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sci.fractals FAQ

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Archive-name: sci/fractals-faq
Posting-Frequency: monthly
Last-modified: March 8, 1998
Version: v5n3
Copyright: Copyright 1997-1998 by Michael C. Taylor and Jean-Pierre Louvet
Maintainer: Michael C. Taylor and Jean-Pierre Louvet

See reader questions & answers on this topic! - Help others by sharing your knowledge
               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.

   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 (USA). It
   is also available from
   .gz (Europe),
   .html (France) and
   Those without FTP or WWW access can obtain the FAQ via email, by
   sending a message to with the _message_:
   send usenet/news.answers/sci/fractals-faq
  Suggestions, Comments, Mistakes
   Please send suggestions and corrections about the sci.fractals FAQ to 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

                               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
   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
   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
   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

   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, The other group
   ( is considered obsolete and may not be carried
   to as many people as 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
   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

   A Primer on How to Work With the Usenet Community

   Frequently Asked Questions about Usenet

   Rules for posting to Usenet

   Emily Postnews Answers Your Questions on Netiquette

   Hints on writing style for Usenet

   What is Usenet?

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

   Fractal Microscope

   Dynamical Systems and Technology Project: a introduction for
          high-school students

   An Introduction to Fractals (Written by Paul Bourke)

   Fractals and Scale (by David G. Green)

   What are fractals? (by Neal Kettler)

   Fract-ED a fractal tutorial for beginners, targeted for high
          school/tech school students.

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

   Godric's fractal gallery. A brief introduction to Fractals clear and
          well illustrated explanations

   Lystad Fractal Info complex numbers and fractals

   Fractal eXtreme: fractal theory theoritical informations

   Frode Gill Fractal pages mathematical and programming data about
          classical fractals and quaternions

   Fractals: a history

   Basic informations about fractals

   Fantastic Fractals a very comprehensive site with tutorials for
          beginners and more advanced readers, workshops etc.

   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

   Mathsnet this site has several pages devoted to fractals and complex

   Fractals in Your Future by Ronald Lewis <>

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_ 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)

   sci.nonlinear FAQ (US)

   Exploring Chaos and Fractals

   Chaos and Complexity Homepage (M. Bourdour)

   The Institute for Nonlinear Science

   _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
   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)

   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. (~26KB)

   _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
   Note: While all chaotic attractors are strange, not all strange
   attractors are chaotic.
    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

   Neal Kettler's Interactive Mandelbrot

   Panagiotis J. Christias' Mandelbrot Explorer

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

   Mandelbrot viewer written in Java (by Simon Arthur)

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

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

   The Mandelbrot Set by Eric Carr

   Java program to view the Mandelbrot Set by Ken Shirriff

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

   _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
   _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
   _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.
   _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 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.
    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
        World Wide Web (in Postscript format)
    3. Jay Hill's Home page which includes his latest updates.
        Jay's Hill Home Page via the World Wide Web.
   _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.
    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.)
    1. M. Shishikura, The Hausdorff Dimension of the Boundary of the
       Mandelbrot Set and Julia Sets, The paper is available from
       anonymous ftp:
   _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
   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_.)
   Douady, A. and Hubbard, J., "Comptes Rendus" (Paris) 294, pp.123-126,
   _Q6i_: What is the Mandelbrot Encyclopedia?
   _A6i_: The Mandelbrot Encyclopedia is a web page by Robert Munafo
   <> about the Mandelbrot Set. It is available
   via WWW at <>.
   _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 :

Subject: Julia sets

   _Q7a_: What is the difference between the Mandelbrot set and a Julia
   _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
   _Q7b_: What is the connection between the Mandelbrot set and Julia
   _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)
 if (random number < .5) then
  z = sqrt(z - c)
  z = -sqrt(z - c)
 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
   (note: a, b are reals, _x_, _y_ are complex, _i_ is the square root of
   Powers of _i_:
          _i_^2 = -1
          (a+_i_*b)+(c+_i_*d) = (a+c)+_i_*(b+d)
          (a+_i_*b)*(c+_i_*d) = a*c-b*d + _i_*(a*d+b*c)
          (a+_i_*b) / (c+_i_*d) = (a+_i_*b)*(c-_i_*d) / (c^2+d^2)
          exp(a+_i_*b) = exp(a)*(cos(b)+_i_*sin(b))
          sin(_x_) = (exp(_i_*_x_) - exp(-_i_*_x_)) / (2*_i_)
          cos(_x_) = (exp(_i_*_x_) + exp(-_i_*_x_)) / 2
          |a+_i_*b| = sqrt(a^2+b^2)
          log(a+_i_*b) = log(|a+_i_*b|)+_i_*arctan(b / a) (Note: log is
   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
   Frode Gill's quaternions page

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
    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
    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

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
   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)

   Frank Rousell's hyperindex of IFS images

   _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
   (, 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"). .
   One of the best online references for Fractal Compress is Yuval
   Fisher's Fractal Image Encoding page
   ( at the Institute for Nonlinear
   Science, University for California, San Diego. It includes references
   to papers, other WWW sites, software, and books about Fractal
   Three major research projects include:
   Waterloo Montreal Verona Fractal Research Initiative


   Bath Scalable Video Software Mk 2

   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
    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: .
    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.
    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
   Andreas Kassler wrote a Fractal Image Compression with WINDOWS package
   for a Fractal Compression thesis. It is available at
   Other references:
   Fractal Compression Bibliography

   Fractal Video Compression

   Many fractal image compression papers are available from

   A review of the literature is in


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

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

   Graphics Archive at the Center for the Computation and Visualization
          of Geometric Structures contains various fractals created from

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 (
   Does anyone know of others? Mail me at
   Some references, many from an unpublished article by Stephanie Mason,
    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

   A fractal music C++ package is available at

   The Fractal Music Project (Claus-Dieter Schulz)

   Chua's Oscillator: Applications of Chaos to Sound and Music

   Fractal Music Lab

   Fractal Music - Phil Thompson

   fractal music in MIDI format by Jose Oscar Marques

   Don Archer's fractal art and music contains several pieces of fractal
          music in MIDI format.

   LMUSe, a DOS program that generates MIDI music and files from 3D

   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:
          To subscribe to the list please send mail to

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
   Fractal landscape information available online:
   EECS News: Fall 1994: Building Fractal Planets by Ken Musgrave

   Gforge and Landscapes (John Beale)

   Java fractal landscapes :
   Fractal landscapes (applet and sources) by Chris Thornborrow


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

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

   The Center for the Computation and Visualization of Geometric
          Structures also has some fractal clouds.

   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
   _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.
    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:
   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, 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
   Chaos MetaLink website
   ( also has
   postcards, CDs, and videos.
   Free fractal posters are available if you send a self-addressed
   stamped envelope to the address given on For foreign requests
   (outside USA) include two IRCs (international reply coupons) to cover
   the weight.
   ReFractal Design ( sells jewelry based on
   Lifesmith Classic Fractals ( claims to be
   the largest fractal art studio in USA. You can contact Jeff Berkowitz
   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
   _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
   Frank Rousell's hyperindex of 3D images

   4D Quaternions by Tom Holroyd

   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
    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 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 .

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)
   19.6 executable via FTP and WWW from SimTel & mirrors world-wide

   19.6 source via FTP and WWW from SimTel & mirrors world-wide

   19.6 executable via FTP from Canada

   19.6 source via FTP from Canada

   (The suffix _196_ will change as new versions are released.)
   Fractint is available on Compuserve: GO GRAPHDEV and look for
   MS-Window FractInt 18.21 via FTP and WWW from SimTel & mirrors

   MS-Window FractInt 18.21 via FTP from Canada

   MS-Windows FractInt 18.21 source via FTP and WWW from SimTel & mirrors

   MS-Windows FractInt 18.21 source via FTP from Canada

   Available on Compuserve in its GRAPHDEV forum. The files are PM*.ZIP.
   These files are also available on many sites, for example
   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

   3.04 source Atlantic Canada

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


   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
   There is an Amiga version also available:
   FracInt 3.2

          There is a collection of map, parameter, etc. files for
          FractInt, called FracXtra. It is available at
        FracXtra Home Page by Dan Goldwater
        FracXtra via FTP and WWW from SimTel & mirrors world-wide
        FracXtra via FTP
          _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
         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?
     * 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 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
   Unique fractal types: Cloud (ff216, cloud surfaces), Fractal (ff052,
   terrain), IMandelVroom (strange attractor contours?), Landscape
   (ff554, scenery), Scenery (ff155, scenery), Plasma (ff573, plasma
   Fractal generators: PolyFractals (ff015), FFEX (ff549)
   Fractint for Amiga

   Lyapunov fractals

   XaoS, by Jan Hubicka, fast portable real-time interactive fractal
          zoomer. 256 workbench displays only.

   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

   Fractal Lab

   The Mandelbrot Set
   The Mandelbrot set (Paton J. Lewis)

   Mark's Java Julia Set Generator

   Fractals by Sun Microsystems

   The Mandelbrot set
   Mandelbrot Java Applet
   Ken Shirriff Java language pages

   example of the plasma method of fractal terrain by Carl Burke,

   Mandelbrot generator in Javascript by Frode Gill.

   Fracula Java Applet. A java applet to glide into the Mandelbrot set
          (best with Pentium and MSIE 3.0). Vince Ruddy

   Chaos and Fractals. Many java applets by Stephen Oswin

   IFS Fractals using javascript (Richard L. Bowman

   A lot of Java applets

   ChaosLab. A nice fully java site with several interactive applets
          showing different types of Mandelbrot, Julia, and strange
          attractors. By Cameron Mckechnie <>

   Fractal landscapes (applet and sources) by Chris Thornborrow

   Forest Echo Farm Fractal Fern

   Fractal java generator by Patrick Charles

   3 interactive java applets by Robert L. Devaney <>

   Interactive java applets by Philip Baker <>
   Chaos and order by Eric Leese
   MB applet by Russ <>
  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.

   Javaquat by Garr Lystad. Can also be run as an applet from Lystad's

  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
  MandelBrowser 2.0
     * by the author of Mandella, 24-bit colour
                                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.
  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.
  SuperMandelZoom 1.0.6
     * useful to those rare individuals who are still using a Mac Plus/SE
       class machine
                           Miscellaneous programs
     * _FDC and FDC 3D_ - Fractal Dimension Calculators
     * _Lsystem, 3D-L-System, IFS, FracHill_
     * _Color Fractal Generator_ 2.12
     * _MandelNet_ (uses several Macs on an AppleTalk network to
       calculate the Mandebrot set!)
     * _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!
     * _Lyapunov_ 1.0.1
     * _Fract_ 1.0 - A fractal-drawing program that uses the IFS
       algorithm. Change parameters to get different self-similar
     * _XaoS_ 2.1 - fast portable real-time interactive fractal zoomer
   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

   Fractal WitchCraft: a very fast fractal design program

   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

 Displays orbits of Mandelbrot Set mapping. C/E/VGA
 Mandelbrot/Julia set 2D/3D EGA/VGA Fractal Gen
 Create Fractal flythroughs with FRACTINT
 Program to visually design IFS fractals
 FRACTINT v19.6 EGA/VGA/XGA fractal generator
 C & ASM src for FRACTINT v19.6
 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.


   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

   3DFract generates 3-D fractals including Sierpinski cheese and 3-D

   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

   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

   ZsManJul 1.0 (requires 386DX+) by Zsolt Zsoldos

   FractMovie 1.62 a real-time 2D/3D IFS fractal movie renderer (requires
          486DX+) with GIF save

   FracZoom Explorer and FracZoom Navigator by Niels Ulrik Reinwald

   RMandel 1.2 80-bit floating point Mandelbrot Set animation generator
          by Marvin R. Lipford

   M24, the new version of TruMand by Mike Freeman 486DX+ True-colour
          Mandelbrot Set generator

   FAE - Fractal Animation Engine shareware by Brian Towles

   XaoS 2.2 fast portable realtime interactive fractal zoomer/morpher for
          MS-DOS (and others) by Jan Hubicka <>
          11 fractal formulas, "Autopilot", solid guessing, zoom up to
          64051194700380384 times

   Ultra Fractal. A DOS program with graphic interface, 256 colors or
          truecolor. Very fast, many formulas. Shareware (Frederik
          Slijkerman <>)

   Fractal worldmap generator. A simple program to generate fractal
          pseudo geographic maps, by John Olsson <>,
          DOS adaptation by Martijn Faassen <>

   Quat - A 3D-Fractal-Generator (Quaternions).

  For MS-Windows:
   dy-syst: Explores Newton's method, Mandelbrot and Julia sets

   bmand 1.1 shareware by Christopher Bare Mandelbrot program

   Quaternion-generator generates Julia-set Quaternions by Frode Gill

   Quat - A 3D-Fractal-Generator (Quaternions).

   A Fractal Experience 32 for Windows 95/NT by David Wright

   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 <>

   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 <>

   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.

   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

   Other fractal programs by Michael Sargent.

   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.

   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

   JuliaSaver : a W95 screen saver that does real-time fractals, by
          Damien M. Jones (<>)

   Mndlzoom W95 or Nt program which iterate the Mandelbrot set within the
          coprocessor stack : very fast, 19-digits significance (Philip
          A. Seeger <>)

   Frang : a real-time zooming Mandelbrot set generator. Needs DirectX
          (can be downloaded from the same URL or from Microsoft).
          Shareware (Michael Baldwin <>)

   Fractal Orbits; A nice implementation of Bubble, Ring, Stalk methods
          by Phil Pickard < >. Very easy to use.
          W95, NT.

   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

          Registered users will receive the full version and a true color
          one. Shareware by Terry W. Gintz <>.

   Set surfer. A nice small program. Draws a variety of fractals of
          Mandelbrot or Julia types. Freeware by Jason Letbetter

   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.

  Screen savers
   Free screen savers : By Philip Baker (<>)

   JuliaSaver : a W95 screen saver that does real-time fractals, by
          Damien M. Jones (<>)

   IFS screen saver: a Windows 3 screen saver, by Bill Decker

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

   Seractal Screen Saver: Windows 3 and Windows 95 time limited versions
          (shareware) (<>

   the Orb series by 'O' from RuneTEK. For MS-Windows 95/NT only.

  For SunView:
   Mandtool: generates Mandelbrot Set

  For Unix/C:
   lsys: L-systems as PostScript (in C++)

   lyapunov: PGM Lyapunov exponent images

   SPD: fractal mountain, tree, recursive tetrahedron

   Fractal Studio: Mandelbrot set; handles distributed computing

   fanal: analysis of fractal dimension for Linux by Jürgen Dollinger

   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

  For X windows :
   xmntns xlmntn: fractal mountains

   xfroot: fractal root window
          X11 distribution
   xmartin: Martin hopalong root window
          X11 distribution
   xmandel: Mandelbrot/Julia sets
          X11 distribution
   lyap: Lyapunov exponent images

   spider: Uses Thurston's algorithm, Kobe algorithm, external angles

   xfractal_explorer: fractal drawing program

   Xmountains: A fractal landscape generator

   xfractint: the Unix version of Fractint : look at XFRACTxxx (xxx being
          the version number)

   xmfract v1.4: Needs Motif 1.2+, based on FractInt

   Fast Julia Set and Mandelbrot for X-Windows by Zsolt Zsoldos

   XaoS realtime fractal zoomer for X11 or SVGAlibs by Jan Hubicka

   AlmondBread-0.2. Fast algorithm ; simultaneous orbit iteration ;
          Fractint-compatible GIF and MAP files ; Tcl/Tk user interface
          (Michael R. Ganss <>)

   Quat - A 3D-Fractal-Generator (Quaternions).

   XFracky 2.5 by Henrik Wann Jensen <> based on Tcl/Tk

  Distributed X systems:
   MandelSpawn: Mandelbrot/Julia on a network

   gnumandel: Mandelbrot on a network

  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.
   _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.
          for an enhanced Grassberger-Procaccia algorithm for correlation
   A MS-DOS version of FP3 is available by request to

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", 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
   In this FAQ, anonymous ftp addresses are given in the URL form
   ftp://name.of.machine/pub/path []. 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". 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 that can retrieve the files for you. To get
   instructions on how to use the ftp gateway send a message to 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

Subject: Archived pictures

   _Q25a_: Where are fractal pictures archived?
  News groups
   _A25a_: Fractal images (GIFs, JPGs...) are posted to (also known as abpf); this newsgroup
   has replaced However, several groups being badly reputed, seems to have some new activity.
  The fractals posted in are recorded daily at
   The following lists are scanty and will evolve soon.
  Other archives and university sites (images, tutorials...)
   Many Mandelbrot set images are available via

   Pictures from 1990 and 1991 are available via anonymous ftp at

   Fractal images including some recent
          images are archived at
   This can also be accessed via WWW at or

   From Paris, France one of the largest collections (>= 820MB) is Frank
          Roussel's at
   Fractal animations in MPEG and FLI format are in

   In Bordeaux (France) there is a mirror of this site,

   and a Canadian mirror at
   Another collection of fractal images is archived at

   Fractal Microscope

   "Contours of the Mind"

   Spanky Fractal Datbase (Noel Giffin)

   Yahoo Index of Fractal Art

   Geometry Centre at University of Minnesota

   Computer Graphics Gallery

  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

          Some images are also available from:

   Cliff Pickover

   Fractal Gallery (J. C. Sprott) Personal images and a thousand of
          fractals collected in abpf

   Fractal from Ojai (Art Baker)

   Skal's 3D-fractal collection (Pascal Massimino)

   3d Fractals (Stewart Dickson) via

   Dirk's 3D-Fractal-Homepage


   Favourite Fractals (Ryan Grant)

   Eric Schol

   Mandelbrot and Julia Sets (David E. Joyce)

   Newton's method

   Gratuitous Fractals (


   Fractal Prairie Page (George Krumins)

   Fractal Gallery (Paul Derbyshire)

   David Finton's fractal homepage

   Algorithmic Image Gallery (Giuseppe Zito)

   Octonion Fractals built using hyper-hyper-complex numbers by Onar Em

   B' Plasma Cloud (animated gif)

   John Bailey's fractal images (<>)

   Fractal Art Parade (Douglas "D" Cootey <>)

   The Fractory (John/Alex <>)

   FracPPC gallery (Dennis C. De Mars <>)

     (Frances Griffin

   J.P. Louvet's Fractal Album
          (Jean-Pierre Louvet <> French and
          English versions)
   Carlson's Fractal Gallery
 (Paul Carlson
   Fractals by Paul Carlson
 (an other Paul
          Carlson's Gallery)
   Daves's Graphics Page
 (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
 (Linda Allison
          <> Delete the dash ("-") in gumbycat to
          send e-mail. It's only purpose is to act as a spam deterent!)
   Sylvie Gallet Gallery

   Sylvie Gallet's Fractal Gallery New pages

          .htm (Sylvie Gallet <>)
   Howard Herscovitch's Home Page

   Fractalus Home. Fractals by Damien M. Jones
 (Damien M. Jones
   Fractopia Home page. Bill Rossi
 (Bill Rossi
   Doug's Gallery. Doug Owen
 (Doug Owen
   TWG's Gallery. Terry W. Gintz
 (Terry W. Gintz
   Fractal Gallery (Mark King
   Julian's fractal page (Julian Adamaitis
   Don Archer's fractal art (Don Archer <>)
   The 4D Julibrot Homepage (Benno Schmid
   The Fractal of the Day Each day Jim Muth
   (<>) post a new fractal !
   The Beauty of Chaos A journey in the
   Mandelbrot set (Uwe Krüger <>)
   The Brian E. Jones Computer Art Gallery (Brian E.
   Jones <>)
   Phractal Phantasies (Margaret
   <> and Jack <> Valero)
   Glimpses of a fugitive Universe (Rollo Silver <>)
   Earl's Computer Art Gallery
   Jacco's Homepage (Jaap Burger <>)
   MOCA: the Museum Of Computer Art The fractal art of Sylvie Gallet, and
   several other artists (Bob Dodson, MOCA curator <> ;
   Don Archer, MOCA director)
   Les St Clair's Fractal Home Page (Les St Clair
  Numerous links to fractal galleries and other fractal subjects can be found
   Spanky fractal database

   Fractal Images / Immagini frattali su Internet

   Chaffey High School's Fractal Image Gallery Links

   Fantastic Fractals. Reference Desk

  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

   _Q25b_: How do I view fractal pictures from
   _A25b_: A detailed explanation is given in the "
   FAQ" (see "pictures-FAQ"). This is posted to the pictures newsgroups
   and is available by ftp:
   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:
   There are also articles on dynamics, including the IMS preprint
   series, available from
   The WWW site has some fractal papers.
   The site has a collection of fractal programs, papers,
   information related to complex systems, and gopher and World Wide Web
   The ftp path is:
 (Look in fractals and chaos)
   via WWW:

   R. Vojak has some papers and preprints available from his home page at
   Université Paris IX Dauphine.
   R. Vojak's home page

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.
   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 containing the sole line of text:
   SUBSCRIBE FRAC-L [email address optional]
   To unsubscribe from frac-l, send LISTPROC (_NOT frac-l_) the message:
   Messages may be posted to frac-l by sending email to:
   Ermel Stepp founded this list; the current listowner is Larry Husch
   and you should contact him ( if there are any
   The Frac-L archives (
   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
   To subscribe Fractal-Art send an email message to
   containing the sole line of text:
     subscribe fractal-art
   Messages may be posted to the fractal-art mailing list by sending
   email to:
   An innovative member of Fractal-Art has created the Unofficial Links
   from Fractal-Art Email Digest
   ( 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 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
   You can contact the fractint list administrator, Tim Wegner, by
   sending e-mail to:
  The Fractal Programmers mailing list
   Subcription/unsubscription/info requests should always be sent to the
   -request address of the mailinglist. This would be:
   <>. To subscribe to the
   mailinglist, simply send a message with the word "subscribe" in the
   _Subject:_ field to <>.

As in:          To:
                Subject: subscribe

   To unsubscribe from the mailinglist, simply send a message with the
   word "unsubscribe" in the _Subject:_ field to

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

   The site has a collection of fractal programs, papers,
   information related to complex systems, and gopher and World Wide Web
   LIFE via WWW

   Center for Complex Systems Research (UIUC)

   Complexity International Journal

   Nonlinear Science Preprints

   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 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
   To subscribe: or send a
   message to with the message "subscribe complex-m" in
   the _body_.
   To send a message to the list, send them to or to

Subject: References

   _Q29a_: What are some general references on fractals, chaos, and
   _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. Jürgens, H. O Peitgen, & D. Saupe. 1990 August, The Language of
   Fractals. _Scientific American_, pp. 60-67.
   H. Jürgens, 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
   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
   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,
   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
   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
   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
   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
   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

   _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
   "Fractal Trans-Light News" published by Roger Bagula
   (<>). 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.
   Email: (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:
   _Fractals, An interdisciplinary Journal On The Complex Geometry of
   _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)

   Scanning huge number of events (G. Zito)

   The Who Is Who Handbook of Nonlinear Dynamics


   _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
   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


   _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
   _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
   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:
   The original submission from Rich Thomson is available from
   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

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?
From: Lee Skinner <LeeHSkinner@CompuServe.COM>
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
   Lee Skinner
From: Javier Barrallo
Date: Sun, 14 Sep 1997 18:06:14 +0200
Subject: Mathematics & Design - 98

  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
   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
   From: John de Rivaz <>
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
CA 90240-2905

   NOTICE from J. C. (Clint) Sprott <>:
   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 <>
   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.
   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, Jürgen 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-Urėa, 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 Potortģ, 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
   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
   ( 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.

   This document is provided as is without any express or implied
  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

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