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Subject: sci.fractals FAQ
This article was archived around: 9 Mar 1998 00:16:03 GMT
Last-modified: March 8, 1998
Copyright: Copyright 1997-1998 by Michael C. Taylor and Jean-Pierre Louvet
Maintainer: Michael C. Taylor and Jean-Pierre Louvet
sci.fractals FAQ (Frequently Asked Questions)
_Volume_ 5 _Number_ 3
_Date_ March 8, 1998
_Copyright_ 1997-1998 by Michael C. Taylor and Jean-Pierre Louvet. All
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
This FAQ is posted monthly to the Usenet newsgroups: sci.fractals
("Objects of non-integral dimension and other chaos"), sci.answers,
and news.answers. Like most FAQs it can be obtained freely with a WWW
browser (such as Mosaic or Netscape), or by anonymous FTP from
ftp://rtfm.mit.edu/pub/usenet/news.answers/sci/fractals-faq (USA). It
is also available from
.html (France) and http://www.mta.ca/~mctaylor/sci.fractals-faq/
Those without FTP or WWW access can obtain the FAQ via email, by
sending a message to firstname.lastname@example.org with the _message_:
Suggestions, Comments, Mistakes
Please send suggestions and corrections about the sci.fractals FAQ to
email@example.com. 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, _alt.binaries.pictures.fractals_. The other group
(alt.fractals.pictures) is considered obsolete and may not be carried
to as many people as _alt.binaries.pictures.fractals._ In fact there
is/was a CancelBot which will delete any binary posts it finds in
sci.fractals (and most other non-binaries newsgroup) so nearly no one
will see it.
_Post only about fractals_, this includes fractal mathematics,
software to generate fractals, where to get fractal posters and
T-shirts, and fractals as art. Do not bother posting about news events
not directly related to fractals, or about which OS / hardware /
language is better. These lead to flame wars.
_Do not post advertisements._ I should not have to mention this, but
people get excited. If you have some _fractal_ software (or posters)
available as shareware or shrink-wrap commercial, post your _brief_
announcement _once_. After than, you should limit yourself to notices
of upgrades and responding _via e-mail_ to people looking for fractal
If you are new to Usenet and/or being online, read the guidelines and
Frequently Asked Questions (FAQ) in news.announce.newusers. They are
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
Dynamical Systems and Technology Project: a introduction for
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 (>100 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 <firstname.lastname@example.org>
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.)
_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
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
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.
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),
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),
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
_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_
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),
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 email@example.com for more
information.) Hill and Fisher used distance estimation techniques to
rigorously bound the area and found the area is between 1.503 and
1.5701. Jay Hill's latest results using Root Solving and Component
Series Evaluation shows the area is at least 1.506302 and less than
1.5613027. See Fractal Horizons edited by Cliff Pickover and Hill's
home page for details about his work.
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
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: ftp://math.sunysb.edu/preprints/ims91-7.ps.Z
_Q6h_: Is the Mandelbrot set connected?
_A6h_: The Mandelbrot set is simply connected. This follows from a
theorem of Douady and Hubbard that there is a conformal isomorphism
from the complement of the Mandelbrot set to the complement of the
unit disk. (In other words, all equipotential curves are simple closed
curves.) It is conjectured that the Mandelbrot set is locally
connected, and thus pathwise connected, but this is currently
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
<firstname.lastname@example.org> about the Mandelbrot Set. It is available
via WWW at <http://home.earthlink.net/~mrob/muency.html>.
_Q6j_: What is the dimension of the Mandelbrot Set?
_A6j_: The Mandelbrot Set has a dimension of 2. The Mandelbrot Set
contains and is contained in a disk. A disk has a dimension of 2, thus
so does the Mandelbrot Set.
The Koch snowflake (Hausdorff dimension 1.2619...) does not satisfy
this condition because it is a thin boundary curve, thus containing no
disk. If you add the region inside the curve then it does have
dimension of 2.
The boundary of the Mandelbrot set and the Julia set of a generic c in
M have Hausdorff dimension 2 and have topological dimension 1. The
proof is based on the study of the bifurcation of parabolic periodic
points. (Since the boundary has empty interior, the topological
dimension is less than 2, and thus is 1.) See reference above
_Q6k_: What are the seahorse and the elephant valleys?
_A6k_: The Mandelbrot set being the most famous fractal, its various
regions are well known and many of them have popular names evoking
graphic details found by zooming into them. The seahorse valley is the
limit border of the main cardioid at the negative side of the x axis
(near to x=-0.75, y=0.0). You can see here convoluted and complex buds
looking more or less like seahorses. The elephant valley is near the
symetry plane on the positive side of the x axis (x=0.25, y=0.0).
Spirals protuding from the border evoke trunks of elephants. By
zooming in these regions many interesting structures can be seen.
A nice guide (by Paul Derbyshire) to explore the various regions of
the Mandelbrot set can be found at :
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.
z = 1 (or any value)
if (random number < .5) then
z = sqrt(z - c)
z = -sqrt(z - c)
_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
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
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
_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
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=3, r=3.45, r=3.54, r=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
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.
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
(http://www.iterated.com/), has a line of products including a Windows
viewer, compressor, magnifier program, and hardware assist board.
Fractal compression is covered in detail in the comp.compression FAQ
file (See "compression-FAQ").
One of the best online references for Fractal Compress is Yuval
Fisher's Fractal Image Encoding page
(http://inls.ucsd.edu/y/Fractals/) at the Institute for Nonlinear
Science, University for California, San Diego. It includes references
to papers, other WWW sites, software, and books about Fractal
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
Andreas Kassler wrote a Fractal Image Compression with WINDOWS package
for a Fractal Compression thesis. It is available at
Fractal Compression Bibliography
Fractal Video Compression
Many fractal image compression papers are available from
A review of the literature is in Guide.ps.gz.
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
_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 (http://www.emf.org/focus_lewisbarry.html).
Does anyone know of others? Mail me at email@example.com.
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: firstname.lastname@example.org
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_,
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
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
_Q17c_: How can Lyapunov exponents be calculated?
_A17c_: For the common periodic forcing pictures, the Lyapunov
lambda = limit as N -> infinity of 1/N times sum from n=1 to N of
log2(abs(dx sub n+1 over dx sub n))
In other words, at each point in the sequence, the derivative of the
iterated equation is evaluated. The Lyapunov exponent is the average
value of the log of the derivative. If the value is negative, the
iteration is stable. Note that summing the logs corresponds to
multiplying the derivatives; if the product of the derivatives has
magnitude < 1, points will get pulled closer together as they go
through the iteration.
MS-DOS and Unix programs for estimating Lyapunov exponents from short
time series are available by ftp: ftp://inls.ucsd.edu/pub/ncsu/
Computing Lyapunov exponents in general is more difficult. Some
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,
Another source is Media Magic; they sell many fractal posters,
calendars, videos, software, t-shirts, ties, and a huge variety of
books on fractals, chaos, graphics, etc. Media Magic is at PO Box 598
Nicasio, CA 94946, 415-662-2426.
A third source is Ultimate Image; they sell fractal t- shirts,
posters, gift cards, and stickers. Ultimate Image is at PO Box 7464,
Nashua, NH 03060-7464.
Yet another source is Dave Kliman (516) 625-2504 email@example.com, 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
(http://www.industrialstreet.com/chaos/metalink.htm) also has
postcards, CDs, and videos.
Free fractal posters are available if you send a self-addressed
stamped envelope to the address given on
http://www.xmission.com/~legalize/gift.html. For foreign requests
(outside USA) include two IRCs (international reply coupons) to cover
ReFractal Design (http://www.refractal.com/) sells jewelry based on
Lifesmith Classic Fractals (http://www.lifesmith.com/) 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
_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 ftp://math.sunysb.edu/preprints/ims90-6.ps.Z. Published in
1991 SIGGRAPH Course Notes #14: Fractal Modeling in 3D Computer
Graphics and Imaging.
Instead of quaternions, you can of course use hypercomplex number such
as in "FractInt", or other functions. For instance, you could use a
map with more than one parameter, which would generate a
Another way of generating 3-D fractals is to use 3-D iterated function
systems (IFS). These are analogous to 2-D IFS, except they generate
points in a 3-D space.
A third way of generating 3-D fractals is to take a 2-D fractal such
as the Mandelbrot set, and convert the pixel values to heights to
generate a 3-D "Mandelbrot mountain". This 3-D object can then be
rendered with normal computer graphics techniques.
POV-Ray 3.0, a freely available ray tracing package, has added 4-D
fractal support. It takes a 3-D slice of a 4-D Julia set based on an
arbitrary 3-D "plane" done at any angle. For more information see the
POV Ray web site at http://www.povray.org/ .
_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
FRAINT.EXE and FRASRC.EXE in LIB 4.
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:
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
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?
* Software to calculate fractal dimension
(all entries marked "ff###" are directories where the inividual
archives of the Fred Fish Disk set available at
ftp://ftp.funet.fi/pub/amiga/fish/ and other sites)
General Mandelbrot generators with many features: Mandelbrot (ff030),
Mandel (ff218), Mandelbrot (ff239), TurboMandel (ff302), MandelBltiz
(ff387), SMan (ff447), MandelMountains (ff383, in 3-D), MandelPAUG
(ff452, MandFXP movies), MandAnim (ff461, anims), ApfelKiste (ff566,
very fast), MandelSquare (ff588, anims)
Mandelbrot and Julia sets generators: MandelVroom (ff215), Fractals
(ff371, also Newton-R and other sets)
With different algorithmic approaches (shown): FastGro (ff188, DLA),
IceFrac (ff303, DLA), DEM (ff303, DEM), CPM (ff303, CPM in 3-D),
FractalLab (ff391, any equation)
Iterated Function System generators (make ferns, etc): FracGen (ff188,
uses "seeds"), FCS (ff465), IFSgen (ff554), IFSLab (ff696, "Collage
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
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.
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,
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
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 <firstname.lastname@example.org>
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 <email@example.com>
Interactive java applets by Philip Baker <firstname.lastname@example.org>
Chaos and order by Eric Leese
MB applet by Russ <RBinNJ@worldnet.att.net>
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
(and PowerPC-based Macintosh compatible computers)
Fractal Domains v. 1.2
* Fractal generator for PowerMacs only, by Dennis C. De Mars
* Generates the Mandelbrot set and associated Julia sets, allows the
user to edit the color map, 24-bit colour
* by the author of Mandella, 24-bit colour
For 68K Macs
* generation of many different types of fractals, allow editing of
the color map, and other display & calculation options. Some
features not available on PowerMacs.
* generation of many different types of fractals, allow editing of
the color map, and other display & calculation options. Some
features not available on PowerMacs.
* useful to those rare individuals who are still using a Mac Plus/SE
* _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)
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,
WL-Plot 2.59 : plots functions including bifurcations and recursive
forb01a.zip: Displays orbits of Mandelbrot Set mapping. C/E/VGA
fract3.zip: Mandelbrot/Julia set 2D/3D EGA/VGA Fractal Gen
fractfly.zip: Create Fractal flythroughs with FRACTINT
fdesi313.zip: Program to visually design IFS fractals
frain196.zip: FRACTINT v19.6 EGA/VGA/XGA fractal generator
frasr196.zip: C & ASM src for FRACTINT v19.6
frcal040.zip: CAL: more than 15 types of fractals including
Lyapunov, IFS, user-defined, logistic, and Quaternion Julia
Vlotkatc uses VESA 640x480x16 Million colour mode to generate
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 <email@example.com>
11 fractal formulas, "Autopilot", solid guessing, zoom up to
Ultra Fractal. A DOS program with graphic interface, 256 colors or
truecolor. Very fast, many formulas. Shareware (Frederik
Fractal worldmap generator. A simple program to generate fractal
pseudo geographic maps, by John Olsson <firstname.lastname@example.org>,
DOS adaptation by Martijn Faassen <email@example.com>
Quat - A 3D-Fractal-Generator (Quaternions).
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 <firstname.lastname@example.org>
IFS Explorer for Windows 95/NT, a companion to Iterate 32, allows
users to explore IFS by changing the IFS parameters. Requires
800x600 screen. Freeware by Jeff Colvin <email@example.com>
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 (<firstname.lastname@example.org>)
Mndlzoom W95 or Nt program which iterate the Mandelbrot set within the
coprocessor stack : very fast, 19-digits significance (Philip
A. Seeger <PASeeger@aol.com>)
Frang : a real-time zooming Mandelbrot set generator. Needs DirectX
(can be downloaded from the same URL or from Microsoft).
Shareware (Michael Baldwin <email@example.com>)
Fractal Orbits; A nice implementation of Bubble, Ring, Stalk methods
by Phil Pickard <firstname.lastname@example.org >. Very easy to use.
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 <email@example.com>.
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.
Free screen savers : By Philip Baker (<firstname.lastname@example.org>)
JuliaSaver : a W95 screen saver that does real-time fractals, by
Damien M. Jones (<email@example.com>)
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 (<firstname.lastname@example.org>)
Seractal Screen Saver: Windows 3 and Windows 95 time limited versions
the Orb series by 'O' from RuneTEK. For MS-Windows 95/NT only.
Mandtool: generates Mandelbrot Set
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
xmartin: Martin hopalong root window
xmandel: Mandelbrot/Julia sets
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 <email@example.com>)
Quat - A 3D-Fractal-Generator (Quaternions).
XFracky 2.5 by Henrik Wann Jensen <firstname.lastname@example.org> 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
_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 fractal.mta.ca", or whatever system you wish to
access. You are prompted for your name and you reply "anonymous". You
are prompted for your password and you reply with your email address.
You then use "ls" to list the files, "cd" to change directories, "get"
to get files, an "quit" to exit. For example, you could say "cd /pub",
"ls", "get README", and "quit"; this would get you the file "README".
See the man page ftp(1) or ask someone at your site for more
In this FAQ, anonymous ftp addresses are given in the URL form
ftp://name.of.machine/pub/path [126.96.36.199]. 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 188.8.131.52". The part after the name:
"/pub/path" is the file or directory to access once you are connected
to the remote machine.
_Q24b_: What if I can't use ftp to access files?
_A24b_: If you don't have access to ftp because you are on a UUCP,
Fidonet, BITNET network there is an e-mail gateway at
email@example.com that can retrieve the files for you. To get
instructions on how to use the ftp gateway send a message to
firstname.lastname@example.org 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?
_A25a_: Fractal images (GIFs, JPGs...) are posted to
alt.binaries.pictures.fractals (also known as abpf); this newsgroup
has replaced alt.fractals.pictures. However, several
alt.binaries.pictures groups being badly reputed,
alt.fractals.pictures seems to have some new activity.
The fractals posted in alt.binaries.pictures.fractals are recorded daily at
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 alt.binaries.pictures.fractals
images are archived at ftp://spanky.triumf.ca/fractals
This can also be accessed via WWW at http://spanky.triumf.ca/ or
From Paris, France one of the largest collections (>= 820MB) is Frank
Roussel's at http://www.cnam.fr/fractals.html
Fractal animations in MPEG and FLI format are in
In Bordeaux (France) there is a mirror of this site,
and a Canadian mirror at http://fractal.mta.ca/cnam/
Another collection of fractal images is archived at
"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:
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 Mathart.com
Favourite Fractals (Ryan Grant)
Mandelbrot and Julia Sets (David E. Joyce)
Gratuitous Fractals (email@example.com)
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 (<firstname.lastname@example.org>)
Fractal Art Parade (Douglas "D" Cootey <D@itsnet.com>)
The Fractory (John/Alex <email@example.com>)
FracPPC gallery (Dennis C. De Mars <firstname.lastname@example.org>)
http://galifrey.triode.net.au/ (Frances Griffin
J.P. Louvet's Fractal Album
(Jean-Pierre Louvet <email@example.com> French and
Carlson's Fractal Gallery
http://sprott.physics.wisc.edu/carlson.htm (Paul Carlson
Fractals by Paul Carlson
http://fractal.mta.ca/fractals/carlson/ (an other Paul
Daves's Graphics Page
http://www.unpronounceable.com/graphics/ (David J. Grossman
<graphics AT unpronounceable DOT com> replace the AT with '@'
and DOT with '.' I apologize that I must take this drastic step
to foil the spammers)
http://www.geocities.com/~gumbycat/index.html (Linda Allison
<firstname.lastname@example.org> 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 <email@example.com>)
Howard Herscovitch's Home Page
Fractalus Home. Fractals by Damien M. Jones
http://www.geocities.com/SoHo/Lofts/2605/ (Damien M. Jones
Fractopia Home page. Bill Rossi
http://members.aol.com/billatny/fractopi.htm (Bill Rossi
Doug's Gallery. Doug Owen
http://www.zenweb.com/rayn/doug/ (Doug Owen
TWG's Gallery. Terry W. Gintz
http://www.zenweb.com/rayn/twg/ (Terry W. Gintz
http://members.aol.com/MKing77043/index.htm (Mark King
Julian's fractal page
http://members.aol.com/julianpa/index.htm (Julian Adamaitis
Don Archer's fractal art
http://www.ingress.com/~arch/ (Don Archer <firstname.lastname@example.org>)
The 4D Julibrot Homepage
http://www.shop.de/priv/hp/3133/fr_4d.htm (Benno Schmid
The Fractal of the Day
http://home.att.net/~Paul.N.Lee/FotD/FotD.html Each day Jim Muth
(<email@example.com>) post a new fractal !
The Beauty of Chaos
http://i30www.ira.uka.de/~ukrueger/fractals/ A journey in the
Mandelbrot set (Uwe Krüger <firstname.lastname@example.org>)
The Brian E. Jones Computer Art Gallery
http://ourworld.compuserve.com/homepages/Brian_E_Jones/ (Brian E.
<email@example.com> and Jack <firstname.lastname@example.org> Valero)
Glimpses of a fugitive Universe
http://www.artvark.com/artvark/ (Rollo Silver <email@example.com>)
Earl's Computer Art Gallery
Jacco's Homepage (Jaap Burger <Jacco.Burger@kabelfoon.nl>)
MOCA: the Museum Of Computer Art The fractal art of Sylvie Gallet, and
several other artists (Bob Dodson, MOCA curator <firstname.lastname@example.org> ;
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 "alt.binaries.pictures
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.
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.
Most of the news readers for Windows or Macintosh, as well as web
browsers such as Netscape or MSIE will automate the decoding for you.
This may not be true of all web browsers.
Subject: Where can I obtain papers about fractals?
_Q26_: Where can I obtain papers about fractals?
_A26_: There are several Internet sites with fractal papers: There is
an ftp archive site for preprints and programs on nonlinear dynamics
and related subjects at: ftp://inls.ucsd.edu/pub.
There are also articles on dynamics, including the IMS preprint
series, available from ftp://math.sunysb.edu/preprints.
The WWW site http://inls.ucsd.edu/y/Complex/ has some fractal papers.
The site life.csu.edu.au has a collection of fractal programs, papers,
information related to complex systems, and gopher and World Wide Web
The ftp path is:
ftp://life.csu.edu.au/pub/complex/ (Look in fractals and chaos)
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.
The FRAC-L mailing list
FRAC-L is a mailing list "Forum on Fractals, Chaos, and Complexity".
The purpose of frac-l is to be a globally networked forum for
discourse and collaboration on fractals, chaos, and complexity in
multiple disciplines, professions, and arts.
To subscribe to frac-l an email message to
email@example.com 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 (firstname.lastname@example.org) if there are any
The Frac-L archives (http://archives.math.utk.edu/hypermail/frac-l/)
go back to Fri 09 Jun 1995.
The Fractal-Art Discussion List
This mailing list is open to all individuals and organizations
interested in all aspects of Fractal Art. This would include fractal
and digital artists, fractal software developers, gallery owners,
museum curators, art marketers and brokers, printers, art collectors,
and simply anybody who just plain likes to look at fractal images.
This should include just about everybody!
Administrator: Jon Noring email@example.com
To subscribe Fractal-Art send an email message to firstname.lastname@example.org
containing the sole line of text:
Messages may be posted to the fractal-art mailing list by sending
email to: email@example.com
An innovative member of Fractal-Art has created the Unofficial Links
from Fractal-Art Email Digest
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
To subscribe you can send a mail to firstname.lastname@example.org with the
following command in the body of your email message:
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: email@example.com
The Fractal Programmers mailing list
Subcription/unsubscription/info requests should always be sent to the
-request address of the mailinglist. This would be:
<firstname.lastname@example.org>. To subscribe to the
mailinglist, simply send a message with the word "subscribe" in the
_Subject:_ field to <email@example.com>.
As in: To: firstname.lastname@example.org
To unsubscribe from the mailinglist, simply send a message with the
word "unsubscribe" in the _Subject:_ field to
_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 life.csu.edu.au 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 email@example.com containing the sole line of text:
The Complexity and Management Mailing List. For more information see
the web archive at http://HOME.EASE.LSOFT.COM/archives/complex-m.html
or their lexicon of terms at http://lissack.com/lexicon/lexicon.html.
http://home.ease.lsoft.com/scripts/wa.exe?SUBED1=complex-m or send a
message to firstname.lastname@example.org with the message "subscribe complex-m" in
To send a message to the list, send them to COMPLEX@lissack.com or to
_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
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
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.
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
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
(<email@example.com>). 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@longevb.demon.co.uk (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: firstname.lastname@example.org
_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
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 ftp://ftp.cs.utah.edu/pub/painter/zoom.tar.gz
Science Fair Projects
_Q32_: Ideas for science fair projects?
_A32_: You should check with your science teacher about any special
rules and restrictions. Fractals are really an area of mathematics and
mathematics may be a difficult topic for science fairs with an
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,
3. _How long is a coastline?_, see The Fractal Geometry of Nature
4. Check books and web sites aimed at high school students.
_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
From: Javier Barrallo
Date: Sun, 14 Sep 1997 18:06:14 +0200
Subject: Mathematics & Design - 98
INVITATION AND CALL FOR PAPERS
Second International Conference on Mathematics & Design 98
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 <John@longevb.demon.co.uk>
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
NOTICE from J. C. (Clint) Sprott <SPROTT@juno.physics.wisc.edu>:
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
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,
From Clifford Pickover <email@example.com>
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.
_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-Urëa, 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 Potortì, 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
(firstname.lastname@example.org) and I will add you to the list. Without the help
of these contributors, the sci.fractals FAQ would be not be possible.
_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.
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
Mount Allison University
49A York Street
Sackville, New Brunswick E4L 1C7