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From: [email protected] (Michael C Taylor (CSD))
Newsgroups: sci.fractals,sci.answers,news.answers
Subject: sci.fractals FAQ
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Date: 9 Mar 1998 00:16:03 GMT
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Summary: Frequently Asked Questions about Fractals
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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 [email protected] with the _message_:

  send usenet/news.answers/sci/fractals-faq

    _________________________________________________________________

 Suggestions, Comments, Mistakes

  Please send suggestions and corrections about the sci.fractals FAQ to
  [email protected]. 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 (&gt100 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 <[email protected]>
         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%[email protected] 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,
      _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%[email protected]> 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 :

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

Subject: Julia sets

  _Q7a_: What is the difference between the Mandelbrot set and a Julia
  set?

  _A7a_: The Mandelbrot set iterates z^2 + c with z starting at 0 and
  varying c. The Julia set iterates z^2 + c for fixed c and varying
  starting z values. That is, the Mandelbrot set is in parameter space
  (c-plane) while the Julia set is in dynamical or variable space
  (z-plane).

  _Q7b_: What is the connection between the Mandelbrot set and Julia
  sets?

  _A7b_: Each point c in the Mandelbrot set specifies the geometric
  structure of the corresponding Julia set. If c is in the Mandelbrot
  set, the Julia set will be connected. If c is not in the Mandelbrot
  set, the Julia set will be a Cantor dust.

  _Q7c_: How is a Julia set actually computed?

  _A7c_: The Julia set can be computed by iteration similar to the
  Mandelbrot computation. The only difference is that the c value is
  fixed and the initial z value varies.

  Alternatively, points on the boundary of the Julia set can be computed
  quickly by using inverse iterations. This technique is particularly
  useful when the Julia set is a Cantor Set. In inverse iteration, the
  equation z1 = z0^2 + c is reversed to give an equation for z0: z0 =
  �sqrt(z1 - c). By applying this equation repeatedly, the resulting
  points quickly converge to the Julia set boundary. (At each step,
  either the positive or negative root is randomly selected.) This is a
  nonlinear iterated function system.

  In pseudocode:

z = 1 (or any value)
loop
if (random number < .5) then
 z = sqrt(z - c)
else
 z = -sqrt(z - c)
endif
plot z
end loop

  _Q7d_: What are some Julia set facts?

  _A7d_: The Julia set of any rational map of degree greater than one is
  perfect (hence in particular uncountable and nonempty), completely
  invariant, equal to the Julia set of any iterate of the function, and
  also is the boundary of the basin of attraction of every attractor for
  the map.

  Julia set references:

   1. A. F. Beardon, _Iteration of Rational Functions : Complex Analytic
      Dynamical Systems_, Springer-Verlag, New York, 1991.
   2. P. Blanchard, Complex Analytic Dynamics on the Riemann Sphere,
      _Bull. of the Amer. Math. Soc_ 11, 1 (July 1984), pp. 85-141.

  This article is a detailed discussion of the mathematics of iterated
  complex functions. It covers most things about Julia sets of rational
  polynomial functions.

Subject: Complex arithmetic and quaternion arithmetic

  _Q8a_: How does complex arithmetic work?

  _A8a_: It works mostly like regular algebra with a couple additional
  formulas:
  (note: a, b are reals, _x_, _y_ are complex, _i_ is the square root of
  -1)

  Powers of _i_:
         _i_^2 = -1

  Addition:
         (a+_i_*b)+(c+_i_*d) = (a+c)+_i_*(b+d)

  Multiplication:
         (a+_i_*b)*(c+_i_*d) = a*c-b*d + _i_*(a*d+b*c)

  Division:
         (a+_i_*b) / (c+_i_*d) = (a+_i_*b)*(c-_i_*d) / (c^2+d^2)

  Exponentiation:
         exp(a+_i_*b) = exp(a)*(cos(b)+_i_*sin(b))

  Sine:
         sin(_x_) = (exp(_i_*_x_) - exp(-_i_*_x_)) / (2*_i_)

  Cosine:
         cos(_x_) = (exp(_i_*_x_) + exp(-_i_*_x_)) / 2

  Magnitude:
         |a+_i_*b| = sqrt(a^2+b^2)

  Log:
         log(a+_i_*b) = log(|a+_i_*b|)+_i_*arctan(b / a) (Note: log is
         multivalued.)

  Log (polar coordinates):
         log(r e^(_i_*a)) = log(r)+_i_*a

  Complex powers:
         _x_^y = exp(y*log(x))

  de Moivre's theorem:
         _x_^n = r^n [cos(n*a) + _i_*sin(n*a)] (where n is an integer)

  More details can be found in any complex analysis book.

  _Q8b_: How does quaternion arithmetic work?

  _A8b_: quaternions have 4 components (a + _i_b + _j_c + _k_d) compared
  to the two of complex numbers. Operations such as addition and
  multiplication can be performed on quaternions, but multiplication is
  not commutative.

  Quaternions satisfy the rules

    * i^2 = j^2 = k^2 = -1
    * ij = -ji = k
    * jk = -kj = i,
    * ki = -ik = j

  See:

  Frode Gill's quaternions page
         http://www.krs.hia.no/~fgill/quatern.html

Subject: Logistic equation

  _Q9_: What is the logistic equation?

  _A9_: It models animal populations. The equation is x -> c x (1 - x),
  where x is the population (between 0 and 1) and c is a growth
  constant. Iteration of this equation yields the period doubling route
  to chaos. For c between 1 and 3, the population will settle to a fixed
  value. At 3, the period doubles to 2; one year the population is very
  high, causing a low population the next year, causing a high
  population the following year. At 3.45, the period doubles again to 4,
  meaning the population has a four year cycle. The period keeps
  doubling, faster and faster, at 3.54, 3.564, 3.569, and so forth. At
  3.57, chaos occurs; the population never settles to a fixed period.
  For most c values between 3.57 and 4, the population is chaotic, but
  there are also periodic regions. For any fixed period, there is some c
  value that will yield that period. See _An Introduction to Chaotic
  Dynamical Systems_, by R. L. Devaney, for more information.

Subject: Feigenbaum's constant

  _Q10_: What is Feigenbaum's constant?

  _A10_: In a period doubling cascade, such as the logistic equation,
  consider the parameter values where period-doubling events occur (e.g.
  r[1]=3, r[2]=3.45, r[3]=3.54, r[4]=3.564...). Look at the ratio of
  distances between consecutive doubling parameter values; let delta[n]
  = (r[n+1]-r[n])/(r[n+2]-r[n+1]). Then the limit as n goes to infinity
  is Feigenbaum's (delta) constant.

  Based on computations by F. Christiansen, P. Cvitanovic and H.H. Rugh,
  it has the value 4.6692016091029906718532038... _Note_: several books
  have published incorrect values starting 4.6692016_6_...; the last
  repeated 6 is a _typographical error_.

  The interpretation of the delta constant is as you approach chaos,
  each periodic region is smaller than the previous by a factor
  approaching 4.669...

  Feigenbaum's constant is important because it is the same for any
  function or system that follows the period-doubling route to chaos and
  has a one-hump quadratic maximum. For cubic, quartic, etc. there are
  different Feigenbaum constants.

  Feigenbaum's alpha constant is not as well known; it has the value
  2.50290787509589282228390287272909. This constant is the scaling
  factor between x values at bifurcations. Feigenbaum says,
  "Asymptotically, the separation of adjacent elements of period-doubled
  attractors is reduced by a constant value [alpha] from one doubling to
  the next". If d[a] is the algebraic distance between nearest elements
  of the attractor cycle of period 2^a, then d[a]/d[a+1] converges to
  -alpha.

  References:

   1. K. Briggs, How to calculate the Feigenbaum constants on your PC,
      _Aust. Math. Soc. Gazette_ 16 (1989), p. 89.
   2. K. Briggs, A precise calculation of the Feigenbaum constants,
      _Mathematics of Computation_ 57 (1991), pp. 435-439.
   3. K. Briggs, G. R. W. Quispel and C. Thompson, Feigenvalues for
      Mandelsets, _J. Phys. A_ 24 (1991), pp. 3363-3368.
   4. F. Christiansen, P. Cvitanovic and H.H. Rugh, "The spectrum of the
      period-doubling operator in terms of cycles", _J. Phys A_ 23, L713
      (1990).
   5. M. Feigenbaum, The Universal Metric Properties of Nonlinear
      Transformations, _J. Stat. Phys_ 21 (1979), p. 69.
   6. M. Feigenbaum, Universal Behaviour in Nonlinear Systems, _Los
      Alamos Sci_ 1 (1980), pp. 1-4. Reprinted in _Universality in
      Chaos_, compiled by P. Cvitanovic.

  Feigenbaum Constants
         http://www.mathsoft.com/asolve/constant/fgnbaum/fgnbaum.html

Subject: Iterated function systems and compression

  _Q11a_: What is an iterated function system (IFS)?

  _A11a_: If a fractal is self-similar, you can specify mappings that
  map the whole onto the parts. Iteration of these mappings will result
  in convergence to the fractal attractor. An IFS consists of a
  collection of these (usually affine) mappings. If a fractal can be
  described by a small number of mappings, the IFS is a very compact
  description of the fractal. An iterated function system is By taking a
  point and repeatedly applying these mappings you end up with a
  collection of points on the fractal. In other words, instead of a
  single mapping x -> F(x), there is a collection of (usually affine)
  mappings, and random selection chooses which mapping is used.

  For instance, the Sierpinski triangle can be decomposed into three
  self-similar subtriangles. The three contractive mappings from the
  full triangle onto the subtriangles forms an IFS. These mappings will
  be of the form "shrink by half and move to the top, left, or right".

  Iterated function systems can be used to make things such as fractal
  ferns and trees and are also used in fractal image compression.
  _Fractals Everywhere_ by Barnsley is mostly about iterated function
  systems.

  The simplest algorithm to display an IFS is to pick a starting point,
  randomly select one of the mappings, apply it to generate a new point,
  plot the new point, and repeat with the new point. The displayed
  points will rapidly converge to the attractor of the IFS.

  Interactive IFS Playground (Otmar Lendl)
         http://www.cosy.sbg.ac.at/rec/ifs/

  Frank Rousell's hyperindex of IFS images
         http://www.cnam.fr/fractals/ifs.html

  _Q11b_: What is the state of fractal compression?

  _A11b_: Fractal compression is quite controversial, with some people
  claiming it doesn't work well, and others claiming it works
  wonderfully. The basic idea behind fractal image compression is to
  express the image as an iterated function system (IFS). The image can
  then be displayed quickly and zooming will generate infinite levels of
  (synthetic) fractal detail. The problem is how to efficiently generate
  the IFS from the image. Barnsley, who invented fractal image
  compression, has a patent on fractal compression techniques
  (4,941,193). Barnsley's company, Iterated Systems Inc
  (http://www.iterated.com/), has a line of products including a Windows
  viewer, compressor, magnifier program, and hardware assist board.

  Fractal compression is covered in detail in the comp.compression FAQ
  file (See "compression-FAQ").
  ftp://rtfm.mit.edu/pub/usenet/comp.compression .

  One of the best online references for Fractal Compress is Yuval
  Fisher's Fractal Image Encoding page
  (http://inls.ucsd.edu/y/Fractals/) at the Institute for Nonlinear
  Science, University for California, San Diego. It includes references
  to papers, other WWW sites, software, and books about Fractal
  Compression.

  Three major research projects include:

  Waterloo Montreal Verona Fractal Research Initiative
         http://links.uwaterloo.ca/

  Groupe FRACTALES
         http://www-syntim.inria.fr/fractales/

  Bath Scalable Video Software Mk 2
         http://dmsun4.bath.ac.uk/bsv-mk2/

  Several books describing fractal image compression are:

   1. M. Barnsley, _Fractals Everywhere_, Academic Press Inc., 1988.
      ISBN 0-12-079062-9. This is an excellent text book on fractals.
      This is probably the best book for learning about the math
      underpinning fractals. It is also a good source for new fractal
      types.
   2. M. Barnsley and L. Anson, _The Fractal Transform_, Jones and
      Bartlett, April, 1993. ISBN 0-86720-218-1. Without assuming a
      great deal of technical knowledge, the authors explain the
      workings of the Fractal Transform(TM).
   3. M. Barnsley and L. Hurd, _Fractal Image Compression_, Jones and
      Bartlett. ISBN 0-86720-457-5. This book explores the science of
      the fractal transform in depth. The authors begin with a
      foundation in information theory and present the technical
      background for fractal image compression. In so doing, they
      explain the detailed workings of the fractal transform. Algorithms
      are illustrated using source code in C.
   4. Y. Fisher (Ed), _Fractal Image Compression: Theory and
      Application_. Springer Verlag, 1995.
   5. Y. Fisher (Ed), _Fractal Image Encoding and Analysis: A NATO ASI
      Series Book_, Springer Verlag, New York, 1996 contains the
      proceedings of the Fractal Image Encoding and Analysis Advanced
      Study Institute held in Trondheim, Norway July 8-17, 1995. The
      book is currently being produced.

  Some introductary articles about fractal compression:

   1. The October 1993 issue of Byte discussed fractal compression. You
      can ftp sample code:
      ftp://ftp.uu.net/published/byte/93oct/fractal.exe .
   2. A Better Way to Compress Images," M.F. Barnsley and A.D. Sloan,
      BYTE, pp. 215-223, January 1988.
   3. "Fractal Image Compression," M.F. Barnsley, Notices of the
      American Mathematical Society, pp. 657-662, June 1996.
      (http://www.ams.org/publications/notices/199606/barnsley.html)
   4. A. E. Jacquin, Image Coding Based on a Fractal Theory of Iterated
      Contractive Image Transformation, _IEEE Transactions on Image
      Processing_, January 1992.
   5. A "Hitchhiker's Guide to Fractal Compression" For Beginners by
      E.R. Vrscay
      ftp://links.uwaterloo.ca/pub/Fractals/Papers/Waterloo/vr95.ps.gz

  Andreas Kassler wrote a Fractal Image Compression with WINDOWS package
  for a Fractal Compression thesis. It is available at
  http://www-vs.informatik.uni-ulm.de/Mitarbeiter/Kassler/papers.htm

  Other references:

  Fractal Compression Bibliography
         http://www.dip.ee.uct.ac.za/imageproc/compression/fractal/fract
         al.bib.html

  Fractal Video Compression
         http://inls.ucsd.edu/y/Fractals/Video/fracvideo.html

  Many fractal image compression papers are available from
         ftp://ftp.informatik.uni-freiburg.de/documents/papers/fractal

  A review of the literature is in Guide.ps.gz.
         ftp://ftp.informatik.uni-freiburg.de/documents/papers/fractal/R
         EADME

Subject: Chaotic demonstrations

  _Q12a_: How can you make a chaotic oscillator?

  _A12a_: Two references are:

   1. T. S. Parker and L. O. Chua, Chaos: a tutorial for engineers,
      _Proceedings IEEE_ 75 (1987), pp. 982-1008.
   2. _New Scientist_, June 30, 1990, p. 37.

  _Q12b_: What are laboratory demonstrations of chaos?

  _A12b_: Robert Shaw at UC Santa Cruz experimented with chaos in
  dripping taps. This is described in:

   1. J. P. Crutchfield, Chaos, _Scientific American_ 255, 6 (Dec.
      1986), pp. 38-49.
   2. I. Stewart, _Does God Play Dice?: the Mathematics of Chaos_, B.
      Blackwell, New York, 1989.

  Two references to other laboratory demonstrations are:

   1. K. Briggs, Simple Experiments in Chaotic Dynamics, _American
      Journal of Physics_ 55, 12 (Dec 1987), pp. 1083-1089.
   2. J. L. Snider, Simple Demonstration of Coupled Oscillations,
      _American Journal of Physics_ 56, 3 (Mar 1988), p. 200.

  See sci.nonlinear FAQ and the sci.nonlinear newsgroup for further
  information.

Subject: L-Systems

  _Q13_: What are L-systems?

  _A13_: A L-system or Lindenmayer system is a formal grammar for
  generating strings. (That is, it is a collection of rules such as
  replace X with XYX.) By recursively applying the rules of the L-system
  to an initial string, a string with fractal structure can be created.
  Interpreting this string as a set of graphical commands allows the
  fractal to be displayed. L-systems are very useful for generating
  realistic plant structures.

  Some references are:

   1. P. Prusinkiewicz and J. Hanan, _Lindenmayer Systems, Fractals, and
      Plants_, Springer-Verlag, New York, 1989.
   2. P. Prusinkiewicz and A. Lindenmayer, _The Algorithmic Beauty of
      Plants_, Springer-Verlag, NY, 1990. ISBN 0-387-97297-8. A very
      good book on L-systems, which can be used to model plants in a
      very realistic fashion. The book contains many pictures.

    _________________________________________________________________

  More information can be obtained via the WWW at:

  L-Systems Tutorial by David Green
         http://life.csu.edu.au/complex/tutorials/tutorial2.html
         http://www.csu.edu.au/complex_systems/tutorial2.html

  Graphics Archive at the Center for the Computation and Visualization
         of Geometric Structures contains various fractals created from
         L-Systems.
         http://www.geom.umn.edu/graphics/pix/General_Interest/Fractals/

Subject: Fractal music

  _Q14_: What are sources of fractal music?

  _A14_: One fractal recording is "The Devil's Staircase: Composers and
  Chaos" on the Soundprint label. A second is "Curves and Jars" by Barry
  Lewis. You can contact MPS Music & Video for further information:
  Rosegarth, Hetton Road, Houghton-le-Spring, DH5 8JN, England or online
  at CDeMUSIC (http://www.emf.org/focus_lewisbarry.html).

  Does anyone know of others? Mail me at [email protected].

  Some references, many from an unpublished article by Stephanie Mason,
  are:

   1. R. Bidlack, Chaotic Systems as Simple (But Complex) Compositional
      Algorithms, _Computer Music Journal_, Fall 1992.
   2. C. Dodge, A Musical Fractal, _Computer Music Journal_ 12, 13 (Fall
      1988), p. 10.
   3. K. J. Hsu and A. Hsu, Fractal Geometry of Music, _Proceedings of
      the National Academy of Science, USA_ 87 (1990), pp. 938-941.
   4. K. J. Hsu and A. Hsu, Self-similatrity of the '1/f noise' called
      music., _Proceedings of the National Academy of Science USA_ 88
      (1991), pp. 3507-3509.
   5. C. Pickover, _Mazes for the Mind: Computers and the Unexpected_,
      St. Martin's Press, New York, 1992.
   6. P. Prusinkiewicz, Score Generation with L-Systems, _International
      Computer Music Conference 86 Proceedings, _1986, pp. 455-457.
   7. _Byte_ 11, 6 (June 1986), pp. 185-196.

  Online resources include:

  Well Tempered Fractal v3.0 by Robert Greenhouse
         http://www-ks.rus.uni-stuttgart.de/people/schulz/fmusic/wtf/

  A fractal music C++ package is available at
         http://hamp.hampshire.edu/~gpzF93/inSanity.html

  The Fractal Music Project (Claus-Dieter Schulz)
         http://www-ks.rus.uni-stuttgart.de/people/schulz/fmusic

  Chua's Oscillator: Applications of Chaos to Sound and Music
         http://www.ccsr.uiuc.edu/People/gmk/Projects/ChuaSoundMusic/Chu
         aSoundMusic.html

  Fractal Music Lab
         http://members.aol.com/strohbeen/fml.html

  Fractal Music - Phil Thompson
         http://easyweb.easynet.co.uk/~cenobyte/

  fractal music in MIDI format by Jose Oscar Marques
         http://midiworld.com/jmarques.htm

  Don Archer's fractal art and music contains several pieces of fractal
         music in MIDI format.
         http://www.dorsai.org/~arch/

  LMUSe, a DOS program that generates MIDI music and files from 3D
         L-systems.
         http://www.interport.net/~dsharp/lmuse.html

  There is now a Fractal Music mailing list. It's purposes are:

   1. To inform people about news, updates, changes on the Fractal Music
      Projects WWW pages.
   2. To encourage discussion between people working in that area.

  The Fractal Music Mailinglist: [email protected]


         To subscribe to the list please send mail to
         [email protected]

Subject: Fractal mountains

  _Q15_: How are fractal mountains generated?

  _A15_: Usually by a method such as taking a triangle, dividing it into
  3 sub-triangles, and perturbing the center point. This process is then
  repeated on the sub-triangles. This results in a 2-d table of heights,
  which can then be rendered as a 3-d image. This is referred to as
  midpoint displacement. Two references are:

   1. M. Ausloos, _Proc. R. Soc. Lond. A_ 400 (1985), pp. 331-350.
   2. H.O. Peitgen, D. Saupe, _The Science of Fractal Images_,
      Springer-Velag, 1988

  Available online is an implementation of fractal Brownian motion (fBm)
  such as described in _The Science of Fractal Images_. Lucasfilm became
  famous for its fractal landscape sequences in _Star Trek II: The Wrath
  of Khan_ the primary one being the _Genesis_ planet transformation.
  Pixar and Digital Productions are have produced fractal landscapes for
  Hollywood.

  Fractal landscape information available online:

  EECS News: Fall 1994: Building Fractal Planets by Ken Musgrave
         http://www.seas.gwu.edu/faculty/musgrave/article.html

  Gforge and Landscapes (John Beale)
         http://www.best.com/~beale/

  Java fractal landscapes :

  Fractal landscapes (applet and sources) by Chris Thornborrow
         http://www-europe.sgi.com/Fun/free/java/chris-thornborrow/index
         .html

Subject: Plasma clouds

  _Q16_: What are plasma clouds?

  _A16_: They are a Fractint fractal and are similar to fractal
  mountains. Instead of a 2-d table of heights, the result is a 2-d
  table of intensities. They are formed by repeatedly subdividing
  squares.

  Robert Cahalan has fractal information about Earth's Clouds including
  how they differ from plasma clouds.

  Fractal Clouds Reference by Robert F. Cahalan
         ([email protected])
         http://climate.gsfc.nasa.gov/~cahalan/FractalClouds/

  Also some plasma-based fractals clouds by John Walker are available.

  Fractal generated clouds
         http://ivory.nosc.mil/html/trancv/html/cloud-fract.html

  The Center for the Computation and Visualization of Geometric
         Structures also has some fractal clouds.
         http://www.geom.umn.edu/graphics/pix/General_Interest/Fractals/

  Two articles about the fractal nature of Earth's clouds:

   1. "Fractal statistics of cloud fields," R. F. Cahalan and J. H.
      Joseph, _Mon. Wea.Rev._ 117, 261-272, 1989
   2. "The albedo of fractal stratocumulus clouds," R. F. Cahalan, W.
      Ridgway, W. J. Wiscombe, T. L. Bell and J. B. Snider, _J. Atmos.
      Sci._ 51, 2434-2455, 1994

Subject: Lyapunov fractals

  _Q17a_: Where are the popular periodically-forced Lyapunov fractals
  described?

  _A17a_: See:

   1. A. K. Dewdney, Leaping into Lyapunov Space, _Scientific American_,
      Sept. 1991, pp. 178-180.
   2. M. Markus and B. Hess, Lyapunov Exponents of the Logistic Map with
      Periodic Forcing, _Computers and Graphics_ 13, 4 (1989), pp.
      553-558.
   3. M. Markus, Chaos in Maps with Continuous and Discontinuous Maxima,
      _Computers in Physics_, Sep/Oct 1990, pp. 481-493.

  _Q17b_: What are Lyapunov exponents?

  _A17b_: Lyapunov exponents quantify the amount of linear stability or
  instability of an attractor, or an asymptotically long orbit of a
  dynamical system. There are as many Lyapunov exponents as there are
  dimensions in the state space of the system, but the largest is
  usually the most important.

  Given two initial conditions for a chaotic system, a and b, which are
  close together, the average values obtained in successive iterations
  for a and b will differ by an exponentially increasing amount. In
  other words, the two sets of numbers drift apart exponentially. If
  this is written e^(n*(lambda) for _n_ iterations, then e^(lambda) is
  the factor by which the distance between closely related points
  becomes stretched or contracted in one iteration. Lambda is the
  Lyapunov exponent. At least one Lyapunov exponent must be positive in
  a chaotic system. A simple derivation is available in:

   1. H. G. Schuster, _Deterministic Chaos: An Introduction_, Physics
      Verlag, 1984.

  _Q17c_: How can Lyapunov exponents be calculated?

  _A17c_: For the common periodic forcing pictures, the Lyapunov
  exponent is:

  lambda = limit as N -> infinity of 1/N times sum from n=1 to N of
  log2(abs(dx sub n+1 over dx sub n))

  In other words, at each point in the sequence, the derivative of the
  iterated equation is evaluated. The Lyapunov exponent is the average
  value of the log of the derivative. If the value is negative, the
  iteration is stable. Note that summing the logs corresponds to
  multiplying the derivatives; if the product of the derivatives has
  magnitude < 1, points will get pulled closer together as they go
  through the iteration.

  MS-DOS and Unix programs for estimating Lyapunov exponents from short
  time series are available by ftp: ftp://inls.ucsd.edu/pub/ncsu/

  Computing Lyapunov exponents in general is more difficult. Some
  references are:

   1. H. D. I. Abarbanel, R. Brown and M. B. Kennel, Lyapunov Exponents
      in Chaotic Systems: Their importance and their evaluation using
      observed data, _International Journal of Modern Physics B_ 56, 9
      (1991), pp. 1347-1375.
   2. A. K. Dewdney, Leaping into Lyapunov Space, _Scientific American_,
      Sept. 1991, pp. 178-180.
   3. M. Frank and T. Stenges, _Journal of Economic Surveys_ 2 (1988),
      pp. 103- 133.
   4. T. S. Parker and L. O. Chua, _Practical Numerical Algorithms for
      Chaotic Systems_, Springer Verlag, 1989.

Subject: Fractal items

  _Q18_: Where can I get fractal T-shirts, posters and other items?

  _A18_: One source is Art Matrix, P.O. box 880, Ithaca, New York,
  14851, 1-800-PAX-DUTY.

  Another source is Media Magic; they sell many fractal posters,
  calendars, videos, software, t-shirts, ties, and a huge variety of
  books on fractals, chaos, graphics, etc. Media Magic is at PO Box 598
  Nicasio, CA 94946, 415-662-2426.

  A third source is Ultimate Image; they sell fractal t- shirts,
  posters, gift cards, and stickers. Ultimate Image is at PO Box 7464,
  Nashua, NH 03060-7464.

  Yet another source is Dave Kliman (516) 625-2504 [email protected], whose
  products are distributed through Spencer Gifts, Posterservice,
  1-800-666-7654, and Scandecor International., this spring, through JC
  Penny, featuring all-over fractal t-shirts, and has fractal umbrellas
  available from Shaw Creations (800) 328-6090.

  Cyber Fiber produces fractal silk scarves, t-shirts, and postcards.
  Contact Robin Lowenthal, Cyber Fiber, 4820 Gallatin Way, San Diego, CA
  92117.

  Chaos MetaLink website
  (http://www.industrialstreet.com/chaos/metalink.htm) also has
  postcards, CDs, and videos.

  Free fractal posters are available if you send a self-addressed
  stamped envelope to the address given on
  http://www.xmission.com/~legalize/gift.html. For foreign requests
  (outside USA) include two IRCs (international reply coupons) to cover
  the weight.

  ReFractal Design (http://www.refractal.com/) sells jewelry based on
  fractals.

  Lifesmith Classic Fractals (http://www.lifesmith.com/) claims to be
  the largest fractal art studio in USA. You can contact Jeff Berkowitz
  at [email protected].

  There is a form of broccoli called Romanesco which is actually
  cauli-brocs, cross between cauliflowers and broccoli. It has a fractal
  like form. It was created in Italy about eight years ago and available
  in many stores in Europe.

Subject: How can I take photos of fractals?

  _Q19_: How can I take photos of fractals?

  _A19_: Noel Giffin gets good results with the following setup: Use 100
  ISO (ASA) Kodak Gold for prints or 64 ISO (ASA) for slides. Use a long
  lens (100mm) to flatten out the field of view and minimize screen
  curvature. Use f/4 stop. Shutter speed must be longer than frame rate
  to get a complete image; 1/4 seconds works well. Use a tripod and
  cable release or timer to get a stable picture. The room should be
  completely blackened, with no light, to prevent glare and to prevent
  the monitor from showing up in the picture.

  You can also obtain high quality images by sending your Targa or GIF
  images to a commercial graphics imaging shop. They can provide much
  higher resolution images. Prices are about $10 for a 35mm slide or
  negative and about $50 for a high quality 4x5 negative.

Subject: Colour Rendering Techniques

  _Q20a_: What are the rendering methods commonly used for 256-colour
  fractals?

  _A20a_: The simplest form of rendering uses escape times. Pixels are
  coloured according to the number of iterations it takes for a pixel to
  _blow-up_ or escape the loop. Different criteria may be chosen to
  speed a pixel to its blow-up point and therefore change the rendering
  of a fractal. These include the biomorph method and epsilon-cross
  method, both developed by Clifford Pickover. Similar to the
  escape-time methods are Fractint's _real_, _imag_ and _summ_ options.
  These add the real and/or imaginary values of a points Z-potential (at
  the blow-up time) to the escape time. Normally, escape-time fractals
  exhibit a flat 2-D appearance with _banding_ quite apparent at the
  lowest escape times. The addition of z-potential to the escape times
  tends to reduce banding and simulate 3-D effects in the outer bands.

  Other traditional rendering methods for 256-colour fractals include
  continuous potential, external decomposition and level-set methods
  like Fractint's Bof60 and Bof61. Here the colour of a point is based
  on its Z-potential and/or exit angle. The potential may be obtained
  for when it is at its lowest or at its last value, or some other
  criteria. The potential is scaled then applied to the palette used.
  Scaling may be linear or logarithmic, as for example palettes are
  defined in Fractint. Orbit-trap fractals make extensive use of level
  curves, which are based on z-potentials scaled linearly. Decomposition
  uses exit angles to define colours. Exit angles are derived from the
  polar notation of a point's complex value. Akin to decomposition is
  Paul Carlson's atan method (which uses an average of the last two
  angles) and the _atan_ (single angle) method in Fractint. All of these
  methods can be used to simulated 3-D effects because of the continuous
  shadings possible.

  _Q20b_: How does rendering differ for true-colour fractals?

  _A20b_: The problem with true-colour rendering is that computers use a
  3D approach to simulating 16 million colours. The basic components for
  addressing true colour are red, green and blue (256 shades each.)
  There is no logical way to determine an one-dimensional index which
  can be used to address all the RGB colours available in true colour.
  Palettes can be simulated in true colour but are limited to about
  65000 colours (256x256). Even so, this is enough to eliminate most
  banding found in 256-colour fractals due to limited colour spread.

  Because of the flexability in choosing colours from an expanded
  "palette", the best rendering methods will use a combination of level
  curves and exit angles. While escape times can be fractionalized using
  interpolated iteration, the result is still very flat. One promising
  addition to true-colour rendering is acheived by accumulating data
  about a point as it is iterated. The data is then used as an offset to
  the colour normally calculated by other methods. Depending on the
  algorithm used, the "filter" (sic: Stephen C. Ferguson) can intensify,
  fragment or add interesting details to a picture.

Subject: 3-D fractals

  _Q21_: How can 3-D fractals be generated?

  _A21_: A common source for 3-D fractals is to compute Julia sets with
  quaternions instead of complex numbers. The resulting Julia set is
  four dimensional. By taking a slice through the 4-D Julia set (e.g. by
  fixing one of the coordinates), a 3-D object is obtained. This object
  can then be displayed using computer graphics techniques such as ray
  tracing.

  Frank Rousell's hyperindex of 3D images
         http://www.cnam.fr/fractals/mandel3D.html

  4D Quaternions by Tom Holroyd
         http://bambi.ccs.fau.edu/~tomh/fractals/fractals.html

  The papers to read on this are:

   1. J. Hart, D. Sandin and L. Kauffman, Ray Tracing Deterministic 3-D
      Fractals, _SIGGRAPH_, 1989, pp. 289-296.
   2. A. Norton, Generation and Display of Geometric Fractals in 3-D,
      _SIGGRAPH_, 1982, pp. 61-67.
   3. A. Norton, Julia Sets in the Quaternions, _Computers and
      Graphics_, 13, 2 (1989), pp. 267-278.

  Two papers on cubic polynomials, which can be used to generate 4-D
  fractals:

   1. B. Branner and J. Hubbard, The iteration of cubic polynomials,
      part I., _Acta Math_ 66 (1988), pp. 143-206.
   2. J. Milnor, Remarks on iterated cubic maps, This paper is available
      from ftp://math.sunysb.edu/preprints/ims90-6.ps.Z. Published in
      1991 SIGGRAPH Course Notes #14: Fractal Modeling in 3D Computer
      Graphics and Imaging.

  Instead of quaternions, you can of course use hypercomplex number such
  as in "FractInt", or other functions. For instance, you could use a
  map with more than one parameter, which would generate a
  higher-dimensional fractal.

  Another way of generating 3-D fractals is to use 3-D iterated function
  systems (IFS). These are analogous to 2-D IFS, except they generate
  points in a 3-D space.

  A third way of generating 3-D fractals is to take a 2-D fractal such
  as the Mandelbrot set, and convert the pixel values to heights to
  generate a 3-D "Mandelbrot mountain". This 3-D object can then be
  rendered with normal computer graphics techniques.

  POV-Ray 3.0, a freely available ray tracing package, has added 4-D
  fractal support. It takes a 3-D slice of a 4-D Julia set based on an
  arbitrary 3-D "plane" done at any angle. For more information see the
  POV Ray web site at http://www.povray.org/ .

Subject: Fractint

  _Q22a_: What is Fractint?

  _A22a_: Fractint is a very popular freeware (not public domain)
  fractal generator. There are DOS, MS-Windows, OS/2, Amiga, and
  Unix/X-Windows versions. The DOS version is the original version, and
  is the most up-to-date.

  _Please note_: sci.fractals is not a product support newsgroup for
  Fractint. Bugs in Fractint/Xfractint should usually go to the authors
  rather than being posted.

  Fractint is on many ftp sites. For example:

  A Guide to getting FractInt by Noel at Spanky (Canada)
  http://spanky.triumf.ca/www/fractint/getting.html

 DOS

  19.6 executable via FTP and WWW from SimTel & mirrors world-wide
         http://www.coast.net/cgi-bin/coast/dwn?msdos/graphics/frain196.
         zip

  19.6 source via FTP and WWW from SimTel & mirrors world-wide
         http://www.coast.net/cgi-bin/coast/dwn?msdos/graphics/frasr196.
         zip

  19.6 executable via FTP from Canada
         ftp://fractal.mta.ca/pub/spanky/programs/ibmpc/frain196.zip

  19.6 source via FTP from Canada
         ftp://fractal.mta.ca/pub/spanky/programs/ibmpc/frasr196.zip

  (The suffix _196_ will change as new versions are released.)

  Fractint is available on Compuserve: GO GRAPHDEV and look for
  FRAINT.EXE and FRASRC.EXE in LIB 4.

 Windows

  MS-Window FractInt 18.21 via FTP and WWW from SimTel & mirrors
         world-wide
         http://www.coast.net/cgi-bin/coast/dwn?win3/graphics/winf1821.z
         ip

  MS-Window FractInt 18.21 via FTP from Canada
         ftp://fractal.mta.ca/pub/spanky/programs/ibmpc/windows/winf1821
         .zip

  MS-Windows FractInt 18.21 source via FTP and WWW from SimTel & mirrors
         world-wide
         http://www.coast.net/cgi-bin/coast/dwn?win3/graphics/wins1821.z
         ip

  MS-Windows FractInt 18.21 source via FTP from Canada
         ftp://fractal.mta.ca/pub/spanky/programs/ibmpc/windows/wins1821
         .zip

 OS/2

  Available on Compuserve in its GRAPHDEV forum. The files are PM*.ZIP.
  These files are also available on many sites, for example
  http://oak.oakland.edu/pub/os2/graphics/

 Unix

  The Unix version of FractInt, called _XFractInt_ requires X-Windows.
  The current version 3.04 is based on FractInt 19.6.

  3.04 source Western Canada
         http://spanky.triumf.ca/pub/fractals/programs/unix/xfract304.tg
         z

  3.04 source Atlantic Canada
         http://fractal.mta.ca/spanky/programs/unix/xfract304.tgz

  XFractInt is also available in LIB 4 of Compuserve's GO GRAPHDEV forum
  in XFRACT.ZIP.

  _Xmfract_ by Darryl House is a port of FractInt to a X/Motif
  multi-window interface. The current version is 1.4 which is compatible
  with FractInt 18.2.

  README
         http://fractal.mta.ca/pub/fractals/programs/unix/xmfract_1-4.re
         adme

  xmfract_1-4_tar.gz
         http://fractal.mta.ca/pub/fractals/programs/unix/xmfract_1-4_ta
         r.gz

 Macintosh

  There is _NO_ Macintosh version of Fractint, although there may be
  several people working on a port. It is possible to run Fractint on
  the Macintosh if you use a PC emulator such as Insignia Software's
  SoftAT.

 Amiga

  There is an Amiga version also available:

  FracInt 3.2
         http://spanky.triumf.ca/pub/fractals/programs/AMIGA/

 FracXtra

         There is a collection of map, parameter, etc. files for
         FractInt, called FracXtra. It is available at

       FracXtra Home Page by Dan Goldwater
               http://fatmac.ee.cornell.edu/~goldwada/fracxtra.html

       FracXtra via FTP and WWW from SimTel & mirrors world-wide
               http://www.coast.net/cgi-bin/coast/dwn?msdos/graphics/fra
               cxtr6.zip

       FracXtra via FTP
               ftp://fractal.mta.ca/pub/spanky/programs/ibmpc/fracxtr6.z
               ip

         _Q22b_: How does Fractint achieve its speed?

         _A22b_: Fractint's speed (such as it is) is due to a
         combination of:

        1. Reducing computation by Periodicity checking and guessing
           solid areas (especially the "lake" area).
        2. Using hand-coded assembler in many places.
        3. Using fixed point math rather than floating point where
           possible (huge improvement for non-coprocessor machine, small
           for 486's, moot for Pentium processors).
        4. Exploiting symmetry of the fractal.
        5. Detecting nearly repeating orbits, avoid useless iteration
           (e.g. repeatedly iterating 02+0 etc. etc.).
        6. Obtaining both sin and cos from one 387 math coprocessor
           instruction.
        7. Using good direct memory graphics writing in 256-color modes.

         The first three are probably the most important. Some of these
         introduce errors, usually quite acceptable.

Subject: Fractal software

  _Q23_: Where can I obtain software packages to generate fractals?

  _A23_:

    * Amiga
    * Java
    * Macintosh
    * MS-DOS
    * MS-Windows
    * SunView
    * UNIX
    * X-Windows

    * Software to calculate fractal dimension

 For Amiga:

  (all entries marked "ff###" are directories where the inividual
  archives of the Fred Fish Disk set available at
  ftp://ftp.funet.fi/pub/amiga/fish/ and other sites)

  General Mandelbrot generators with many features: Mandelbrot (ff030),
  Mandel (ff218), Mandelbrot (ff239), TurboMandel (ff302), MandelBltiz
  (ff387), SMan (ff447), MandelMountains (ff383, in 3-D), MandelPAUG
  (ff452, MandFXP movies), MandAnim (ff461, anims), ApfelKiste (ff566,
  very fast), MandelSquare (ff588, anims)

  Mandelbrot and Julia sets generators: MandelVroom (ff215), Fractals
  (ff371, also Newton-R and other sets)

  With different algorithmic approaches (shown): FastGro (ff188, DLA),
  IceFrac (ff303, DLA), DEM (ff303, DEM), CPM (ff303, CPM in 3-D),
  FractalLab (ff391, any equation)

  Iterated Function System generators (make ferns, etc): FracGen (ff188,
  uses "seeds"), FCS (ff465), IFSgen (ff554), IFSLab (ff696, "Collage
  Theorem")

  Unique fractal types: Cloud (ff216, cloud surfaces), Fractal (ff052,
  terrain), IMandelVroom (strange attractor contours?), Landscape
  (ff554, scenery), Scenery (ff155, scenery), Plasma (ff573, plasma
  clouds)

  Fractal generators: PolyFractals (ff015), FFEX (ff549)

  Fractint for Amiga
         http://spanky.triumf.ca/pub/fractals/programs/AMIGA/

  Lyapunov fractals
         http://www.itsnet.com/~bug/fractals/Lyapunovia.html

  XaoS, by Jan Hubicka, fast portable real-time interactive fractal
         zoomer. 256 workbench displays only.
         http://www.paru.cas.cz/~hubicka/XaoS/

  Commercial packages: Fractal Pro 5.0, Scenery Animator 2.0, Vista
  Professional, Fractuality (reviewed in April '93 Amiga User
  International). MathVISION 2.4. Generates Julia, Mandelbrot, and
  others. Includes software for image processing, complex arithmetic,
  data display, general equation evaluation. Available for $223 from
  Seven Seas Software, Box 1451, Port Townsend WA 98368.

 Java applets

  Chaos!
         http://www.vt.edu:10021/B/bwn/Chaos.html

  Fractal Lab
         http://www.wmin.ac.uk/~storyh/fractal/frac.html

  The Mandelbrot Set
         http://www.mindspring.com/~chroma/m andelbrot.html

  The Mandelbrot set (Paton J. Lewis)
         http://numinous.com/_private/people/pjl/graphics/mandelbrot/man
         delbrot.html

  Mark's Java Julia Set Generator
         http://www.stolaf.edu/people/mcclure/java/Julia/

  Fractals by Sun Microsystems
         http://java.sun.com/jav
         a.sun.com/applets/applets/Fractal/example1.html

  The Mandelbrot set
         http://www.franceway.com/java/fractale /mandel_b.htm

  Mandelbrot Java Applet
         http://www.mit.edu:8001/people/m kgray/java/Mandel.html

  Ken Shirriff Java language pages
         http://www.sunlabs.com/~shirriff/java/

  example of the plasma method of fractal terrain by Carl Burke,
         <[email protected]>
         http://www.geocities.com/Area51/6902/t_sd_app.html

  Mandelbrot generator in Javascript by Frode Gill.
         http://www.krs.hia.no/~fgill/javascript/mandscr.htm

  Fracula Java Applet. A java applet to glide into the Mandelbrot set
         (best with Pentium and MSIE 3.0). Vince Ruddy
         <[email protected]>
         http://www.geocities.com/SiliconValley/Pines/5788/index.html

  Chaos and Fractals. Many java applets by Stephen Oswin
         <[email protected]>
         www.ukmail.org/~oswin/

  IFS Fractals using javascript (Richard L. Bowman
         <[email protected]>)
         http://www.bridgewater.edu/departments/physics/ISAW/FracMain.ht
         ml

  A lot of Java applets
         http://java.developer.com/pages/tmp-Gamelan.mm.graphics.fractal
         s.html

  ChaosLab. A nice fully java site with several interactive applets
         showing different types of Mandelbrot, Julia, and strange
         attractors. By Cameron Mckechnie <[email protected]>
         http://www.actrix.gen.nz/users/chaoslab/chaoslab.html

  Fractal landscapes (applet and sources) by Chris Thornborrow
         http://www-europe.sgi.com/Fun/free/java/chris-thornborrow/index
         .html

  Forest Echo Farm Fractal Fern
         http://www.forestecho.com/ferns.html

  Fractal java generator by Patrick Charles
         http://www.csn.org/~pcharles/classes/FractalApp.html

  3 interactive java applets by Robert L. Devaney <[email protected]>
         http://math.bu.edu/DYSYS/applets/index.html

  Interactive java applets by Philip Baker <[email protected]>
  http://www.pjbsware.demon.co.uk/java/index.htm

  Chaos and order by Eric Leese
  http://www.geocities.com/CapeCanaveral/Hangar/7959/

  MB applet by Russ <[email protected]>
  http://home.att.net/~RBinNJ/mbapplet.htm

 Stand alone application

  Filmer by Julian Haight. Filmer is a front-end program for Fractint
         that generates amazing fractal animation. Fractint is a program
         for calculating still fractal images (you need Fractint
         installed to use Filmer). Filmer uses Fractint parameter (.par)
         files to specify the coordinates and other parameters of a
         fractal. It then calculates the intermediate frames and calls
         Fractint to make a continuous animation. Filmer also has many
         options for pallete rotation and generation.
         http://www.julianhaight.com/filmer/
  Javaquat by Garr Lystad. Can also be run as an applet from Lystad's
         page.
         http://www.iglobal.net/lystad/fractal-top.html

 For Macs:

                              For PowerMacs
            (and PowerPC-based Macintosh compatible computers)

 Fractal Domains v. 1.2

    * Fractal generator for PowerMacs only, by Dennis C. De Mars
      (formerly FracPPC)
    * Generates the Mandelbrot set and associated Julia sets, allows the
      user to edit the color map, 24-bit colour
         + http://members.aol.com/ddemars/fracppc.html

 MandelBrowser 2.0

    * by the author of Mandella, 24-bit colour
         +
           ftp://mirrors.aol.com/pub/mac/graphics/fractal/mandelbrowser2.
           0.sit.hqx

    _________________________________________________________________

                               For 68K Macs

 Mandella 8.7

    * generation of many different types of fractals, allow editing of
      the color map, and other display & calculation options. Some
      features not available on PowerMacs.
         +
           ftp://mirrors.aol.com/pub/mac/graphics/fractal/mandella8.7.cpt
           .hqx

 Mandelzot 4.0.1

    * generation of many different types of fractals, allow editing of
      the color map, and other display & calculation options. Some
      features not available on PowerMacs.
         +
           ftp://mirrors.aol.com/pub/mac/graphics/fractal/mandelzot4.01.c
           pt.hqx

 SuperMandelZoom 1.0.6

    * useful to those rare individuals who are still using a Mac Plus/SE
      class machine
         +
           ftp://mirrors.aol.com/pub/mac/graphics/fractal/supermandelzoom
           1.06.cpt.hqx

    _________________________________________________________________

                          Miscellaneous programs

    * _FDC and FDC 3D_ - Fractal Dimension Calculators
         + http://www.mhri.edu.au/~pdb/software/
    * _Lsystem, 3D-L-System, IFS, FracHill_
         + http://www.mhri.edu.au/~pdb/fractals/
    * _Color Fractal Generator_ 2.12
         +
           ftp://mirrors.aol.com/pub/mac/graphics/fractal/colorfractalgen
           2.12.sit.hqx
    * _MandelNet_ (uses several Macs on an AppleTalk network to
      calculate the Mandebrot set!)
         +
           ftp://mirrors.aol.com/pub/mac/graphics/fractal/mandelnet1.2.si
           t.hqx
    * _Julia's Nightmare_ - original and cool program, as you drag the
      mouse about the complex plane, the corresponding Julia set is
      generated in real time!
         +
           ftp://mirrors.aol.com/pub/mac/graphics/fractal/juliasnightmare
           .sit.hqx
    * _Lyapunov_ 1.0.1
         +
           ftp://mirrors.aol.com/pub/mac/graphics/fractal/lyapunov1.01.cp
           t.hqx
    * _Fract_ 1.0 - A fractal-drawing program that uses the IFS
      algorithm. Change parameters to get different self-similar
      patterns.
         +
           ftp://mirrors.aol.com/pub/mac/graphics/fractal/fract1.0.cpt.hq
           x
    * _XaoS_ 2.1 - fast portable real-time interactive fractal zoomer
         + http://www.paru.cas.cz/~hubicka/XaoS/

    _________________________________________________________________

                                Commerical

  There are also commercial programs: _IFS Explorer_ and _Fractal Clip
  Art_ (published by Koyn Software (314) 878-9125), _Kai's Fractal
  Explorer_ (part of the Kai's Power Tools package)

 For MSDOS:

  DEEPZOOM: a high-precision Mandelbrot Set program for displaying
         highly zoomed fractals
         http://spanky.triumf.ca/pub/fractals/programs/ibmpc/depzm13.zip

  Fractal WitchCraft: a very fast fractal design program
         ftp://garbo.uwasa.fi/pc/demo/fw1-08.zip
         ftp://ftp.cdrom.com/pub/garbo/garbo_pc/show/fw1-08.zip

  Fractal Discovery Laboratory: designed for use in a science museum or
         school setting. The Lab has five sections: Art Gallery,
         Microscope, Movies, Tools, and Library
         Sampler available from Compuserve GRAPHDEV Lib 4 in DISCOV.ZIP,
         or send high-density disk and self-addressed, stamped envelope
         to: Earl F. Glynn, 10808 West 105th Street, Overland Park,
         Kansas 66214-3057.

  WL-Plot 2.59 : plots functions including bifurcations and recursive
         relations
         ftp://archives.math.utk.edu/software/msdos/graphing/wlplt/wlplt
         259.zip

  From http://www.simtel.net/pub/simtelnet/msdos/graphics/
         forb01a.zip: Displays orbits of Mandelbrot Set mapping. C/E/VGA

         fract3.zip: Mandelbrot/Julia set 2D/3D EGA/VGA Fractal Gen
         fractfly.zip: Create Fractal flythroughs with FRACTINT
         fdesi313.zip: Program to visually design IFS fractals
         frain196.zip: FRACTINT v19.6 EGA/VGA/XGA fractal generator
         frasr196.zip: C & ASM src for FRACTINT v19.6
         frcal040.zip: CAL: more than 15 types of fractals including
         Lyapunov, IFS, user-defined, logistic, and Quaternion Julia

  Vlotkatc uses VESA 640x480x16 Million colour mode to generate
         Volterra-Lotka images.
         http://spanky.triumf.ca/pub/fractals/programs/ibmpc/vlotkatc.zi
         p
         http://spanky.triumf.ca/pub/fractals/programs/ibmpc/vlotkatc.do
         c

  Fast FPU Fractal Fun 2.0 (FFFF2.0) is the first Mandelbrot Set
         generator working in hicolor gfx modes thus using up to 32768
         different colors on screen by Daniele Paccaloni requires 386DX+
         and VESA support
         http://spanky.triumf.ca/pub/fractals/programs/IBMPC/FFFF20.ZIP

  3DFract generates 3-D fractals including Sierpinski cheese and 3-D
         snowflake
         http://www.cstp.umkc.edu/users/bhugh/home.html

  FracTrue 2.10 - Hi/TrueColor Generator including a formular parser.
         286+ VGA by Bernd Hemmerling

  LyapTrue 2.10 Lyapunov generator

  ChaosTrue 2.00 - 18 types

  Atractor 1.00 256 colour
         http://www.cs.tu-berlin.de/~hemmerli/fractal.html

  HOP based on the HOPALONG fractal type. Math coprocessor (386DX and
         above) and SuperVGA required. shareware ($30) Places to
         download HOPZIP.EXE from:
         Compuserve GRAPHDEV forum, lib 4
         The Well under ibmpc/graphics
         http://ourworld.compuserve.com/homepages/mpeters/hop.htm
         ftp://ftp.uni-heidelberg.de/pub/msdos/graphics/
         http://spanky.triumf.ca/pub/fractals/programs/ibmpc/

  ZsManJul 1.0 (requires 386DX+) by Zsolt Zsoldos
         http://www.chem.leeds.ac.uk/ICAMS/people/zsolt/zsmanjul.html

  FractMovie 1.62 a real-time 2D/3D IFS fractal movie renderer (requires
         486DX+) with GIF save
         http://pub.vse.cz/pub/msdos/SAC/pc/graph/frcmv162.zip

  FracZoom Explorer and FracZoom Navigator by Niels Ulrik Reinwald
         386DX+
         http://www.softorange.com/software.html

  RMandel 1.2 80-bit floating point Mandelbrot Set animation generator
         by Marvin R. Lipford
         ftp://fractal.mta.ca/pub/cnam/anim/FRACSOFT/rmandel.zip

  M24, the new version of TruMand by Mike Freeman 486DX+ True-colour
         Mandelbrot Set generator
         http://www.capcollege.bc.ca/~mfreeman/mand.html

  FAE - Fractal Animation Engine shareware by Brian Towles
         http://spanky.triumf.ca/pub/fractals/programs/ibmpc/FAE210B.ZIP

  XaoS 2.2 fast portable realtime interactive fractal zoomer/morpher for
         MS-DOS (and others) by Jan Hubicka <[email protected]>
         11 fractal formulas, "Autopilot", solid guessing, zoom up to
         64051194700380384 times
         http://www.paru.cas.cz/~hubicka/XaoS/

  Ultra Fractal. A DOS program with graphic interface, 256 colors or
         truecolor. Very fast, many formulas. Shareware (Frederik
         Slijkerman <[email protected]>)
         http://ourworld.compuserve.com/homepages/slijkerman/

  Fractal worldmap generator. A simple program to generate fractal
         pseudo geographic maps, by John Olsson <[email protected]>,
         DOS adaptation by Martijn Faassen <[email protected]>
         http://www.lysator.liu.se/~johol/fwmg/fwmg.html

  Quat - A 3D-Fractal-Generator (Quaternions).
         http://wwwcip.rus.uni-stuttgart.de/~phy11733/quat_e.html

 For MS-Windows:

  dy-syst: Explores Newton's method, Mandelbrot and Julia sets
         ftp://cssun.mathcs.emory.edu/pub/riddle/

  bmand 1.1 shareware by Christopher Bare Mandelbrot program
         http://www.ualberta.ca/~jdawe/mandelbrot/bmand11.zip

  Quaternion-generator generates Julia-set Quaternions by Frode Gill
         http://www.krs.hia.no/~fgill/fractal.html

  Quat - A 3D-Fractal-Generator (Quaternions).
         http://wwwcip.rus.uni-stuttgart.de/~phy11733/quat_e.html

  A Fractal Experience 32 for Windows 95/NT by David Wright
         <[email protected]>
         http://www.mnsinc.com/wgwright/fracexp/

  Iterate 32 for Windows 95/NT written in VisualBasic. Generates IFS,
         includes 10 built-in attractors, plots via chaos algorithm or
         MRCM (multiple reduction copy machine), includes MS-Word
         document about IFS and fractal compression in easy to
         understand terms. Freeware by Jeff Colvin <[email protected]>
         http://hamnetcenter.com/jeffc/fractal.html

  IFS Explorer for Windows 95/NT, a companion to Iterate 32, allows
         users to explore IFS by changing the IFS parameters. Requires
         800x600 screen. Freeware by Jeff Colvin <[email protected]>
         http://hamnetcenter.com/jeffc/fractal.html

  DFRAC 1.4 by John Ratcliff a Windows 95 DirectDraw Mandelbrot explorer
         with movie feature. Requires DirectDraw, FPU, and
         monitor/graphics card capable of 800x600 graphic mode.
         Freeware.
         http://www.inlink.com/~jratclif/john.htm

  QS W95 Fractals generates several fractals types in 24-bit colour
         includind Volterra-Lotka, enhanced sine, "Escher-like tiling"
         of Julia Set, magnetism formulae, and "self-squared dragons".
         Supports FractInt MAP files, saves 24-bit Targa or 8-bit GIF,
         several colour options. Freeware by Michael Sargent
         <[email protected]>.
         http://www.uvm.edu/~msargent/

  Other fractal programs by Michael Sargent.
         http://www.uvm.edu/~msargent/fractals.htm

  Fractal eXtreme for 32-bit Windows 1.01c. A fast interactive fractal
         explorer of Mandelbrot, Julia Set, and Mandelbrot to various
         powers, Newton, "Hidden Mandelbrot", and Auto Quadratic.
         Movies, curve-based palette editor, deep zoom (>2000 digits
         precision for some types), Auto-Explore. Shareware, with
         ability to register online, by Cygnus Software.
         http://www.cygnus-software.com/

  Iterations, Flarium24 and Inkblot Kaos Original programs : Now
         Iterations is true color as are Flarium 24 and Inkblot Kaos.
         For W95 or NT. Freeware by Stephen C. Ferguson
         (<[email protected]>)
         http://home1.gte.net/itriazon/

  JuliaSaver : a W95 screen saver that does real-time fractals, by
         Damien M. Jones (<[email protected]>)
         http://www.icd.com/tsd/juliasaver/

  Mndlzoom W95 or Nt program which iterate the Mandelbrot set within the
         coprocessor stack : very fast, 19-digits significance (Philip
         A. Seeger <[email protected]>)
         http://members.aol.com/paseeger/

  Frang : a real-time zooming Mandelbrot set generator. Needs DirectX
         (can be downloaded from the same URL or from Microsoft).
         Shareware (Michael Baldwin <[email protected]>)
         http://www.servtech.com/public/baldwin/frang/frang.html

  Fractal Orbits; A nice implementation of Bubble, Ring, Stalk methods
         by Phil Pickard <[email protected] >. Very easy to use.
         W95, NT.
         ftp://ftp-hs.iuta.u-bordeaux.fr/fractorb/

  Fractal Commander and Fractal Elite (formerly Zplot) Very
         comprehensive programs which gather several powerful methods
         (original or found in other programs). Now only 32 bits version
         is supported. You can download a free simplified version
         (Fractal Agent) at
         http://www.simtel.net/pub/simtelnet/win95/math/fa331.zip.
         Registered users will receive the full version and a true color
         one. Shareware by Terry W. Gintz <[email protected]>.
         http://www.geocities.com/SoHo/Lofts/5601/gallery.htm

  Set surfer. A nice small program. Draws a variety of fractals of
         Mandelbrot or Julia types. Freeware by Jason Letbetter
         <[email protected]>.
         http://www.flash.net/~redbeard/

  Kai Power Tools 2 and 3 include Fractal Explorer.
         MetaCreations will mail a replacement CD to early KPT 3.0
         owners which didn't include Fractal Explorer.

  Fantastic Fractals. This program can draw several sorts of fractals
         (IFS, L-system, Julia...). Well designed for IFS.
         http://library.advanced.org/12740/

 Screen savers

  Free screen savers : By Philip Baker (<[email protected]>)
         http://www.pjbsware.demon.co.uk/snsvdsp.htm

  JuliaSaver : a W95 screen saver that does real-time fractals, by
         Damien M. Jones (<[email protected]>)
         http://www.icd.com/tsd/juliasaver/

  IFS screen saver: a Windows 3 screen saver, by Bill Decker
         (<[email protected]>)
         http://www.geocities.com/SoHo/Studios/1450/

  Fractint Screen Saver: a Windows 95 - NT screen saver, by Thore
         Berntsen ; needs the DOS program Fractint (<[email protected]>)

         http://home.sol.no/~thbernt/fintsave.htm

  Seractal Screen Saver: Windows 3 and Windows 95 time limited versions
         (shareware) (<[email protected])>
         http://www.seraline.com/seractal.htm

  the Orb series by 'O' from RuneTEK. For MS-Windows 95/NT only.
         http://www.hypermart.net/runetek/

 For SunView:

  Mandtool: generates Mandelbrot Set
         http://fractal.mta.ca/spanky/programs/mandtool/m_tar.z
         ftp://spanky.triumf.ca/fractals/programs/mandtool/M_TAR.Z

 For Unix/C:

  lsys: L-systems as PostScript (in C++)
         ftp://ftp.cs.unc.edu/pub/users/leech/lsys.tar.gz

  lyapunov: PGM Lyapunov exponent images
         ftp://ftp.uu.net/usenet/comp.sources.misc/volume23/lyapunov/

  SPD: fractal mountain, tree, recursive tetrahedron
         ftp://ftp.povray.org/pub/povray/spd/

  Fractal Studio: Mandelbrot set; handles distributed computing
         ftp://archive.cs.umbc.edu/pub/peter/fractal-studio

  fanal: analysis of fractal dimension for Linux by J�rgen Dollinger
         ftp://ftp.uni-stuttgart.de/pub/systems/linux/local/math/fanal-0
         1b.tar.gz

  XaoS, by Jan Hubicka, fast portable real-time interactive fractal
         zoomer. supports X11 (8,15,16,24,31-bit colour, StaticGray,
         StaticColor), Curses, Linux/SVGAlib
         http://www.paru.cas.cz/~hubicka/XaoS/

 For X windows :

  xmntns xlmntn: fractal mountains
         ftp://ftp.uu.net/usenet/comp.sources.x/volume8/xmntns

  xfroot: fractal root window
         X11 distribution

  xmartin: Martin hopalong root window
         X11 distribution

  xmandel: Mandelbrot/Julia sets
         X11 distribution

  lyap: Lyapunov exponent images
         ftp://ftp.uu.net/usenet/comp.sources.x/volume17/lyapunov-xlib

  spider: Uses Thurston's algorithm, Kobe algorithm, external angles
         http://inls.ucsd.edu/y/Complex/spider.tar.Z

  xfractal_explorer: fractal drawing program
         ftp://ftp.x.org/contrib/applications/xfractal_explorer-v1.0.tar
         .gz

  Xmountains: A fractal landscape generator
         ftp://ftp.epcc.ed.ac.uk/pub/personal/spb/xmountains

  xfractint: the Unix version of Fractint : look at XFRACTxxx (xxx being
         the version number)
         http://spanky.triumf.ca/www/fractint/getting.html

  xmfract v1.4: Needs Motif 1.2+, based on FractInt
         http://hpftp.cict.fr/hppd/hpux/X11/Misc/xmfract-1.4/

  Fast Julia Set and Mandelbrot for X-Windows by Zsolt Zsoldos
         http://www.chem.leeds.ac.uk/ICAMS/people/zsolt/mandel.html

  XaoS realtime fractal zoomer for X11 or SVGAlibs by Jan Hubicka
         <[email protected]>
         http://www.paru.cas.cz/~hubicka/XaoS/

  AlmondBread-0.2. Fast algorithm ; simultaneous orbit iteration ;
         Fractint-compatible GIF and MAP files ; Tcl/Tk user interface
         (Michael R. Ganss <[email protected]>)
         http://www.cs.tu-berlin.de/~rms/AlmondBread/

  Quat - A 3D-Fractal-Generator (Quaternions).
         http://wwwcip.rus.uni-stuttgart.de/~phy11733/quat_e.html

  XFracky 2.5 by Henrik Wann Jensen <[email protected]> based on Tcl/Tk
         http://www.gk.dtu.dk/~hwj/
         http://sunsite.unc.edu/pub/Linux/X11/apps/math/fractals/

 Distributed X systems:

  MandelSpawn: Mandelbrot/Julia on a network
         ftp://ftp.x.org/R5contrib/mandelspawn-0.07.tar.Z
         ftp://ftp.funet.fi/pub/X11/R5contrib/mandelspawn-0.07.tar.Z

  gnumandel: Mandelbrot on a network
         ftp://ftp.elte.hu/pub/software/unix/gnu/gnumandel.tar.Z

 Software for computing fractal dimension:

  _Fractal Dimension Calculator_ is a Macintosh program which uses the
  box-counting method to compute the fractal dimension of planar
  graphical objects.

  http://wuarchive.wustl.edu/edu/math/software/mac/fractals/FDC/

  http://wuarchive.wustl.edu/packages/architec/Fractals/FDC2D.sea.hqx

  http://wuarchive.wustl.edu/packages/architec/Fractals/FDC3D.sea.hqx

  _FD3_: estimates capacity, information, and correlation dimension from
  a list of points. It computes log cell sizes, counts, log counts, log
  of Shannon statistics based on counts, log of correlations based on
  counts, two-point estimates of the dimensions at all scales examined,
  and over-all least-square estimates of the dimensions.

  ftp://inls.ucsd.edu/pub/cal-state-stan
         for an enhanced Grassberger-Procaccia algorithm for correlation
         dimension.

  A MS-DOS version of FP3 is available by request to
  [email protected].

Subject: FTP questions

  _Q24a_: How does anonymous ftp work?

  _A24a_: Anonymous ftp is a method of making files available to anyone
  on the Internet. In brief, if you are on a system with ftp (e.g.
  Unix), you type "ftp fractal.mta.ca", or whatever system you wish to
  access. You are prompted for your name and you reply "anonymous". You
  are prompted for your password and you reply with your email address.
  You then use "ls" to list the files, "cd" to change directories, "get"
  to get files, an "quit" to exit. For example, you could say "cd /pub",
  "ls", "get README", and "quit"; this would get you the file "README".
  See the man page ftp(1) or ask someone at your site for more
  information.

  In this FAQ, anonymous ftp addresses are given in the URL form
  ftp://name.of.machine/pub/path [138.73.1.18]. The first part is the
  protocol, FTP, rather than say "gopher", the second part
  "name.of.machine" is the machine you must ftp to. If your machine
  cannot determine the host from the name, you can try the numeric
  Internet address: "ftp 138.73.1.18". The part after the name:
  "/pub/path" is the file or directory to access once you are connected
  to the remote machine.

  _Q24b_: What if I can't use ftp to access files?

  _A24b_: If you don't have access to ftp because you are on a UUCP,
  Fidonet, BITNET network there is an e-mail gateway at
  [email protected] that can retrieve the files for you. To get
  instructions on how to use the ftp gateway send a message to
  [email protected] with one line containing the word "help".

  Warning, these archives can be very large, sometimes several megabytes
  (MB) of data which will be sent to your e-mail address. If you have a
  disk quota for incoming mail, often 1MB or less, be careful not exceed
  it.

Subject: Archived pictures

  _Q25a_: Where are fractal pictures archived?

 News groups

  _A25a_: Fractal images (GIFs, JPGs...) are posted to
  alt.binaries.pictures.fractals (also known as abpf); this newsgroup
  has replaced alt.fractals.pictures. However, several
  alt.binaries.pictures groups being badly reputed,
  alt.fractals.pictures seems to have some new activity.

 The fractals posted in alt.binaries.pictures.fractals are recorded daily at

  http://www.xmission.com/~legalize/fractals/index.html
  http://galaxy.uci.agh.edu.pl/pictures//alt.binaries.pictures.fractals/
         last.html
  http://www.cs.uni-magdeburg.de/pictures/Usenet/fractals/summary/

  The following lists are scanty and will evolve soon.

 Other archives and university sites (images, tutorials...)

  Many Mandelbrot set images are available via
         ftp://ftp.ira.uka.de/pub/graphic/fractals
  Pictures from 1990 and 1991 are available via anonymous ftp at
         ftp://csus.edu/pub/alt.fractals.pictures
  Fractal images including some recent alt.binaries.pictures.fractals
         images are archived at ftp://spanky.triumf.ca/fractals
  This can also be accessed via WWW at http://spanky.triumf.ca/ or
         http://fractal.mta.ca/spanky/
  From Paris, France one of the largest collections (>= 820MB) is Frank
         Roussel's at http://www.cnam.fr/fractals.html
  Fractal animations in MPEG and FLI format are in
         http://www.cnam.fr/fractals/anim.html
  In Bordeaux (France) there is a mirror of this site,
         http://graffiti.cribx1.u-bordeaux.fr/MAPBX/roussel/fractals.htm
         l
  and a Canadian mirror at http://fractal.mta.ca/cnam/
  Another collection of fractal images is archived at
         ftp://ftp.maths.tcd.ie/pub/images/Computer
  Fractal Microscope
         http://www.ncsa.uiuc.edu/Edu/Fractal/Fractal_Home.html
  "Contours of the Mind"
         http://online.anu.edu.au/ITA/ACAT/contours/contours.html
  Spanky Fractal Datbase (Noel Giffin)
         http://spanky.triumf.ca/www/spanky.html
  Yahoo Index of Fractal Art
         http://www.yahoo.com/Arts/Visual_Arts/Computer_Generated/Fracta
         ls/
  Geometry Centre at University of Minnesota
         http://www.geom.umn.edu/graphics/pix/General_Interest/Fractals/
  Computer Graphics Gallery
         http://www.maths.tcd.ie/pub/images/images.html

 Many fractal creators have personal web pages showing images, tutorials...

  Flame Index A collection of interesting smoke- and flame-like jpeg
         iterated function system images
         http://www.cs.cmu.edu/~spot/flame.htm
         Some images are also available from:
         ftp://hopeless.mess.cs.cmu.edu/spot/film/

  Cliff Pickover
         http://sprott.physics.wisc.edu/pickover/home.htm

  Fractal Gallery (J. C. Sprott) Personal images and a thousand of
         fractals collected in abpf
         http://sprott.physics.wisc.edu/fractals.htm

  Fractal from Ojai (Art Baker)
         http://www.bhs.com/ffo/

  Skal's 3D-fractal collection (Pascal Massimino)
         http://www.eleves.ens.fr:8080/home/massimin/quat/f_gal.ang.html

  3d Fractals (Stewart Dickson) via Mathart.com
         http://www.wri.com/~mathart/portfolio/SPD_Frac_portfolio.html

  Dirk's 3D-Fractal-Homepage
         http://wwwcip.rus.uni-stuttgart.de/~phy11733/index_e.html

  Softsource
         http://www.softsource.com/softsource/fractal.html

  Favourite Fractals (Ryan Grant)
         http://www.ncsa.uiuc.edu/SDG/People/rgrant/fav_pics.html

  Eric Schol
         http://snt.student.utwente.nl/~schol/gallery/

  Mandelbrot and Julia Sets (David E. Joyce)
         http://aleph0.clarku.edu/~djoyce/home.html

  Newton's method
         http://aleph0.clarku.edu/~djoyce/newton/newton.html

  Gratuitous Fractals ([email protected])
         http://www.vanderbilt.edu/VUCC/Misc/Art1/fractals.html

  Xmorphia
         http://www.ccsf.caltech.edu/ismap/image.html

  Fractal Prairie Page (George Krumins)
         http://www.prairienet.org/astro/fractal.html

  Fractal Gallery (Paul Derbyshire)
         http://chat.carleton.ca/~pderbysh/fractgal.html

  David Finton's fractal homepage
         http://www.d.umn.edu/~dfinton/fractals/

  Algorithmic Image Gallery (Giuseppe Zito)
         http://www.ba.infn.it/gallery

  Octonion Fractals built using hyper-hyper-complex numbers by Onar Em
         http://www.stud.his.no/~onar/Octonion.html

  B' Plasma Cloud (animated gif)
         http://www.az.com/~rsears/fractp1.html

  John Bailey's fractal images (<[email protected]>)
         http://www.frontiernet.net/~jmb184/interests/fractals/

  Fractal Art Parade (Douglas "D" Cootey <[email protected]>)
         http://www.itsnet.com/~bug/fractals.html

  The Fractory (John/Alex <[email protected]>)
         http://tqd.advanced.org/3288/

  FracPPC gallery (Dennis C. De Mars <[email protected]>)
         http://members.aol.com/ddemars/gallery.html

  http://galifrey.triode.net.au/ (Frances Griffin
         <[email protected]>)
         http://galifrey.triode.net.au/

  J.P. Louvet's Fractal Album
         http://graffiti.cribx1.u-bordeaux.fr/MAPBX/louvet/jpl0a.html )
         (Jean-Pierre Louvet <[email protected]> French and
         English versions)

  Carlson's Fractal Gallery
         http://sprott.physics.wisc.edu/carlson.htm (Paul Carlson
         <[email protected]>)

  Fractals by Paul Carlson
         http://fractal.mta.ca/fractals/carlson/ (an other Paul
         Carlson's Gallery)

  Daves's Graphics Page
         http://www.unpronounceable.com/graphics/ (David J. Grossman
         <graphics AT unpronounceable DOT com> replace the AT with '@'
         and DOT with '.' I apologize that I must take this drastic step
         to foil the spammers)

  Gumbycat's cyberhome
         http://www.geocities.com/~gumbycat/index.html (Linda Allison
         <[email protected]> Delete the dash ("-") in gumbycat to
         send e-mail. It's only purpose is to act as a spam deterent!)

  Sylvie Gallet Gallery
         http://spanky.triumf.ca/www/fractint/SYLVIE/GALLET.HTML

  Sylvie Gallet's Fractal Gallery New pages
         http://ourworld.compuserve.com/homepages/Sylvie_Gallet/homepage
         .htm (Sylvie Gallet <[email protected]>)

  Howard Herscovitch's Home Page
         http://home.echo-on.net/~hnhersco/

  Fractalus Home. Fractals by Damien M. Jones
         http://www.geocities.com/SoHo/Lofts/2605/ (Damien M. Jones
         <[email protected]>)

  Fractopia Home page. Bill Rossi
         http://members.aol.com/billatny/fractopi.htm (Bill Rossi
         <[email protected]>)

  Doug's Gallery. Doug Owen
         http://www.zenweb.com/rayn/doug/ (Doug Owen
         <[email protected]>)

  TWG's Gallery. Terry W. Gintz
         http://www.zenweb.com/rayn/twg/ (Terry W. Gintz
         <[email protected]>)

  Fractal Gallery
  http://members.aol.com/MKing77043/index.htm (Mark King
  <[email protected]>)

  Julian's fractal page
  http://members.aol.com/julianpa/index.htm (Julian Adamaitis
  <[email protected]>)

  Don Archer's fractal art
  http://www.ingress.com/~arch/ (Don Archer <[email protected]>)

  The 4D Julibrot Homepage
  http://www.shop.de/priv/hp/3133/fr_4d.htm (Benno Schmid
  <[email protected]>)

  The Fractal of the Day
  http://home.att.net/~Paul.N.Lee/FotD/FotD.html Each day Jim Muth
  (<[email protected]>) post a new fractal !

  The Beauty of Chaos
  http://i30www.ira.uka.de/~ukrueger/fractals/ A journey in the
  Mandelbrot set (Uwe Kr�ger <[email protected]>)

  The Brian E. Jones Computer Art Gallery
  http://ourworld.compuserve.com/homepages/Brian_E_Jones/ (Brian E.
  Jones <[email protected]>)

  Phractal Phantasies
  http://www.globalserve.net/~jval/intro.htm (Margaret
  <[email protected]> and Jack <[email protected]> Valero)

  Glimpses of a fugitive Universe
  http://www.artvark.com/artvark/ (Rollo Silver <[email protected]>)

  Earl's Computer Art Gallery
  http://computerart.org/

  Jacco's Homepage (Jaap Burger <[email protected]>)
  http://wwwserv.caiw.nl/~jaccobu/index.htm

  MOCA: the Museum Of Computer Art The fractal art of Sylvie Gallet, and
  several other artists (Bob Dodson, MOCA curator <[email protected]> ;
  Don Archer, MOCA director)
  http://www.dorsai.org/~moca/

  Les St Clair's Fractal Home Page (Les St Clair
  <[email protected]>)
  http://ourworld.compuserve.com/homepages/Les_StClair/

 Numerous links to fractal galleries and other fractal subjects can be found
 at

  Spanky fractal database
         http://spanky.triumf.ca/www/welcome1.html

  Fractal Images / Immagini frattali su Internet
         http://www.ba.infn.it/www/fractal.html

  Chaffey High School's Fractal Image Gallery Links
         http://www.chaffey.org/fractals/galleries.html

  Fantastic Fractals. Reference Desk
         http://library.advanced.org/12740/cgi-bin/linking.cgi?browser=m
         sie&language=enu

 The Infinite Fractal Loop

  The Infinite Fractal Loop was initiated by Douglas Cootey ; it is now
  managed by Damien M. Jones. It is a link between a number of personal
  fractal galleries. The home page of the subscribers display the logo
  of the Infinite Fractal Loop. By clicking on selected areas of this
  logo the server of the loop will call an other site of this loop and
  from this new page, you can go to an other gallery... There are nearly
  40 members in the loop.

  You can have more information and subscribe at
  http://www.emi.net/~dmj/ifl/

  _Q25b_: How do I view fractal pictures from
  alt.binaries.pictures.fractals?

  _A25b_: A detailed explanation is given in the "alt.binaries.pictures
  FAQ" (see "pictures-FAQ"). This is posted to the pictures newsgroups
  and is available by ftp:
  ftp://rtfm.mit.edu:/pub/usenet/news.answers/pictures-faq/.

  In brief, there is a series of things you have to do before viewing
  these posted images. It will depend a little on the system you are
  working with, but there is much in common. Some newsreaders have
  features to automatically extract and decode images ready to display
  ("e" in trn) but if you don't you can use the following manual method.

  Manual method

   1. Save/append all posted parts sequentially to one file.
   2. Edit this file and delete all text segments except what is between
      the BEGIN-CUT and END-CUT portions. This means that BEGIN-CUT and
      END-CUT lines will disappear as well. There will be a section to
      remove for each file segment as well as the final END-CUT line.
      What is left in the file after editing will be bizarre garbage
      starting with begin 660 imagename.GIF and then about 6000 lines
      all starting with the letter "M" followed by a final "end" line.
      This is called a uuencoded file.
   3. You must uudecode the uuencoded file. There should be an
      appropriate utility at your site; "uudecode filename " should work
      under Unix. Ask a system person or knowledgeable programming type.
      It will decode the file and produce another file called
      imagename.GIF. This is the image file.
   4. You must use another utility to view these GIF images. It must be
      capable of displaying color graphic images in GIF format. (If you
      get a JPG or JPEG format file, you may have to convert it to a GIF
      file with yet another utility.) In the XWindows environment, you
      may be able to use "xv", "xview", or "xloadimage" to view GIF
      files. If you aren't using X, then you'll either have to find a
      comparable utility for your system or transfer your file to some
      other system. You can use a file transfer utility such as Kermit
      to transfer the binary file to an IBM-PC.

  Automated method

  Most of the news readers for Windows or Macintosh, as well as web
  browsers such as Netscape or MSIE will automate the decoding for you.
  This may not be true of all web browsers.

Subject: Where can I obtain papers about fractals?

  _Q26_: Where can I obtain papers about fractals?

  _A26_: There are several Internet sites with fractal papers: There is
  an ftp archive site for preprints and programs on nonlinear dynamics
  and related subjects at: ftp://inls.ucsd.edu/pub.

  There are also articles on dynamics, including the IMS preprint
  series, available from ftp://math.sunysb.edu/preprints.

  The WWW site http://inls.ucsd.edu/y/Complex/ has some fractal papers.

  The site life.csu.edu.au has a collection of fractal programs, papers,
  information related to complex systems, and gopher and World Wide Web
  connections.

  The ftp path is:
         ftp://life.csu.edu.au/pub/complex/ (Look in fractals and chaos)

  via WWW:
         http://life.csu.edu.au/complex/

  R. Vojak has some papers and preprints available from his home page at
  Universit� Paris IX Dauphine.

  R. Vojak's home page
  http://www.ceremade.dauphine.fr/~vojak/

Subject: How can I join fractal mailing lists?

  _Q27_: How can I join fractal mailing lists?

  _A27_: There are now 4 mailing lists devoted to fractals.

  FRAC-L

  Fractal-Art

  Fractint

  Fractal Programmers

 The FRAC-L mailing list

  FRAC-L is a mailing list "Forum on Fractals, Chaos, and Complexity".
  The purpose of frac-l is to be a globally networked forum for
  discourse and collaboration on fractals, chaos, and complexity in
  multiple disciplines, professions, and arts.

  To subscribe to frac-l an email message to
  [email protected] containing the sole line of text:
  SUBSCRIBE FRAC-L [email address optional]

  To unsubscribe from frac-l, send LISTPROC (_NOT frac-l_) the message:
  UNSUBSCRIBE FRAC-L

  Messages may be posted to frac-l by sending email to:
  [email protected]

  Ermel Stepp founded this list; the current listowner is Larry Husch
  and you should contact him ([email protected]) if there are any
  difficulties.

  The Frac-L archives (http://archives.math.utk.edu/hypermail/frac-l/)
  go back to Fri 09 Jun 1995.

 The Fractal-Art Discussion List

  This mailing list is open to all individuals and organizations
  interested in all aspects of Fractal Art. This would include fractal
  and digital artists, fractal software developers, gallery owners,
  museum curators, art marketers and brokers, printers, art collectors,
  and simply anybody who just plain likes to look at fractal images.
  This should include just about everybody!

  Administrator: Jon Noring [email protected]

  To subscribe Fractal-Art send an email message to [email protected]
  containing the sole line of text:

    subscribe fractal-art

  Messages may be posted to the fractal-art mailing list by sending
  email to: [email protected]

  An innovative member of Fractal-Art has created the Unofficial Links
  from Fractal-Art Email Digest
  (http://www.ee.calpoly.edu/~jcline/fractalart-links.htm) which
  collects all the URLs posted to the Fractal-Art mailing list and makes
  them into a web page. Created by Jonathan Cline.

 The Fractint mailing list

  This mailing list is for the discussion of fractals, fractal art,
  fractal algorithms, fractal software, and fractal programming.
  Specific discussion related to the freeware MS-DOS program Fractint
  and it's ports to other platforms is welcome, but discussion need not
  be Fractint related. Technical discussion is welcome, but so are
  beginner's questions, so don't be shy. This is a good place to share
  Fractint tips, tricks, and techniques, or to wax poetic about other
  fractal software.

  To subscribe you can send a mail to [email protected] with the
  following command in the body of your email message:

  subscribe fractint

  Messages may be posted to the fractint mailing list by sending email
  to: [email protected]

  You can contact the fractint list administrator, Tim Wegner, by
  sending e-mail to: [email protected]

 The Fractal Programmers mailing list

  Subcription/unsubscription/info requests should always be sent to the
  -request address of the mailinglist. This would be:
  <[email protected]>. To subscribe to the
  mailinglist, simply send a message with the word "subscribe" in the
  _Subject:_ field to <[email protected]>.

As in:          To: [email protected]
               Subject: subscribe

  To unsubscribe from the mailinglist, simply send a message with the
  word "unsubscribe" in the _Subject:_ field to
  <[email protected]>.

Subject: Complexity

  _Q28_: What is complexity?

  _A28_: Emerging paradigms of thought encompassing fractals, chaos,
  nonlinear science, dynamic systems, self-organization, artificial
  life, neural networks, and similar systems comprise the science of
  complexity. Several helpful online resources on complexity are:

  Institute for Research on Complexity
         http://webpages.marshall.edu/~stepp/vri/irc/irc.html

  The site life.csu.edu.au has a collection of fractal programs, papers,
  information related to complex systems, and gopher and World Wide Web
  connections.

  LIFE via WWW
         http://life.csu.edu.au/complex/

  Center for Complex Systems Research (UIUC)
         http://www.ccsr.uiuc.edu/

  Complexity International Journal
         http://www.csu.edu.au/ci/ci.html

  Nonlinear Science Preprints
         http://xxx.lanl.gov/archive/nlin-sys

  Nonlinear Science Preprints via email:

  To subscribe to public bulletin board to receive announcements of the
  availability of preprints from Los Alamos National Laboratory, send
  email to [email protected] containing the sole line of text:
  subscribe your-real-name

  The Complexity and Management Mailing List. For more information see
  the web archive at http://HOME.EASE.LSOFT.COM/archives/complex-m.html
  or their lexicon of terms at http://lissack.com/lexicon/lexicon.html.

  To subscribe:
  http://home.ease.lsoft.com/scripts/wa.exe?SUBED1=complex-m or send a
  message to [email protected] with the message "subscribe complex-m" in
  the _body_.

  To send a message to the list, send them to [email protected] or to
  [email protected].

Subject: References

  _Q29a_: What are some general references on fractals, chaos, and
  complexity?

  _A29a_: Some references are:

  M. Barnsley, _Fractals Everywhere_, Academic Press Inc., 1988, 1993.
  ISBN 0-12-079062-9. This is an excellent text book on fractals. This
  is probably the best book for learning about the math underpinning
  fractals. It is also a good source for new fractal types.

  M. Barnsley, _The Desktop Fractal Design System_ Versions 1 and 2.
  1992, 1988. Academic Press. Available from Iterated Systems.

  M. Barnsley and P H Lyman, _Fractal Image Compression_. 1993. AK
  Peters Limited. Available from Iterated Systems.

  M. Barnsley and L. Anson, _The Fractal Transform_, Jones and Bartlett,
  April, 1993. ISBN 0-86720-218-1. This book is a sequel to _Fractals
  Everywhere_. Without assuming a great deal of technical knowledge, the
  authors explain the workings of the Fractal Transform(tm). The Fractal
  Transform is the compression tool for storing high-quality images in a
  minimal amount of space on a computer. Barnsley uses examples and
  algorithms to explain how to transform a stored pixel image into its
  fractal representation.

  R. Devaney and L. Keen, eds., _Chaos and Fractals: The Mathematics
  Behind the Computer Graphics_, American Mathematical Society,
  Providence, RI, 1989. This book contains detailed mathematical
  descriptions of chaos, the Mandelbrot set, etc.

  R. L. Devaney, _An Introduction to Chaotic Dynamical Systems_,
  Addison- Wesley, 1989. ISBN 0-201-13046-7. This book introduces many
  of the basic concepts of modern dynamical systems theory and leads the
  reader to the point of current research in several areas. It goes into
  great detail on the exact structure of the logistic equation and other
  1-D maps. The book is fairly mathematical using calculus and topology.

  R. L. Devaney, _Chaos, Fractals, and Dynamics_, Addison-Wesley, 1990.
  ISBN 0-201-23288-X. This is a very readable book. It introduces chaos
  fractals and dynamics using a combination of hands-on computer
  experimentation and precalculus math. Numerous full-color and black
  and white images convey the beauty of these mathematical ideas.

  R. Devaney, _A First Course in Chaotic Dynamical Systems, Theory and
  Experiment_, Addison Wesley, 1992. A nice undergraduate introduction
  to chaos and fractals.

  A. K. Dewdney, (1989, February). Mathematical Recreations. _Scientific
  American_, pp. 108-111.

  G. A. Edgar, _Measure Topology and Fractal Geometry_, Springer-Verlag
  Inc., 1990. ISBN 0-387-97272-2. This book provides the math necessary
  for the study of fractal geometry. It includes the background material
  on metric topology and measure theory and also covers topological and
  fractal dimension, including the Hausdorff dimension.

  K. Falconer, _Fractal Geometry: Mathematical Foundations and
  Applications_, Wiley, New York, 1990.

  J. Feder, _Fractals_, Plenum Press, New York, 1988. This book is
  recommended as an introduction. It introduces fractals from
  geometrical ideas, covers a wide variety of topics, and covers things
  such as time series and R/S analysis that aren't usually considered.

  Y. Fisher (ed), _Fractal Image Compression: Theory and Application_.
  Springer Verlag, 1995.

  L. Gardini (ed), _Chaotic Dynamics in Two-Dimensional Noninvertive
  Maps_. World Scientific 1996, ISBN: 9810216475

  J. Gleick, _Chaos: Making a New Science_, Penguin, New York, 1987.

  B. Hao, ed., _Chaos_, World Scientific, Singapore, 1984. This is an
  excellent collection of papers on chaos containing some of the most
  significant reports on chaos such as "Deterministic Nonperiodic Flow"
  by E.N. Lorenz.

  I. Hargittai and C. Pickover. _Spiral Symmetry_ 1992 World Scientific
  Publishing, River Edge, New Jersey 07661. ISBN 981-02-0615-1. Topics:
  Spirals in nature, art, and mathematics. Fractal spirals, plant
  spirals, artist's spirals, the spiral in myth and literature... Loads
  of images.

  H. J�rgens, H. O Peitgen, & D. Saupe. 1990 August, The Language of
  Fractals. _Scientific American_, pp. 60-67.

  H. J�rgens, H. O. Peitgen, H.O., & D. Saupe, 1992, _Chaos and
  Fractals: New Frontiers of Science_. New York: Springer-Verlag.

  S. Levy, _Artificial life : the quest for a new creation_, Pantheon
  Books, New York, 1992. This book takes off where Gleick left off. It
  looks at many of the same people and what they are doing post-Gleick.

  B. Mandelbrot, _The Fractal Geometry of Nature_, W. H. FreeMan, New
  York. ISBN 0-7167-1186-9. In this book Mandelbrot attempts to show
  that reality is fractal-like. He also has pictures of many different
  fractals.

  B. Mandelbrot, _Les objets fractals_, Flammarion, Paris. ISBN
  2-08-211188-1. The French Mandelbrot's book, where the word _fractal_
  has been used for the first time.

  J.L. McCauley, _Chaos, dynamics, and fractals : an algorithmic
  approach to deterministic chaos_, Cambridge University Press, 1993.

  E. R. MacCormac (ed), M. Stamenov (ed), _Fractals of Brain, Fractals
  of Mind : In Search of a Symmetry Bond (Advances in Consciousness
  Research, No 7)_, John Benjamins, ISBN: 1556191871, Subjects include:
  Neural networks (Neurobiology), Mathematical models, Fractals, and
  Consciousness

  G.V. Middleton, (ed), _1991: Nonlinear Dynamics, Chaos and Fractals
  (w/ application to geological systems)_ Geol. Assoc. Canada, Short
  Course Notes Vol. 9, 235 p. This volume contains a disk with some
  examples (also as pascal source code) ($25 CDN)

  T.F. Nonnenmacher, G.A Losa, E.R Weibel (ed.) _Fractals in Biology and
  Medicine_ ISBN 0817629890, Springer Verlag, 1994

  L. Nottale, _Fractal Space-Time and Microphysics, Towards a Theory of
  Scale Relativity_, World Scientific (1993).

  E. Ott, _Chaos in dynamical systems_, Cambridge University Press,
  1993.

  E. Ott, T. Sauer, J.A. Yorke (ed.) _Coping with chaos : analysis of
  chaotic data and the exploitation of chaotic systems_, New York, J.
  Wiley, 1994.

  D. Peak and M. Frame, _Chaos Under Control: The Art and Science of
  Complexity_, W.H. Freeman and Company, New York 1994, ISBN
  0-7167-2429-4 "The book is written at the perfect level to help a
  beginner gain a solid understanding of both basic and subtler appects
  of chaos and dynamical systems." - a review from the back cover

  H. O. Peitgen and P. H. Richter, _The Beauty of Fractals_,
  Springer-Verlag, New York, 1986. ISBN 0-387-15851-0. This book has
  lots of nice pictures. There is also an appendix giving the
  coordinates and constants for the color plates and many of the other
  pictures.

  H. Peitgen and D. Saupe, eds., _The Science of Fractal Images_,
  Springer-Verlag, 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.

  H. Peitgen, H. Juergens and D. Saupe, _Fractals for the Classroom_,
  Springer-Verlag, New York, 1992. These two volumes are aimed at
  advanced secondary school students (but are appropriate for others
  too), have lots of examples, explain the math well, and give BASIC
  programs.

  H. Peitgen, H. Juergens and D. Saupe, _Chaos and Fractals: New
  Frontiers of Science_, Springer-Verlag, New York, 1992.

  E. Peters, _Fractal Market Analysis - Applying Chaos Theory to
  Investment & Economics_, John Wiley & Sons, 1994, ISBN 0-471-58524-6.

  C. Pickover, _Computers, Pattern, Chaos, and Beauty: Graphics from an
  Unseen World_, St. Martin's Press, New York, 1990. This book contains
  a bunch of interesting explorations of different fractals.

  C. Pickover, _Keys to Infinity_, (1995) John Wiley: NY. ISBN
  0-471-11857-5.

  C. Pickover, (1995) _Chaos in Wonderland: Visual Adventures in a
  Fractal World._ St. Martin's Press: New York. ISBN 0-312-10743-9.
  (Devoted to the Lyapunov exponent.)

  C. Pickover, _Computers and the Imagination_ (Subtitled: Visual
  Adventures from Beyond the Edge) (1993) St. Martin's Press: New York.

  C. Pickover. _The Pattern Book: Fractals, Art, and Nature_ (1995)
  World Scientific. ISBN 981-02-1426-X Some of the patterns are
  ultramodern, while others are centuries old. Many of the patterns are
  drawn from the universe of mathematics.

  C. Pickover, _Visualizing Biological Information_ (1995) World
  Scientific: Singapore, New Jersey, London, Hong Kong.
  on the use of computer graphics, fractals, and musical techniques to
  find patterns in DNA and amino acid sequences.

  C. Pickover, _Fractal Horizons: The Future Use of Fractals._ (1996)
  St. Martin's Press, New York.
  Speculates on advances in the 21st Century. Six broad sections:
  Fractals in Education, Fractals in Art, Fractal Models and Metaphors,
  Fractals in Music and Sound, Fractals in Medicine, and Fractals and
  Mathematics. Topics include: challenges of using fractals in the
  classroom, new ways of generating art and music, the use of fractals
  in clothing fashions of the future, fractal holograms, fractals in
  medicine, fractals in boardrooms of the future, fractals in chess.

  J. Pritchard, _The Chaos Cookbook: A Practical Programming Guide_,
  Butterworth-Heinemann, Oxford, 1992. ISBN 0-7506-0304-6. It contains
  type in and go listings in BASIC and Pascal. It also eases you into
  some of the mathematics of fractals and chaos in the context of
  graphical experimentation. So it's more than just a
  type-and-see-pictures book, but rather a lab tutorial, especially good
  for those with a weak or rusty (or even nonexistent) calculus
  background.

  P. Prusinkiewicz and A. Lindenmayer, _The Algorithmic Beauty of
  Plants_, Springer-Verlag, NY, 1990. ISBN 0-387-97297-8. A very good
  book on L-systems, which can be used to model plants in a very
  realistic fashion. The book contains many pictures.

  Edward R. Scheinerman, _Invitation to Dynamical Systems_,
  Prentice-Hall, 1996, ISBN 0-13-185000-8, xvii + 373 pages

  M. Schroeder, _Fractals, Chaos, and Power Laws: Minutes from an
  Infinite Paradise_, W. H. Freeman, New York, 1991. This book contains
  a clearly written explanation of fractal geometry with lots of puns
  and word play.

  J. Sprott, _Strange Attractors: Creating Patterns in Chaos_, M&T Books
  (subsidary of Henry Holt and Co.), New York. ISBN 1-55851-298-5. This
  book describes a new method for generating beautiful fractal patterns
  by iterating simple maps and ordinary differential equations. It
  contains over 350 examples of such patterns, each producing a
  corresponding piece of fractal music. It also describes methods for
  visualizing objects in three and higher dimensions and explains how to
  produce 3-D stereoscopic images using the included red/blue glasses.
  The accompanying 3.5" IBM-PC disk contain source code in BASIC, C,
  C++, Visual BASIC for Windows, and QuickBASIC for Macintosh as well as
  a ready-to-run IBM-PC executable version of the program. Available for
  $39.95 + $3.00 shipping from M&T Books (1-800-628-9658).

  D. Stein (ed), _Proceedings of the Santa Fe Institute's Complex
  Systems Summer School_, Addison-Wesley, Redwood City, CA, 1988. See
  especially the first article by David Campbell: "Introduction to
  nonlinear phenomena".

  R. Stevens, _Fractal Programming in C_, M&T Publishing, 1989 ISBN
  1-55851-038-9. This is a good book for a beginner who wants to write a
  fractal program. Half the book is on fractal curves like the Hilbert
  curve and the von Koch snow flake. The other half covers the
  Mandelbrot, Julia, Newton, and IFS fractals.

  I. Stewart, _Does God Play Dice?: the Mathematics of Chaos_, B.
  Blackwell, New York, 1989.

  Y. Takahashi, _Algorithms, Fractals, and Dynamics_, Plenum Pub Corp,
  (May) 1996, ISBN: 0306451271 Subjects: Differentiable dynamical syste,
  Congresses, Fractals, Algorithms, Differentiable Dynamical Systems,
  Algorithms (Computer Programming)

  T. Wegner and B. Tyler, _Fractal Creations_, 2nd ed. The Waite Group,
  1993. ISBN 1-878739-34-4 This is the book describing the Fractint
  program.

  _Q29b_: What are some relevant journals?

  _A29b_: Some relevant journals are:

  "Chaos and Graphics" section in the quarterly journal _Computers and
  Graphics_. This contains recent work in fractals from the graphics
  perspective, and usually contains several exciting new ideas.

  "Mathematical Recreations" section by I. Stewart in _Scientific
  American_.

  "Fractal Trans-Light News" published by Roger Bagula
  (<[email protected]>). Roger Bagula 11759 Waterhill Road, Lakeside,
  CA 92040 USA. Fractal Trans-Light News is a newsletter of mathematics,
  computer programs, art and poetry. To subscribe, send USD $20 (USD $50
  for overseas delivery) to the address above.

  _Fractal Report_. Reeves Telecommunication Labs.
  West Towan House, Porthtowan, TRURO, Cornwall TR4 8AX, U.K.
  WWW: http://ourworld.compuserve.com/homepages/JohndeR/fractalr.htm
  Email: [email protected] (John de Rivaz)

  _FRAC'Cetera_. This is a gazetteer of the world of fractals and
  related areas, supplied on IBM PC format HD disk. FRACT'Cetera is the
  home of FRUG - the Fractint User Group. For more information, contact:
  Jon Horner, Editor,
  FRAC'Cetera Le Mont Ardaine, Rue des Ardains, St. Peters Guernsey GY7
  9EU Channel Islands, United Kingdom. Email: [email protected]

  _Fractals, An interdisciplinary Journal On The Complex Geometry of
  Nature
  _This is a new journal published by World Scientific. B.B Mandelbrot
  is the Honorary Editor and T. Vicsek, M.F. Shlesinger, M.M Matsushita
  are the Managing Editors). The aim of this first international journal
  on fractals is to bring together the most recent developments in the
  research of fractals so that a fruitful interaction of the various
  approaches and scientific views on the complex spatial and temporal
  behavior could take place.

    _________________________________________________________________

  _Q28c_: What are some other Internet references?

  _A28c_: Some other Internet references:

  Web references to nonlinear dynamics

  Dynamical Systems (G. Zito)
         http://alephwww.cern.ch/~zito/chep94sl/sd.html

  Scanning huge number of events (G. Zito)
         http://alephwww.cern.ch/~zito/chep94sl/chep94sl.html

  The Who Is Who Handbook of Nonlinear Dynamics
         http://www.nonlin.tu-muenchen.de/chaos/Dokumente/WiW/wiw.html

Multifractals

  _Q30_: What are multifractals?

  _A30_: It is not easy to give a succinct definition of multifractals.
  Following Feder (1988) one may distinguish a measure (of probability,
  or some physical quantity) from its geometric support - which might or
  might not have fractal geometry. Then if the measure has different
  fractal dimension on different parts of the support, the measure is a
  multifractal.

  Hastings and Sugihara (1993) distinguish multifractals from
  multiscaling fractals - which have different fractal dimensions at
  different scales (e.g. show a break in slope in a dividers plot, or
  some other power law). I believe different authors use different names
  for this phenomenon, which is often confused with true multifractal
  behaviour.

Aliasing

  _Q31a_: What is aliasing?

  _A31a_: In computer graphics circles, "aliasing" refers to the
  phenomenon of a high frequency in a continuous signal masquerading as
  a lower frequency in the sampled output of the continuous signal. This
  is a consequence of the discrete sampling used by the computer.

  Put another way, it is the appearance of "chuckiness" in an still
  image. Because of the finite resolution of a computer screen, a single
  pixel has an associate width, whereas in mathematics each point is
  infintesimely small, with _no width_. So a single pixel on the screen
  actually visually represents an infinite number of mathematical
  points, each of which may have a different correct visual
  representation.

  _Q31b_: What does aliasing have to do with fractals?

  _A31b_: Fractals, are very strange objects indeed. Because they have
  an infinite amount of arbitrarily small detail embedded inside them,
  they have an infinite number of frequencies in the images. When we use
  a program to compute an image of a fractal, each pixel in the image is
  actually a sample of the fractal. Because the fractal itself has
  arbitrarily high frequencies inside it, we can never sample high
  enough to reveal the "true" nature of the fractal. _Every_ fractal
  ever computed has aliasing in it. (A special kind of aliasing is
  called "Moire' patterns" and are often visible in fractals as well.)

  _Q31c_: How Do I "Anti-Alias" Fractals?

  _A31c_: We can't eliminate aliasing entirely from a fractal but we can
  use some tricks to reduce the aliasing present in the fractal. This is
  what is called "anti-aliasing." The technique is really quite simple.
  We decide what size we want our final image to be, and we take our
  samples at a higher resolution than our final size. So if we want a
  100x100 image, we use at least 3 times the number of pixels in our
  "supersampled" image - 300x300, or 400x400 for even better results.

  But wait, we want a 100x100 image, right? Right. So far, we haven't
  done anything special. The anti-aliasing part comes in when we take
  our supersampled image and use a filter to combine several adjacent
  pixels in our supersampled image into a single pixel in our final
  image. The choice of the filter is very important if you want the best
  results! Most image manipulation and paint programs have a resize with
  anti-aliasing option. You can try this and see if you like the
  results. Unfortunately, most programs don't tell you exactly what
  filter they are applying when they "anti-alias," so you have to
  subjectively compare different tools to see which one gives you the
  best results.

  The most obvious filter is a simple averaging of neighbouring pixels
  in the supersampled image. Being the most obvious choice, it is
  generally the one most widely implemented in programs. Unfortunately
  it gives poor results. However, many fractal programs are now
  beginning to incorporate anti-aliasing directly in the fractal
  generation process along with a high quality filter. Unless you are a
  programmer, your best bet is to take your supersampled image and try
  different programs and filters to see which one gives you the best
  results.

  An example of such filtering in a fractal program can be found on
  Dennis C. De Mars' web page on anti-aliasing in his FracPPC program:
  http://members.aol.com/dennisdema/anti-alias/anti-alias.html

   References

  The original submission from Rich Thomson is available from
  http://www.mta.ca/~mctaylor/fractals/aliasing.html

  To read more about Digital Signal Processing, a good but technical
  book is "Digital Signal Processing", by Alan V. Oppenheim and Ronald
  W. Schafer, ISBN 0-13-214635-5, Prentice-Hall, 1975.

  For more on anti-aliasing filters and their application to computer
  graphics, you can read "Reconstruction Filters in Computer Graphics",
  Don P. Mitchell, Arun N. Netravali, Computer Graphics, Volume 22,
  Number 4, August 1988. (SIGGRAPH 1988 Proceedings).

  If you're a programmer type and want to experiment with lots of
  different filters on images, or if you're looking for an efficient
  sample implementation of digital filtering, check out Paul Heckbert's
  zoom program at ftp://ftp.cs.utah.edu/pub/painter/zoom.tar.gz

Science Fair Projects

  _Q32_: Ideas for science fair projects?

  _A32_: You should check with your science teacher about any special
  rules and restrictions. Fractals are really an area of mathematics and
  mathematics may be a difficult topic for science fairs with an
  experimental bias.

   1. Modelling real-world phenomena with fractals, e.g. Lorenz's
      weathers models or fractal plants and landscapes
   2. Calculate the fractal (box-counting) dimension of a leaf, stone,
      river bed
   3. _How long is a coastline?_, see The Fractal Geometry of Nature
   4. Check books and web sites aimed at high school students.

Subject: Notices

  _Q33_: Are there any special notices?

  _A33_:

From: Lee Skinner <[email protected]>
Date: Sun, 26 Oct 1997 12:37:33 -0500
Subject: Explora Science Exhibit

 Explora Science Exhibit

  The newly combined Explora Science Center and Children's Museum of
  Albuquerque had its Grand Opening on Saturday October 25 1997. One of
  the best exhibits is one illustrating fractals and fractal art.
  Posters made by Doug Czor illustrate how fractals are computed.
  Fractal-art images were exhibited by Lee Skinner, Jon Noring, Rollo
  Silver and Bob Hill. The exhibit will probably be on display for about
  6 months. Channel 13 News had a brief story about the opening and
  broadcasted some of the fractal-art images. The museum's gift shop is
  selling Rollo's Fractal Universe calendars and 4 different mouse-pad
  designs of fractals by Lee Skinner. Two of the art pieces are
  18432x13824/65536 Cibachrome prints using images recalculated by Jon
  Noring.

  Lee Skinner

    _________________________________________________________________

From: Javier Barrallo
Date: Sun, 14 Sep 1997 18:06:14 +0200
Subject: Mathematics & Design - 98

 INVITATION AND CALL FOR PAPERS
 Second International Conference on Mathematics & Design 98

  Dear friend,

  This is to invite you to participate in the Second International
  Conference on Mathematics & Design 98 to be held at San Sebastian,
  Spain, 1-4 June 1998.

  The main objective of these Conferences is to bring together
  mathematicians, engineers, architects, designers and scientists
  interested on the interaction between Mathematics and Design, where
  the world design is understood in its more broad sense, including all
  types of design.

  Further information and a regularly updated program is available
  under:

  http://www.sc.ehu.es/md98

  We will be pleased if you kindly forward this message to colleagues of
  yours who might be interested in this announcement.

  Hoping to be able to have your valuable collaboration and assistance
  to the Conference,

  The Organising Committee
  E-mail: [email protected]

    _________________________________________________________________

  From: John de Rivaz <[email protected]>

Mr Roger Bagula, publisher of The Fractal Translight Newsletter, is seeking
new articles. Write to him for a sample copy - he is not on the Internet -
and he appreciates something for materials and postage.

Mr Roger Bagula,
11759 Waterhill Road
Lakeside
CA 90240-2905
USA

    _________________________________________________________________

  NOTICE from J. C. (Clint) Sprott <[email protected]>:

  The program, Chaos Data Analyzer, which I authored is a research and
  teaching tool containing 14 tests for detecting hidden determinism in
  a seemingly random time series of up to 16,382 points provided by the
  user in an ASCII data file. Sample data files are included for model
  chaotic systems. When chaos is found, calculations such as the
  probability distribution, power spectrum, Lyapunov exponent, and
  various measures of the fractal dimension enable you to determine
  properties of the system Underlying the behavior. The program can be
  used to make nonlinear predictions based on a novel technique
  involving singular value decomposition. The program is menu-driven,
  very easy to use, and even contains an automatic mode in which all the
  tests are performed in succession and the results are provided on a
  one-page summary.

  Chaos Data Analyzer requires an IBM PC or compatible with at least
  512K of memory. A math coprocessor is recommended (but not required)
  to speed some of the calculations. The program is available on 5.25 or
  3.5" disk and includes a 62-page User's Manual. Chaos Data Analyzer is
  peer-reviewed software published by Physics Academic Software, a
  cooperative Project of the American Institute of Physics, the American
  Physical Society, And the American Association of Physics Teachers.

  Chaos Data Analyzer and other related programs are available from The
  Academic Software Library, North Carolina State University, Box 8202,
  Raleigh, NC 27695-8202, Tel: (800) 955-TASL or (919) 515-7447 or Fax:
  (919) 515-2682. The price is $99.95. Add $3.50 for shipping in U.S. or
  $12.50 for foreign airmail. All TASL programs come with a 30-day,
  money-back guarantee.

    _________________________________________________________________

  From Clifford Pickover <[email protected]>

  You are cordially invited to submit interesting, well-written articles
  for the "Chaos and Graphics Section" of the international journal
  Computers and Graphics. I edit this on-going section which appears in
  each issue of the journal. Topics include the mathematical,
  scientific, and artistic application of fractals, chaos, and related.
  Your papers can be quite short if desired, for example, often a page
  or two is sufficient to convey an idea and a pretty graphic. Longer,
  technical papers are also welcome. The journal is peer-reviewed. I
  publish color, where appropriate. Write to me for guidelines. Novelty
  of images is often helpful.

 Goals

  The goal of my section is to provide visual demonstrations of
  complicated and beautiful structures which can arise in systems based
  on simple rules. The section presents papers on the seemingly
  paradoxical combinations of randomness and structure in systems of
  mathematical, physical, biological, electrical, chemical, and artistic
  interest. Topics include: iteration, cellular automata, bifurcation
  maps, fractals, dynamical systems, patterns of nature created from
  simple rules, and aesthetic graphics drawn from the universe of
  mathematics and art.

                          Subject: Acknowledgements

  _Q34_: Who has contributed to the sci.fractals FAQ?

  _A34_: Former editors, participants in the Usenet group sci.fractals
  and the listserv forum frac-l have provided most of the content of
  sci.fractals FAQ. For their help with this FAQ, "thank you" to:

  Alex Antunes, Donald Archer, Simon Arthur, Roger Bagula, John Beale,
  Matthew J. Bernhardt, Steve Bondeson, Erik Boman, Jacques Carette,
  John Corbit, Douglas Cootey, Charles F. Crocker, Michael Curl, Predrag
  Cvitanovic, Paul Derbyshire, John de Rivaz, Abhijit Deshmukh, Tony
  Dixon, J�rgen Dollinger, Robert Drake, Detlev Droege, Gerald Edgar,
  Glenn Elert, Gordon Erlebacher, Yuval Fisher, Duncan Foster, David
  Fowler, Murray Frank, Jean-loup Gailly, Noel Giffin, Frode Gill, Terry
  W. Gintz, Earl Glynn, Lamont Granquist, John Holder, Jon Horner, Luis
  Hernandez-Ur�a, Jay Hill, Arto Hoikkala, Carl Hommel, Robert Hood,
  Larry Husch, Oleg Ivanov, Henrik Wann Jensen, Simon Juden, J.
  Kai-Mikael, Leon Katz, Matt Kennel, Robert Klep, Dave Kliman, Pavel
  Kotulsky, Tal Kubo, Per Olav Lande, Paul N. Lee, Jon Leech, Otmar
  Lendl, Ronald Lewis, Jean-Pierre Louvet, Garr Lystad, Jose Oscar
  Marques, Douglas Martin, Brian Meloon, Tom Menten, Guy Metcalfe,
  Eugene Miya, Lori Moore, Robert Munafo, Miriam Nadel, Ron Nelson, Tom
  Parker, Dale Parson, Matt Perry, Cliff Pickover, Francois Pitt, Olaf
  G. Podlaha, Francesco Potort�, Kevin Ring, Michael Rolenz, Tom Scavo,
  Jeffrey Shallit, Ken Shirriff, Rollo Silver, Lee H Skinner, David
  Sharp, J. C. Sprott, Gerolf Starke, Bruce Stewart, Dwight Stolte,
  Michael C. Taylor, Rich Thomson, Tommy Vaske, Tim Wegner, Andrea
  Whitlock, David Winsemius, Erick Wong, Wayne Young, Giuseppe Zito, and
  others.

  A special thanks to Jean-Pierre Louvet, who has taken on the task of
  maintaining the sections for fractal software and where fractal
  pictures are archived.

  If I have missed you, I am very sorry, let me know
  ([email protected]) and I will add you to the list. Without the help
  of these contributors, the sci.fractals FAQ would be not be possible.

                             Subject: Copyright

  _Q35_: Copyright?

  _A35_: This document, "sci.fractals FAQ", is _Copyright � 1997-1998 by
  Michael C. Taylor and Jean-Pierre Louvet._ All Rights Reserved. This
  document is published in New Brunswick, Canada.

  Previous versions:
         Copyright 1995-1997 Michael Taylor
         Copyright 1995 Ermel Stepp (edition v2n1)
         Copyright 1993-1994 Ken Shirriff

  The Fractal FAQ was created by Ken Shirriff and edited by him through
  September 26, 1994. The second editor of the Fractal FAQ is Ermel
  Stepp (Feb 13, 1995). Since December 2, 1995 the acting editor has
  been Michael C. Taylor.

  Permission is granted for _non-profit_ reproduction and distribution
  of this issue of the sci.fractals FAQ as a complete document. You may
  product complete copies, including this notice, of the sci.fractals
  FAQ for classroom use. This _does not_ mean automatic permission for
  usage in CD-ROM collections or commercial educational products. If you
  would like to include sci.fractals FAQ, in whole or in part, in a
  commercial product contact Michael C. Taylor.

Warranty

  This document is provided as is without any express or implied
  warranty.

 Contacting the editors

  If you would like to contact the editors, you may do so in writing at
  the following addresses:

  Attn: Michael Taylor
  Computing Services
  Mount Allison University
  49A York Street
  Sackville, New Brunswick E4L 1C7
  CANADA

  email: [email protected]

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