic Systems_, Springer Verlag, 1989.
4. H. Peitgen and D. Saupe, eds., _The Science of Fractal Images_,
Springer-Verlag Inc., New York, 1988. ISBN 0-387-96608-0.
This book contains many color and black and white photographs,
high level math, and several pseudocoded algorithms.
5. G. Procaccia, _Physica D_ 9 (1983), pp. 189-208.
6. J. Theiler, _Physical Review A_ 41 (1990), pp. 3038-3051.
References on how to estimate fractal dimension:
1. S. Jaggi, D. A. Quattrochi and N. S. Lam, Implementation and
operation of three fractal measurement algorithms for analysis of
remote-sensing data., _Computers & Geosciences _19, 6 (July 1993),
pp. 745-767.
2. E. Peters, _Chaos and Order in the Capital Markets _, New York,
1991. ISBN 0-471-53372-6
Discusses methods of computing fractal dimension. Includes several
short programs for nonlinear analysis.
3. J. Theiler, Estimating Fractal Dimension, _Journal of the Optical
Society of America A-Optics and Image Science_ 7, 6 (June 1990),
pp. 1055-1073.
There are some programs available to compute fractal dimension. They
are listed in a section below (see Q22 "Fractal software").
Reference on the Hausdorff-Besicovitch dimension
A clear and concise (2 page) write-up of the definition of the
Hausdorff-Besicovitch dimension in MS-Word 6.0 format is available in
zip format.
hausdorff.zip (~26KB)
http://www.newciv.org/jhs/hausdorff.zip
_Q4b_ : What is topological dimension?
_A4b_: Topological dimension is the "normal" idea of dimension; a
point has topological dimension 0, a line has topological dimension 1,
a surface has topological dimension 2, etc.
For a rigorous definition:
A set has topological dimension 0 if every point has arbitrarily small
neighborhoods whose boundaries do not intersect the set.
A set S has topological dimension k if each point in S has arbitrarily
small neighborhoods whose boundaries meet S in a set of dimension k-1,
and k is the least nonnegative integer for which this holds.
Subject: Strange attractors
_Q5_: What is a strange attractor?
_A5_: A strange attractor is the limit set of a chaotic trajectory. A
strange attractor is an attractor that is topologically distinct from
a periodic orbit or a limit cycle. A strange attractor can be
considered a fractal attractor. An example of a strange attractor is
the Henon attractor.
Consider a volume in phase space defined by all the initial conditions
a system may have. For a dissipative system, this volume will shrink
as the system evolves in time (Liouville's Theorem). If the system is
sensitive to initial conditions, the trajectories of the points
defining initial conditions will move apart in some directions, closer
in others, but there will be a net shrinkage in volume. Ultimately,
all points will lie along a fine line of zero volume. This is the
strange attractor. All initial points in phase space which ultimately
land on the attractor form a Basin of Attraction. A strange attractor
results if a system is sensitive to initial conditions and is not
conservative.
Note: While all chaotic attractors are strange, not all strange
attractors are chaotic.
Reference:
1. Grebogi, et al., Strange Attractors that are not Chaotic, _Physica
D_ 13 (1984), pp. 261-268.
Subject: 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).
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
The Mandelbrot Set by Eric Carr
http://www.cs.odu.edu/~carr/fractals/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 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:
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. 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:
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.
3. Jay Hill's Home page which includes his latest updates.
Jay's Hill Home Page via the World Wide Web.
http://www.geocities.com/CapeCanaveral/Lab/3825/
_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,
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