From: aa459@ccn.cs.dal.ca (Michael Charles Taylor)
Followup-To: poster
Newsgroups: sci.fractals,sci.answers,news.answers
Subject: sci.fractals FAQ
Approved: news-answers-request@MIT.EDU
Summary: Frequently Asked Questions about Fractals
Keywords: fractals Mandelbrot Julia chaos IFS
Archive-name: sci/fractals-faq
Posting-Frequency: monthly
Last-modified: July 8 1996
Version: v3n7
URL: http://www.ccn.cs.dal.ca/~aa459/sci/fractals-faq-html/
sci.fractals FAQ
(Frequently Asked Questions)
_________________________________________________________________
Volume 3 Number 7
Date July 8, 1996
_________________________________________________________________
Copyright 1995-1996 by Michael C. Taylor. All Rights Reserved.
_________________________________________________________________
Introduction
This FAQ is posted monthly to sci.fractals, a Usenet newsgroup about
fractals; mathematics and software. This document is aimed at being a
reference about fractals, including answers to commonly asked
questions, archive listings of fractal software, images, and papers
that can be accessed via the Internet using FTP, gopher, or
World-Wide-Web (WWW), and a bibliography for further readings.
The FAQ does not give a textbook approach to learning about fractals,
but a summary of information from which you can learn more about and
explore fractals.
This FAQ is posted monthly to the Usenet newsgroups: sci.fractals
("Objects of non-integral dimension and other chaos") , sci.answers,
and news.answers. Like most FAQs it can be obtained freely with a WWW
browser (such as Mosaic or Netscape), or by anonymous FTP from
ftp://rtfm.mit.edu/pub/usenet/news.answers/sci/fractals-faq (USA)
(which is also
ftp://18.181.0.24/pub/usenet/news.answers/sci/fractals-faq if you have
Domain Name lookup problems). It is also available from
ftp://ftp.Germany.EU.net/pub/newsarchive/news.answers (Europe),
http://spanky.triumf.ca/pub/fractals/docs/SCI_FRACTALS.FAQ (Canada),
and http://www.chebucto.ns.ca/~aa459/sci/fractals-faq (Canada).
The hypertext version is available from
http://www.chebucto.ns.ca/~aa459/sci/fractals-faq-html/.
Those without FTP or WWW access can obtain the FAQ via email, by
sending a message to mail-server@rtfm.mit.edu with the message:
send usenet/news.answers/sci/fractals-faq
_________________________________________________________________
Suggestions, Comments, Mistakes
Please send suggestions and corrections about the sci.fractals FAQ to
aa459@chebucto.ns.ca. Without your contributions, the FAQ for
sci.fractals will not grow in its wealth. "For the readers, by the
readers." Rather than calling me a fool behind my back, if you find a
mistake, whether spelling or factual, please send me a note. That way
readers of future versions of the FAQ will not be misled. Also if you
have problems with the appearance of the hypertext version. There
should not be any Netscape only markup tags contained in the hypertext
verion, but I have not followed strict HTML 2.0 specifications. If the
appearance is "incorrect" let me know what problems you experience.
Why the different name?
The old FAQ about fractals has not not been updated for over a year
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. Hence it is now the
sci.fractals FAQ.
If you are viewing this file with a newsreader such as "rn" or "trn",
you can search for a particular question by using "g^Qn" (that's
lower-case g, up-arrow, Q, and n, the number of the question you
wish). Or you may browse forward using to search for a
Subject: line.
The questions which are answered are:
Q0: I am new to the 'Net what should I know about being online? NEW
Q1: I want to learn about fractals. What should I read first?
Q2: What is a fractal? What are some examples of fractals?
Q3: What is chaos?
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? NEW
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?
Q7a: What is the difference between the Mandelbrot set and a Julia
set?
Q7b: What is the connection between the Mandelbrot set and Julia sets?
Q7c: How is a Julia set actually computed?
Q7d: What are some Julia set facts?
Q8a: How does complex arithmetic work?
Q8b: How does quaternion arithmetic work?
Q9: What is the logistic equation?
Q10: What is Feigenbaum's constant? NEW
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 is some information on fractal music?
Q15: How are fractal mountains generated?
Q16: What are plasma clouds?
Q17a: Where are the popular periodically-forced Lyapunov fractals
described?
Q17b: What are Lyapunov exponents?
Q17c: How can Lyapunov exponents be calculated?
Q18: Where can I get fractal T-shirts and posters?
Q19: How can I take photos of fractals?
Q20: How can 3-D fractals be generated?
Q21a: What is Fractint? NEW
Q21b: How does Fractint achieve its speed?
Q22: Where can I obtain software packages to generate fractals? NEW
Q23a: How does anonymous ftp work?
Q23b: What if I can't use ftp to access files?
Q24a: Where are fractal pictures archived? NEW
Q24b: How do I view fractal pictures from
alt.binaries.pictures.fractals?
Q25: Where can I obtain fractal papers?
Q26: How can I join the FRAC-L fractal discussion?
Q27: What is complexity?
Q28a: What are some general references on fractals and chaos? NEW
Q28b: What are some relevant journals?
Q28c: What are some other Internet references?
Q29: What is a multifractal?
Q30: Are there any special notices? NEW
Q31: Who has contributed to the Fractal FAQ? NEW
Q32: Copyright?
_________________________________________________________________
Subject: USENET and Netiquette
Q0: I am new to the 'Net what should I know about being online? NEW
A0: Read the guidelines and Frequently Asked Questions (FAQ) in
news.announce.newusers. They are available from:
Welcome to news.newusers.questions
ftp://rtfm.mit.edu/pub/usenet/news.answers/news-newusers-intro
ftp://garbo.uwasa.fi/pc/doc-net/usenews.zip
A Primer on How to Work With the Usenet Community
ftp://rtfm.mit.edu/pub/usenet/news.answers/usenet/primer/part1
Frequently Asked Questions about Usenet
ftp://rtfm.mit.edu/pub/usenet/news.answers/usenet/faq/part1
Rules for posting to Usenet
ftp://rtfm.mit.edu/pub/usenet/news.answers/usenet/posting-rules/part1
Emily Postnews Answers Your Questions on Netiquette
ftp://rtfm.mit.edu/pub/usenet/news.answers/usenet/emily-postnews/part1
Hints on writing style for Usenet
ftp://rtfm.mit.edu/pub/usenet/news.answers/usenet/writing-style/part1
What is Usenet?
ftp://rtfm.mit.edu/pub/usenet/news.answers/usenet/what-is/part1
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 is considered
obsolete and may not be carried to as many people as a.b.p.f. In fact
there is CancelBot which will delete any 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
I will. If you have some fractal software available as shareware or
shrink-wrap do not post your brief announcement more than once. After
than, you should limit yourself to notices of upgrades and responding
via e-mail to people looking for fractal software.
______________________________________________________________________
Subject: Learning about fractals
Q1: I want to learn about fractals. What should I read/view first?
A1: Chaos is a good book to get a general overview and history that
does not require an extensive math background. Fractals Everywhere is
a textbook on fractals that describes what fractals are and how to
generate them, but it requires knowing intermediate analysis. Chaos,
Fractals, and Dynamics is also a good start. There is a longer book
list at the end of this file (see "What are some general
references?").
Also, there are networked resources available, such as:
Exploring Fractals and Chaos
http://www.lib.rmit.edu.au/fractals/exploring.html
Fractal Microscope
http://www.ncsa.uiuc.edu/Edu/Fractal/Fractal_Home.html
Dynamical Systems and Technology Project: a introduction for
high-school students
http://math.bu.edu/DYSYS/dysys.html
An Introduction to Fractals (Written by Paul Bourke)
http://www.auckland.ac.nz/arch/pdbourke/fractals/fracintro.html
Fractals and Scale (by David G. Green)
http://life.csu.edu.au/complex/tutorials/tutorial3.html
What are fractals? (by Neal Kettler)
http://www.vis.colostate.edu/~user1209/fractals/fracinfo.html
Fract-ED a fractal tutorial for beginners, targeted for high
school/tech school students.
http://www.ealnet.com/ealsoft/fracted.htm
Mandelbrot Questions & Answers (without any scary details) by Paul
Derbyshire
http://chat.carleton.ca/~pderbysh/mandlfaq.html
_________________________________________________________________
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, and coastlines, that do not
correspond to simple geometric shapes.
Benoit B. Mandelbrot gives a mathematical definition of a fractal as a
set of which the Hausdorff Besicovich dimension strictly exceeds the
topological dimension. However, he is not satisfied with this
definition as it excludes sets one would consider fractals.
According to Mandelbrot, who invented the word: "I coined fractal from
the Latin adjective fractus. The corresponding Latin verb frangere
means "to break:" to create irregular fragments. It is therefore
sensible - and how appropriate for our needs! - that, in addition to
"fragmented" (as in fraction or refraction), fractus should also mean
"irregular," both meanings being preserved in fragment." (The Fractal
Geometry of Nature, page 4.)
_________________________________________________________________
Subject: Chaos
Q3: What is chaos?
A3: 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.
The following resources may be helpful to understand chaos:
sci.nonlinear FAQ (UK)
http://www.fen.bris.ac.uk/engmaths/research/nonlinear/faq.html
sci.nonlinear FAQ (US)
http://amath.colorado.edu/appm/faculty/jdm/faq.html
Exploring Chaos and Fractals
http://www.lib.rmit.edu.au/fractals/exploring.html
Chaos and Complexity Homepage (M. Bourdour)
http://www.cc.duth.gr/~mboudour/nonlin.html
The Institute for Nonlinear Science
http://inls.ucsd.edu/
_________________________________________________________________
Subject: Fractal dimension
Q4a : What is fractal dimension? How is it calculated?
A4a: A common type of fractal dimension is the Hausdorff-Besicovich
Dimension, but there are several different ways of computing fractal
dimension.
Roughly, fractal dimension can be calculated by taking the limit of
the quotient of the log change in object size and the log change in
measurement scale, as the measurement scale approaches zero. The
differences come in what is exactly meant by "object size" and what is
meant by "measurement scale" and how to get an average number out of
many different parts of a geometrical object. Fractal dimensions
quantify the static geometry of an object.
For example, consider a straight line. Now blow up the line by a
factor of two. The line is now twice as long as before. Log 2 / Log 2
= 1, corresponding to dimension 1. Consider a square. Now blow up the
square by a factor of two. The square is now 4 times as large as
before (i.e. 4 original squares can be placed on the original square).
Log 4 / log 2 = 2, corresponding to dimension 2 for the square.
Consider a snowflake curve formed by repeatedly replacing ___ with
_/\_, where each of the 4 new lines is 1/3 the length of the old line.
Blowing up the snowflake curve by a factor of 3 results in a snowflake
curve 4 times as large (one of the old snowflake curves can be placed
on each of the 4 segments _/\_).
Log 4 / log 3 = 1.261... Since the dimension 1.261 is larger than the
dimension 1 of the lines making up the curve, the snowflake curve is a
fractal.
For more information on fractal dimension and scale, via the WWW
Fractals and Scale (by David G. Green)
http://life.csu.edu.au/complex/tutorials/tutorial3.html
Fractal dimension references:
1. J. P. Eckmann and D. Ruelle, Reviews of Modern Physics 57, 3
(1985), pp. 617-656.
2. K. J. Falconer, The Geometry of Fractal Sets, Cambridge Univ.
Press, 1985.
3. T. S. Parker and L. O. Chua, Practical Numerical Algorithms for
Chaotic Systems, Springer Verlag, 1989.
4. H. Peitgen and D. Saupe, eds., The Science of Fractal Images,
Springer-Verlag Inc., New York, 1988. ISBN 0-387-96608-0.
This book contains many color and black and white photographs,
high level math, and several pseudocoded algorithms.
5. G. Procaccia, Physica D 9 (1983), pp. 189-208.
6. J. Theiler, Physical Review A 41 (1990), pp. 3038-3051.
References on how to estimate fractal dimension:
1. S. Jaggi, D. A. Quattrochi and N. S. Lam, Implementation and
operation of three fractal measurement algorithms for analysis of
remote-sensing data., Computers & Geosciences 19, 6 (July 1993),
pp. 745-767.
2. E. Peters, Chaos and Order in the Capital Markets , New York,
1991. ISBN 0-471-53372-6
Discusses methods of computing fractal dimension. Includes several
short programs for nonlinear analysis.
3. J. Theiler, Estimating Fractal Dimension, Journal of the Optical
Society of America A-Optics and Image Science 7, 6 (June 1990),
pp. 1055-1073.
There are some programs available to compute fractal dimension. They
are listed in a section below (see Q22 "Fractal software").
Reference on the Hausdorff-Besicovitch dimension
A clear and concise (2 page) write-up of the definition of the
Hausdorff-Besicovitch dimension in MS-Word 6.0 format is available in
zip format.
hausdorff.zip (~26KB)
http://www.newciv.org/jhs/hausdorff.zip
Q4b : What is topological dimension?
A4b: Topological dimension is the "normal" idea of dimension; a point
has topological dimension 0, a line has topological dimension 1, a
surface has topological dimension 2, etc.
For a rigorous definition:
A set has topological dimension 0 if every point has arbitrarily small
neighborhoods whose boundaries do not intersect the set.
A set S has topological dimension k if each point in S has arbitrarily
small neighborhoods whose boundaries meet S in a set of dimension k-1,
and k is the least nonnegative integer for which this holds.
_________________________________________________________________
Subject: Strange attractors
Q5: What is a strange attractor?
A5: A strange attractor is the limit set of a chaotic trajectory. A
strange attractor is an attractor that is topologically distinct from
a periodic orbit or a limit cycle. A strange attractor can be
considered a fractal attractor. An example of a strange attractor is
the Henon attractor.
Consider a volume in phase space defined by all the initial conditions
a system may have. For a dissipative system, this volume will shrink
as the system evolves in time (Liouville's Theorem). If the system is
sensitive to initial conditions, the trajectories of the points
defining initial conditions will move apart in some directions, closer
in others, but there will be a net shrinkage in volume. Ultimately,
all points will lie along a fine line of zero volume. This is the
strange attractor. All initial points in phase space which ultimately
land on the attractor form a Basin of Attraction. A strange attractor
results if a system is sensitive to initial conditions and is not
conservative.
Note: While all chaotic attractors are strange, not all strange
attractors are chaotic.
Reference:
1. Grebogi, et al., Strange Attractors that are not Chaotic, Physica
D 13 (1984), pp. 261-268.
_________________________________________________________________
Subject: The Mandelbrot set
Q6a : What is the Mandelbrot set?
A6a: The Mandelbrot set is the set of all complex c such that
iterating z -> z² + c does not go to infinity (starting with z = 0).
Other images and resources are:
Frank Rousell's hyperindex of clickable/retrievable Mandelbrot images
http://www.cnam.fr/fractals/mandel.html
Neal Kettler's Interactive Mandelbrot
http://www.vis.colostate.edu/~user1209/fractals/explorer/
Panagiotis J. Christias' Mandelbrot Explorer
http://www.softlab.ntua.gr/mandel/mandel.html
2D & 3D Mandelbrot fractal explorer (set up by Robert Keller)
http://reality.sgi.com/employees/rck/hydra/
Mandelbrot viewer written in Java (by Simon Arthur)
http://www.mindspring.com/~chroma/mandelbrot.html
Mandelbrot Questions & Answers (without any scary details) by Paul
Derbyshire
http://chat.carleton.ca/~pderbysh/mandlfaq.html
Quick Guide to the Mandelbrot Set (includes a tourist map) by Paul
Derbyshire
http://chat.carleton.ca/~pderbysh/manguide.html
Beginner's guide to the Mandelbrot Set by Eric Carr
http://www.cs.odu.edu/~carr/mandelbr.html
Java program to view the Mandelbrot Set by Ken Shirriff
ftp://ftp.cs.berkeley.edu/ucb/sprite/www/java/mandel.html
Q6b : How is the Mandelbrot set actually computed?
A6b: The basic algorithm is: For each pixel c, start with z = 0.
Repeat z = z² + c up to N times, exiting if the magnitude of z gets
large. If you finish the loop, the point is probably inside the
Mandelbrot set. If you exit, the point is outside and can be colored
according to how many iterations were completed. You can exit if
|z| > 2, since if z gets this big it will go to infinity. The maximum
number of iterations, N, can be selected as desired, for instance 100.
Larger N will give sharper detail but take longer.
Frode Gill has some information about generating the Mandelbrot Set at
http://www.krs.hia.no/~fgill/mandel.html.
Q6c : Why do you start with z = 0?
A6c: Zero is the critical point of z = z² + c, that is, a point where
d/dz (z² + c) = 0. If you replace z² + c with a different function,
the starting value will have to be modified. E.g. for z -> z² + z,
the critical point is given by 2z + 1 = 0, so start with z = -½. In
some cases, there may be multiple critical values, so they all should
be tested.
Critical points are important because by a result of Fatou: every
attracting cycle for a polynomial or rational function attracts at
least one critical point. Thus, testing the critical point shows if
there is any stable attractive cycle. See also:
1. M. Frame and J. Robertson, A Generalized Mandelbrot Set and the
Role of Critical Points, Computers and Graphics 16, 1 (1992), pp.
35-40.
Note that you can precompute the first Mandelbrot iteration by
starting with z = c instead of z = 0, since 0² + c = c.
Q6d: What are the bounds of the Mandelbrot set? When does it diverge?
A6d: The Mandelbrot set lies within |c| <= 2. If |z| exceeds 2, the z
sequence diverges.
Proof: if |z| > 2, then |z² + c| >= |z ²| - |c| > 2|z| - |c|. If
|z| >= |c|, then 2|z| - |c| > |z|. So, if |z| > 2 and |z| >= c, then
|z² + c| > |z|, so the sequence is increasing. (It takes a bit more
work to prove it is unbounded and diverges.) Also, note that |z| = c,
so if |c| > 2, the sequence diverges.
Q6e : How can I speed up Mandelbrot set generation?
A6e: See the information on speed below (see "Fractint"). Also see:
1. R. Rojas, A Tutorial on Efficient Computer Graphic Representations
of the Mandelbrot Set, Computers and Graphics 15, 1 (1991), pp.
91-100.
Q6f: What is the area of the Mandelbrot set?
A6f: Ewing and Schober computed an area estimate using 240,000 terms
of the Laurent series. The result is 1.7274... However, the Laurent
series converges very slowly, so this is a poor estimate. A project to
measure the area via counting pixels on a very dense grid shows an
area around 1.5066. (Contact rpm%mrob.uucp@spdcc.com for more
information.) Hill and Fisher used distance estimation techniques to
rigorously bound the area and found the area is between 1.503 and
1.5701.
References:
1. J. H. Ewing and G. Schober, The Area of the Mandelbrot Set, Numer.
Math. 61 (1992), pp. 59-72.
2. Y. Fisher and J. Hill, Bounding the Area of the Mandelbrot Set,
Numerische Mathematik,. (Submitted for publication). Available via
World Wide Web (in Postscript format)
http://inls.ucsd.edu/y/Complex/area.ps.Z.
Q6g: What can you say about the structure of the Mandelbrot set?
A6g: Most of what you could want to know is in Branner's article in
Chaos and Fractals: The Mathematics Behind the Computer Graphics.
Note that the Mandelbrot set in general is not strictly self-similar;
the tiny copies of the Mandelbrot set are all slightly different,
mainly because of the thin threads connecting them to the main body of
the Mandelbrot set. However, the Mandelbrot set is quasi-self-similar.
However, the Mandelbrot set is self-similar under magnification in
neighborhoods of Misiurewicz points (e.g. -.1011 + .9563i). The
Mandelbrot set is conjectured to be self-similar around generalized
Feigenbaum points (e.g. -1.401155 or -.1528 + 1.0397i), in the sense
of converging to a limit set.
References:
1. T. Lei, Similarity between the Mandelbrot set and Julia Sets,
Communications in Mathematical Physics 134 (1990), pp. 587-617.
2. J. Milnor, Self-Similarity and Hairiness in the Mandelbrot Set, in
Computers in Geometry and Topology, M. Tangora (editor), Dekker,
New York, pp. 211-257.
The "external angles " of the Mandelbrot set (see Douady and Hubbard
or brief sketch in "Beauty of Fractals") induce a Fibonacci partition
onto it.
The boundary of the Mandelbrot set and the Julia set of a generic c in
M have Hausdorff dimension 2 and have topological dimension 1. The
proof is based on the study of the bifurcation of parabolic periodic
points. (Since the boundary has empty interior, the topological
dimension is less than 2, and thus is 1.)
Reference:
1. M. Shishikura, The Hausdorff Dimension of the Boundary of the
Mandelbrot Set and Julia Sets, The paper is available from
anonymous ftp: ftp://math.sunysb.edu/preprints/ims91-7.ps.Z [IP:
129.49.18.1]
Q6h: Is the Mandelbrot set connected?
A6h: The Mandelbrot set is simply connected. This follows from a
theorem of Douady and Hubbard that there is a conformal isomorphism
from the complement of the Mandelbrot set to the complement of the
unit disk. (In other words, all equipotential curves are simple closed
curves.) It is conjectured that the Mandelbrot set is locally
connected, and thus pathwise connected, but this is currently
unproved.
Connectedness definitions:
Connected: X is connected if there are no proper closed subsets A and
B of X such that A union B = X, but A intersect B is empty. I.e. X is
connected if it is a single piece.
Simply connected: X is simply connected if it is connected and every
closed curve in X can be deformed in X to some constant closed curve.
I.e. X is simply connected if it has no holes.
Locally connected: X is locally connected if for every point p in X,
for every open set U containing p, there is an open set V containing p
and contained in the connected component of p in U. I.e. X is locally
connected if every connected component of every open subset is open in
X. Arcwise (or path) connected: X is arcwise connected if every two
points in X are joined by an arc in X.
(The definitions are from Encyclopedic Dictionary of Mathematics.)
Reference:
Douady, A. and Hubbard, J., "Comptes Rendus" (Paris) 294, pp.123-126,
1982.
Q6i: What is the Mandelbrot Encyclopedia?
A6i: The Mandelbrot Encyclopedia is a mail server which contains
information about the Mandelbrot Set. It was setup by Robert Munafo
but is not currently available. Further
information will be available once it is available again.
Q6j: What is the dimension of the Mandelbrot Set?
A6j: The Mandelbrot Set has a dimension of 2. The Mandelbrot Set
contains and is contained in a disk. A disk has a dimension of 2, thus
so does the Mandelbrot Set. The Koch snowflake (dimension 1.2619...)
does not satify 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.
_________________________________________________________________
Subject: Julia sets
Q7a: What is the difference between the Mandelbrot set and a Julia
set?
A7a: The Mandelbrot set iterates z² + c with z starting at 0 and
varying c. The Julia set iterates z² + c for fixed c and varying
starting z values. That is, the Mandelbrot set is in parameter space
(c-plane) while the Julia set is in dynamical or variable space
(z-plane).
Q7b: What is the connection between the Mandelbrot set and Julia sets?
A7b: Each point c in the Mandelbrot set specifies the geometric
structure of the corresponding Julia set. If c is in the Mandelbrot
set, the Julia set will be connected. If c is not in the Mandelbrot
set, the Julia set will be a Cantor dust.
Q7c: How is a Julia set actually computed?
A7c: The Julia set can be computed by iteration similar to the
Mandelbrot computation. The only difference is that the c value is
fixed and the initial z value varies.
Alternatively, points on the boundary of the Julia set can be computed
quickly by using inverse iterations. This technique is particularly
useful when the Julia set is a Cantor Set. In inverse iteration, the
equation z1 = z0² + c is reversed to give an equation for z0: z0 = ±
sqrt(z1 - c). By applying this equation repeatedly, the resulting
points quickly converge to the Julia set boundary. (At each step,
either the positive or negative root is randomly selected.) This is a
nonlinear iterated function system.
In pseudocode:
z = 1 (or any value)
loop
if (random number < .5) then
z = sqrt(z - c)
else
z = -sqrt(z - c)
endif
plot z
end loop
Q7d: What are some Julia set facts?
A7d: The Julia set of any rational map of degree greater than one is
perfect (hence in particular uncountable and nonempty), completely
invariant, equal to the Julia set of any iterate of the function, and
also is the boundary of the basin of attraction of every attractor for
the map.
Julia set references:
1. A. F. Beardon, Iteration of Rational Functions : Complex Analytic
Dynamical Systems, Springer-Verlag, New York, 1991.
2. P. Blanchard, Complex Analytic Dynamics on the Riemann Sphere,
Bull. of the Amer. Math. Soc 11, 1 (July 1984), pp. 85-141.
This article is a detailed discussion of the mathematics of iterated
complex functions. It covers most things about Julia sets of rational
polynomial functions.
_________________________________________________________________
Subject: Complex arithmetic and quaternion arithmetic
Q8a: How does complex arithmetic work?
A8a: It works mostly like regular algebra with a couple additional
formulas:
(note: a,b are reals, x ,y are complex, i is the square root of -1)
Powers of i:
i² = -1
Addition:
(a+i·b)+(c+i ·d) = (a+c)+i·(b+d)
Multiplication:
(a+i ·b)·(c+i·d) = a ·c-b·d + i·(a·d+b·c)
Division:
(a+i·b) ÷ (c+i·d) = (a+i·b)·(c-i·d) ÷ (c²+d²)
Exponentiation:
exp(a+i·b) = exp(a)(cos(b)+i ·sin(b))
Sine:
sin(x) = (exp(i·x) - exp(-i·x))÷(2·i)
Cosine:
cos(x) = (exp(i·x) + exp(-i·x)) ÷ 2
Magnitude:
|a+i·b| = sqrt(a²+b²)
Log:
log(a+i·b) = log(|a+i·b|)+i·arctan(b ÷ a) (Note: log is
multivalued.)
Log (polar coordinates):
log(r·e^(i ·ø)) = log(r)+i·ø
Complex powers:
x^y = exp(y·log(x))
de Moivre's theorem:
x^n = r^n · [cos(n ·ø) + i · sin(n·ø)] (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 + ib + jc + kd) compared to the
two of complex numbers. Operations such as addition and multiplication
can be performed on quaternions, but multiplication is not
commutative.
Quaternions satisfy the rules
* i² = j² = k² = -1
* ij = -ji = k
* jk = -kj = i,
* ki = -ik = j
See:
Frode Gill's quaternions page
http://www.krs.hia.no/~fgill/quatern.html
_________________________________________________________________
Subject: Logistic equation
Q9: What is the logistic equation?
A9: It models animal populations. The equation is x -> c·x·(1 - x),
where x is the population (between 0 and 1) and c is a growth
constant. Iteration of this equation yields the period doubling route
to chaos. For c between 1 and 3, the population will settle to a fixed
value. At 3, the period doubles to 2; one year the population is very
high, causing a low population the next year, causing a high
population the following year. At 3.45, the period doubles again to 4,
meaning the population has a four year cycle. The period keeps
doubling, faster and faster, at 3.54, 3.564, 3.569, and so forth. At
3.57, chaos occurs; the population never settles to a fixed period.
For most c values between 3.57 and 4, the population is chaotic, but
there are also periodic regions. For any fixed period, there is some c
value that will yield that period. See "An Introduction to Chaotic
Dynamical Systems" for more information.
_________________________________________________________________
Subject: Feigenbaum's constant
Q10: What is Feigenbaum's constant? NEW
A10: In a period doubling cascade, such as the logistic equation,
consider the parameter values where period-doubling events occur (e.g.
r[1]=3, r[2]=3.45, r[3]=3.54, r[4]=3.564...). Look at the ratio of
distances between consecutive doubling parameter values; let delta[n]
= (r[n+1]-r[n])/(r[n+2]-r[n+1]). Then the limit as n goes to infinity
is Feigenbaum's (delta) constant.
Based on computations by F. Christiansen, P. Cvitanovic and H.H. Rugh,
it has the value 4.6692016091029906718532038... Note: several books
have published incorrect values starting 4.66920166...; the last
repeated 6 is a typographical error.
The interpretation of the delta constant is as you approach chaos,
each periodic region is smaller than the previous by a factor
approaching 4.669...
Feigenbaum's constant is important because it is the same for any
function or system that follows the period-doubling route to chaos and
has a one-hump quadratic maximum. For cubic, quartic, etc. there are
different Feigenbaum constants.
Feigenbaum's alpha constant is not as well known; it has the value
2.50290787509589282228390287272909. This constant is the scaling
factor between x values at bifurcations. Feigenbaum says,
"Asymptotically, the separation of adjacent elements of period-doubled
attractors is reduced by a constant value [alpha] from one doubling to
the next". If d[a] is the algebraic distance between nearest elements
of the attractor cycle of period 2ª, then d[a]/d[a+1] converges to
-alpha.
References:
1. K. Briggs, How to calculate the Feigenbaum constants on your PC,
Aust. Math. Soc. Gazette 16 (1989), p. 89.
2. K. Briggs, A precise calculation of the Feigenbaum constants,
Mathematics of Computation 57 (1991), pp. 435-439.
3. K. Briggs, G. R. W. Quispel and C. Thompson, Feigenvalues for
Mandelsets, J. Phys. A 24 (1991), pp. 3363-3368.
4. F. Christiansen, P. Cvitanovic and H.H. Rugh, "The spectrum of the
period-doubling operator in terms of cycles", J. Phys A 23, L713
(1990).
5. M. Feigenbaum, The Universal Metric Properties of Nonlinear
Transformations, J. Stat. Phys 21 (1979), p. 69.
6. M. Feigenbaum, Universal Behaviour in Nonlinear Systems, Los
Alamos Sci 1 (1980), pp. 1-4. Reprinted in Universality in Chaos,
compiled by P. Cvitanovic.
Feigenbaum Constants
http://www.mathsoft.com/asolve/constant/fgnbaum/fgnbaum.h
tml
_________________________________________________________________
Subject: Iterated function systems and compression
Q11a: What is an iterated function system (IFS)?
A11a: If a fractal is self-similar, you can specify mappings that map
the whole onto the parts. Iteration of these mappings will result in
convergence to the fractal attractor. An IFS consists of a collection
of these (usually affine) mappings. If a fractal can be described by a
small number of mappings, the IFS is a very compact description of the
fractal. An iterated function system is By taking a point and
repeatedly applying these mappings you end up with a collection of
points on the fractal. In other words, instead of a single mapping x
-> F(x), there is a collection of (usually affine) mappings, and
random selection chooses which mapping is used.
For instance, the Sierpinski triangle can be decomposed into three
self-similar subtriangles. The three contractive mappings from the
full triangle onto the subtriangles forms an IFS. These mappings will
be of the form "shrink by half and move to the top, left, or right".
Iterated function systems can be used to make things such as fractal
ferns and trees and are also used in fractal image compression.
Fractals Everywhere by Barnsley is mostly about iterated function
systems.
The simplest algorithm to display an IFS is to pick a starting point,
randomly select one of the mappings, apply it to generate a new point,
plot the new point, and repeat with the new point. The displayed
points will rapidly converge to the attractor of the IFS.
Interactive IFS Playground (Otmar Lendl)
http://www.cosy.sbg.ac.at/rec/ifs/
Frank Rousell's hyperindex of IFS images
http://www.cnam.fr/fractals/ifs.html
Q11b: What is the state of fractal compression?
A11b: Fractal compression is quite controversial, with some people
claiming it doesn't work well, and others claiming it works
wonderfully. The basic idea behind fractal image compression is to
express the image as an iterated function system (IFS). The image can
then be displayed quickly and zooming will generate infinite levels of
(synthetic) fractal detail. The problem is how to efficiently generate
the IFS from the image. Barnsley, who invented fractal image
compression, has a patent on fractal compression techniques
(4,941,193). Barnsley's company, Iterated Systems Inc
(http://www.iterated.com/), has a line of products including a Windows
viewer, compressor, magnifier program, and hardware assist board.
Fractal compression is covered in detail in the comp.compression FAQ
file (See "compression-FAQ").
ftp://rtfm.mit.edu/pub/usenet/comp.compression [18.181.0.24].
One of the best online references for Fractal Compress is Yuval
Fisher's Fractal Image Encoding page
(http://inls.ucsd.edu/y/Fractals/) at the Institute for Nonlinear
Science, University for California, San Diego. It includes references
to papers, other WWW sites, software, and books about Fractal
Compression.
Three major research projects include
Waterloo Montreal Verona Fractal Research Initiative
http://links.uwaterloo.ca/
Groupe FRACTALES
http://www-syntim.inria.fr/fractales/
Bath Scalable Video Demo Software
http://dmsun4.bath.ac.uk/bsvdemo/
Several books describing fractal image compression are:
1. M. Barnsley, Fractals Everywhere, Academic Press Inc., 1988. ISBN
0-12-079062-9. This is an excellent text book on fractals. This is
probably the best book for learning about the math underpinning
fractals. It is also a good source for new fractal types.
2. M. Barnsley and L. Anson, The Fractal Transform, Jones and
Bartlett, April, 1993. ISBN 0-86720-218-1. Without assuming a
great deal of technical knowledge, the authors explain the
workings of the Fractal Transform(TM).
3. M. Barnsley and L. Hurd, Fractal Image Compression, Jones and
Bartlett. ISBN 0-86720-457-5. This book explores the science of
the fractal transform in depth. The authors begin with a
foundation in information theory and present the technical
background for fractal image compression. In so doing, they
explain the detailed workings of the fractal transform. Algorithms
are illustrated using source code in C.
4. Y. Fisher (Ed), Fractal Image Compression: Theory and Application.
Springer Verlag, 1995.
5. Y. Fisher (Ed), Fractal Image Encoding and Analysis: A NATO ASI
Series Book, Springer Verlag, New York, 1996 contains the
proceedings of the Fractal Image Encoding and Analysis Advanced
Study Institute held in Trondheim, Norway July 8-17, 1995. The
book is currently being produced.
The October 1993 issue of Byte discussed fractal compression. You can
ftp sample code: ftp://ftp.uu.net/published/byte/93oct/fractal.exe.
Andreas Kassler wrote a Fractal Image Compression with WINDOWS package
for a Fractal Compression thesis. It is available at
http://www-vs.informatik.uni-ulm.de/Mitarbeiter/Kassler.html
An introductory paper is:
1. A. E. Jacquin, Image Coding Based on a Fractal Theory of Iterated
Contractive Image Transformation, IEEE Transactions on Image
Processing, January 1992.
Many fractal image compression papers are available from
ftp://ftp.informatik.uni-freiburg.de/documents/papers/fractal [IP
132.230.150.1].
A review of the literature is in Guide.ps.gz. See the README file for
an overview of the available documents.
Other references:
Fractal Compression Bibliography
http://dip1.ee.uct.ac.za/fractal.bib.html
Fractal Video Compression
http://inls.ucsd.edu/y/Fractals/Video/fracvideo.html
_________________________________________________________________
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.
_________________________________________________________________
Subject: L-Systems
Q13: What are L-systems?
A13: A L-system or Lindenmayer system is a formal grammar for
generating strings. (That is, it is a collection of rules such as
replace X with XYX.) By recursively applying the rules of the L-system
to an initial string, a string with fractal structure can be created.
Interpreting this string as a set of graphical commands allows the
fractal to be displayed. L-systems are very useful for generating
realistic plant structures.
Some references are:
1. P. Prusinkiewicz and J. Hanan, Lindenmayer Systems, Fractals, and
Plants, Springer-Verlag, New York, 1989.
2. P. Prusinkiewicz and A. Lindenmayer, The Algorithmic Beauty of
Plants, Springer-Verlag, NY, 1990. ISBN 0-387-97297-8. A very good
book on L-systems, which can be used to model plants in a very
realistic fashion. The book contains many pictures.
_________________________________________________________________
More information can be obtained via the WWW at:
L-Systems Tutorial by David Green
http://life.csu.edu.au/complex/tutorials/tutorial2.html
L-system leaf
http://www.csu.edu.au/complex_systems/iconfern.gif
3 Dim. L-system Tree program (P.J.Drinkwater)
http://hill.lut.ac.uk/TestStuff/trees/
L-system images from the Center for the Computation and Visualization
of Geometric Structures
http://www.geom.umn.edu/pix/archive/subjects/L-systems.html
_________________________________________________________________
Subject: Fractal music
Q14: What is some information on fractal music?
A14: One fractal recording is "The Devil's Staircase: Composers and
Chaos" on the Soundprint label.
Some references, many from an unpublished article by Stephanie Mason,
are:
1. R. Bidlack, Chaotic Systems as Simple (But Complex) Compositional
Algorithms, Computer Music Journal, Fall 1992.
2. C. Dodge, A Musical Fractal, Computer Music Journal 12, 13 (Fall
1988), p. 10.
3. K. J. Hsu and A. Hsu, Fractal Geometry of Music, Proceedings of
the National Academy of Science, USA 87 (1990), pp. 938-941.
4. K. J. Hsu and A. Hsu, Self-similatrity of the '1/f noise' called
music., Proceedings of the National Academy of Science USA 88
(1991), pp. 3507-3509.
5. C. Pickover, Mazes for the Mind: Computers and the Unexpected, St.
Martin's Press, New York, 1992.
6. P. Prusinkiewicz, Score Generation with L-Systems, International
Computer Music Conference 86 Proceedings, 1986, pp. 455-457.
7. Byte 11, 6 (June 1986), pp. 185-196.
Online resources include:
Well Tempered Fractal v3.0 from Spanky via FTP by Robert Greenhouse
ftp://spanky.triumf.ca/pub/fractals/programs/ibmpc/wtf30.zip
A fractal music C++ package is available at
http://hamp.hampshire.edu/~gpzF93/inSanity.html
The Fractal Music Project (Claus-Dieter Schulz)
http://www-ks.rus.uni-stuttgart.de/people/schulz/fmusic
Chua's Oscillator: Applications of Chaos to Sound and Music
http://www.ccsr.uiuc.edu/People/gmk/Projects/ChuaSoundMusic/Chu
aSoundMusic.html
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: fmusic@kssun7.rus.uni-stuttgart.de
To subscribe to the list please send mail to
fmusic-request@kssun7.rus.uni-stuttgart.de
_________________________________________________________________
Subject: Fractal mountains
Q15: How are fractal mountains generated?
A15: Usually by a method such as taking a triangle, dividing it into 3
subtriangles, and perturbing the center point. This process is then
repeated on the subtriangles. This results in a 2-d table of heights,
which can then be rendered as a 3-d image. Two references are:
1. M. Ausloos, Proc. R. Soc. Lond. A 400 (1985), pp. 331-350.
2. H.O. Peitgen, D. Saupe, The Science of Fractal Images,
Springer-Velag, 1988
Available online is an implementation of fractal Brownian motion (fBm)
such as described in The Science of Fractal Images.
Gforge (John Beale)
http://jump.stanford.edu:8080/beale/land/index.html
Other fractal landscape
EECS News: Fall 1994: Building Fractal Planets by Ken Musgrave
http://www.seas.gwu.edu/faculty/musgrave/article.html
_________________________________________________________________
Subject: Plasma clouds
Q16: What are plasma clouds?
A16: They are a Fractint fractal and are similar to fractal mountains.
Instead of a 2-d table of heights, the result is a 2-d table of
intensities. They are formed by repeatedly subdividing squares.
Robert Cahalan has fractal information about Earth's Clouds including
how they differ from plasma clouds.
Fractal Clouds Reference by Robert F. Cahalan
(cahalan@clouds.gsfc.nasa.gov)
http://climate.gsfc.nasa.gov/~cahalan/FractalClouds/
Also some plasma-based fractals clouds by John Walker are
available.
Fractal generated clouds
http://ivory.nosc.mil/html/trancv/html/cloud-fract.html
Two articles about the fractal nature of Earth's clouds:
1. "Fractal statistics of cloud fields," R. F. Cahalan and J. H.
Joseph, Mon. Wea.Rev. 117, 261-272, 1989
2. "The albedo of fractal stratocumulus clouds," R. F. Cahalan, W.
Ridgway, W. J. Wiscombe, T. L. Bell and J. B. Snider, J. Atmos.
Sci. 51, 2434-2455, 1994
_________________________________________________________________
Subject: Lyapunov fractals
Q17a: Where are the popular periodically-forced Lyapunov fractals
described?
A17a: See:
1. A. K. Dewdney, Leaping into Lyapunov Space, Scientific American,
Sept. 1991, pp. 178-180.
2. M. Markus and B. Hess, Lyapunov Exponents of the Logistic Map with
Periodic Forcing, Computers and Graphics 13, 4 (1989), pp.
553-558.
3. M. Markus, Chaos in Maps with Continuous and Discontinuous Maxima,
Computers in Physics, Sep/Oct 1990, pp. 481-493.
Q17b: What are Lyapunov exponents?
A17b: Lyapunov exponents quantify the amount of linear stability or
instability of an attractor, or an asymptotically long orbit of a
dynamical system. There are as many lyapunov exponents as there are
dimensions in the state space of the system, but the largest is
usually the most important.
Given two initial conditions for a chaotic system, a and b, which are
close together, the average values obtained in successive iterations
for a and b will differ by an exponentially increasing amount. In
other words, the two sets of numbers drift apart exponentially. If
this is written e^(n*(lambda)) for n iterations, then e^(lambda) is
the factor by which the distance between closely related points
becomes stretched or contracted in one iteration. Lambda is the
Lyapunov exponent. At least one Lyapunov exponent must be positive in
a chaotic system. A simple derivation is available in:
1. H. G. Schuster, Deterministic Chaos: An Introduction, Physics
Verlag, 1984.
Q17c: How can Lyapunov exponents be calculated?
A17c: For the common periodic forcing pictures, the lyapunov exponent
is:
lambda = limit as N -> infinity of 1/N times sum from n=1 to N of
log2(abs(dx sub n+1 over dx sub n))
In other words, at each point in the sequence, the derivative of the
iterated equation is evaluated. The Lyapunov exponent is the average
value of the log of the derivative. If the value is negative, the
iteration is stable. Note that summing the logs corresponds to
multiplying the derivatives; if the product of the derivatives has
magnitude < 1, points will get pulled closer together as they go
through the iteration.
MS-DOS and Unix programs for estimating Lyapunov exponents from short
time series are available by ftp: ftp://inls.ucsd.edu/pub/ncsu/
Computing Lyapunov exponents in general is more difficult. Some
references are:
1. H. D. I. Abarbanel, R. Brown and M. B. Kennel, Lyapunov Exponents
in Chaotic Systems: Their importance and their evaluation using
observed data, International Journal of Modern Physics B 56, 9
(1991), pp. 1347-1375.
2. A. K. Dewdney, Leaping into Lyapunov Space, Scientific American,
Sept. 1991, pp. 178-180.
3. M. Frank and T. Stenges, Journal of Economic Surveys 2 (1988), pp.
103- 133.
4. T. S. Parker and L. O. Chua, Practical Numerical Algorithms for
Chaotic Systems, Springer Verlag, 1989.
_________________________________________________________________
Subject: Fractal items
Q18: Where can I get fractal T-shirts and posters?
A18: One source is Art Matrix, P.O. box 880, Ithaca, New York, 14851,
1-800-PAX-DUTY.
Another source is Media Magic; they sell many fractal posters,
calendars, videos, software, t-shirts, ties, and a huge variety of
books on fractals, chaos, graphics, etc. Media Magic is at PO Box 598
Nicasio, CA 94946, 415-662-2426.
A third source is Ultimate Image; they sell fractal t- shirts,
posters, gift cards, and stickers. Ultimate Image is at PO Box 7464,
Nashua, NH 03060-7464.
Yet another source is Dave Kliman (516) 625-2504 dkliman@pb.net, whose
products are distributed through Spencer Gifts, Posterservice,
1-800-666-7654, and Scandecor International., this spring, through JC
Penny, featuring all-over fractal t-shirts, and has fractal umbrellas
available from Shaw Creations (800) 328-6090.
Cyber Fiber produces fractal silk scarves, t-shirts, and postcards.
Contact Robin Lowenthal, Cyber Fiber, 4820 Gallatin Way, San Diego, CA
92117.
Chaos MetaLink website
(http://www.industrialstreet.com/chaos/metalink.htm) also has
postcards, CDs, and videos.
Free fractal posters are available if you send a self-addressed
stamped envelope to the address given on
http://www.xmission.com/~legalize/. For foreign requests (outside USA)
include two IRCs (international reply coupons) to cover the weight.
ReFractal Design (http://www.refractal.com/) sells jewelry based on
fractals.
_______________________________________________________________________________
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: 3-D fractals
Q20: How can 3-D fractals be generated?
A20: A common source for 3-D fractals is to compute Julia sets with
quaternions instead of complex numbers. The resulting Julia set is
four dimensional. By taking a slice through the 4-D Julia set (e.g. by
fixing one of the coordinates), a 3-D object is obtained. This object
can then be displayed using computer graphics techniques such as ray
tracing.
Frank Rousell's hyperindex of 3D images
http://www.cnam.fr/fractals/mandel3D.html
4D Quaternions by Tom Holroyd
http://bambi.ccs.fau.edu/~tomh/fractals/fractals.html
The papers to read on this are:
1. J. Hart, D. Sandin and L. Kauffman, Ray Tracing Deterministic 3-D
Fractals, SIGGRAPH, 1989, pp. 289-296.
2. A. Norton, Generation and Display of Geometric Fractals in 3-D,
SIGGRAPH, 1982, pp. 61-67.
3. A. Norton, Julia Sets in the Quaternions, Computers and Graphics,
13, 2 (1989), pp. 267-278.
Two papers on cubic polynomials, which can be used to generate 4-D
fractals:
1. B. Branner and J. Hubbard, The iteration of cubic polynomials,
part I., Acta Math 66 (1988), pp. 143-206.
2. J. Milnor, Remarks on iterated cubic maps, This paper is available
from ftp://math.sunysb.edu/preprints/ims90-6.ps.Z. Published in
1991 SIGGRAPH Course Notes #14: Fractal Modeling in 3D Computer
Graphics and Imaging.
Instead of quaternions, you can of course use hypercomplex number such
as in "FractInt", or other functions. For instance, you could use a
map with more than one parameter, which would generate a
higher-dimensional fractal.
Another way of generating 3-D fractals is to use 3-D iterated function
systems (IFS). These are analogous to 2-D IFS, except they generate
points in a 3-D space.
A third way of generating 3-D fractals is to take a 2-D fractal such
as the Mandelbrot set, and convert the pixel values to heights to
generate a 3-D "Mandelbrot mountain". This 3-D object can then be
rendered with normal computer graphics techniques.
POV-Ray 3.0, a freely available ray tracing package, has added 4-D
fractal support. It takes a 3-D slice of a 4-D Julia set based on an
arbitrary 3-D "plane" done at any angle. For more information see the
POV Ray web site at http://www.povray.org/.
_________________________________________________________________
Subject: Fractint
Q21a: What is Fractint?
A21a: Fractint is a very popular freeware (not public domain) fractal
generator. There are DOS, Windows, OS/2, Amiga, and Unix/X 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:
DOS
19.3 source via WWW from USA
http://www.coast.net/cgi-bin/coast/dwn?msdos/graphics/frasr192.
zip
19.3 executable via WWW from USA
http://www.coast.net/cgi-bin/coast/dwn?msdos/graphics/frain192.
zip
19.3 source via FTP from Canada
ftp://spanky.triumf.ca/fractals/programs/ibmpc/frasr193.zip
19.3 executable via FTP from Canada
ftp://spanky.triumf.ca/fractals/programs/ibmpc/frain193.zip
(The suffix 193 will change as new versions are released.)
Fractint is available on Compuserve: GO GRAPHDEV and look for
FRAINT.EXE and FRASRC.EXE in LIB 4.
Windows
MS-Window FractInt 18.21 via FTP from Canada
ftp://spanky.triumf.ca/fractals/programs/ibmpc/windows/winf1821
.zip
MS-Window FractInt 18.21 via WWW from USA
http://www.coast.net/cgi-bin/coast/dwn?win3/graphics/winf1821.z
ip
MS-Windows FractInt 18.21 source via FTP from Canada
ftp://spanky.triumf.ca/fractals/programs/ibmpc/windows/wins1821
.zip
MS-Windows FractInt 18.21 source via WWW from USA
http://www.coast.net/cgi-bin/coast/dwn?win3/graphics/wins1821.z
ip
OS/2
Available on Compuserve in its GRAPHDEV forum. The files are PM*.ZIP.
These files are also available by
ftp://ftp-os2.nmsu.edu/os2/graphics/pmfra2.zip
Unix
The Unix version of FractInt, called XFractInt requires X-Windows.
3.02 source
ftp://ftp.cs.berkeley.edu/pub/sprite/xfract302.shar.Z
XFractInt is also available in LIB 4 of Compuserve's GO GRAPHDEV forum
in XFRACT.ZIP.
Macintosh
There is NO Macintosh version of Fractint, although there may be
several people working on a port. It is possibleto run Fractint on the
Macintosh if you use Insignia Software's SoftAT, which is a PC AT
emulator.
Amiga
There is an Amiga version also available:
FractInt 2.6 via FTP from an AmiNET archive in USA
ftp://wuarchive.wustl.edu/pub/aminet/gfx/fract/fractint26.lha
FracInt 2.6 via WWW from an AmiNET archive in USA
http://wuarchive.wustl.edu/pub/aminet/gfx/fract/fractint26.lha
The latest version (3.02) via WWW from Norway
http://login.eunet.no/~terjepe/aboutfractint.html
FracXtra
There is a collection of map, parameter, etc. files for FractInt,
called FracXtra. It is available
FracXtra Home Page by Dan Goldwater
http://fatmac.ee.cornell.edu/~goldwada/fracxtra.html
FracXtra via WWW (preferred)
http://www.coast.net/cgi-bin/coast/dwn?mdos/graphics/fracxtr6.z
ip
FracXtra via FTP
ftp://spanky.triumf.ca/fractals/programs/ibmpc/fracxtr6.zip
FractInt PAR Exchange
by Landon Kuhn "for all the fans of Fractint and fractal creation."
Its purpose is the trading of parameter files created by Fractint.
FractInt PAR Exchange
http://www.hevanet.com/lkuhn/px
For European users, these files are available from
ftp://ftp.uni-koeln.de/. If you can't use ftp, see the mail server
information below.
Q21b: How does Fractint achieve its speed?
A21b: Fractint's speed (such as it is) is due to a combination of:
1. Using fixed point math rather than floating point where possible
(huge improvement for non-coprocessor machine, small for 486's).
2. Exploiting symmetry of the fractal.
3. Detecting nearly repeating orbits, avoid useless iteration (e.g.
repeatedly iterating 0²+0 etc. etc.).
4. Reducing computation by guessing solid areas (especially the
"lake" area).
5. Using hand-coded assembler in many places.
6. Obtaining both sin and cos from one 387 math coprocessor
instruction.
7. Using good direct memory graphics writing in 256-color modes.
The first four are probably the most important. Some of these
introduce errors, usually quite acceptable.
_________________________________________________________________
Subject: Fractal software NEW
Q22: Where can I obtain software packages to generate fractals?
A22:
For X windows:
xmntns xlmntn: fractal mountains
ftp://ftp.uu.net/usenet/comp.sources.x/volume8/xmntns
xfroot: fractal root window
X11 distribution
xmartin: Martin hopalong root window
X11 distribution
xmandel: Mandelbrot/Julia sets
X11 distribution
lyap: Lyapunov exponent images
ftp://ftp.uu.net/usenet/comp.sources.x/volume17/lyapunov-xlib
spider: Uses Thurston's algorithm, Kobe algorithm, external angles
http://inls.ucsd.edu/y/Complex/spider.tar.Z
xfractal_explorer: fractal drawing program
ftp://ftp.x.org/contrib/applications/xfractal_explorer-v1.0.tar
.gz
Xmountains: A fractal landscape generator
ftp://ftp.epcc.ed.ac.uk/pub/personal/spb/xmountains
xfract: Mandelbrot with a color-cycling feature
ftp://charm.il.ft.hse.nl/pub/X11/src/xfract.tar.gz
xmfract v1.4: Needs Motif 1.2+, based on FractInt
ftp://ftp.x.org/contrib/graphics/xmfract_1.4.tar.gz
Fast Julia Set and Mandelbrot for X-Windows by Zsolt Zsoldos
http://www.chem.leeds.ac.uk/ICAMS/people/zsolt/mandel.html
Distributed X systems:
MandelSpawn: Mandelbrot/Julia on a network
ftp://ftp.x.org/R5contrib/mandelspawn-0.07.tar.Z
ftp://ftp.funet.fi/pub/X11/R5contrib/mandelspawn-0.07.tar.Z
gnumandel: Mandelbrot on a network
ftp://ftp.elte.hu/pub/software/unix/gnu/gnumandel.tar.Z
For SunView:
Mandtool: Mandelbrot
ftp://spanky.triumf.ca/fractals/programs/mandtool/M_TAR.Z
For Unix/C:
lsys: L-systems as PostScript (in C++)
ftp://ftp.cs.unc.edu/pub/users/leech/lsys.tar.gz
lyapunov: PGM Lyapunov exponent images
ftp://ftp.uu.net/usenet/comp.sources.misc/volume23/lyapunov/
SPD: fractal mountain, tree, recursive tetrahedron
ftp://ftp.povray.org/pub/povray/spd/
Fractal Studio: Mandelbrot set; handles distributed computing
ftp://archive.cs.umbc.edu/pub/peter/fractal-studio
fanal: analysis of fractal dimension by Jürgen Dollinger
ftp://ftp.uni-stuttgart.de/pub/systems/linux/local/math/fanal-0
1b.tar.gz
For Mac:
LSystem, 3D-L-System, IFS, FracHill, Mandella
http://wuarchive.wustl.edu/edu/math/software/mac/fractals/
ftp://ftp.auckland.ac.nz/
fractal-wizard.hqx, julias-dream-107.hqx, mandella-87.hqx
ftp://mirrors.aol.com/pub/info-mac/app/
ftp://plaza.aarnet.edu.au/micros/mac/info-mac/app/
mandel-tv: a very fast Mandelbrot generator.
ftp://mirrors.aol.com/pub/info-mac/sci/
ftp://plaza.aarnet.edu.au/micros/mac/info-mac/sci/
mandelzot, powerexplorer
ftp://mirrors.aol.com/pub/info-mac/
There are also commercial programs, such as IFS Explorer and Fractal
Clip Art, which are published by Koyn Software (314) 878-9125. Kai's
Fractal Explorer (part of the Kai's Power Tools package for Adobe
Photoshop)
Note: This listing is quite old. If you have a Mac (especially a
PowerMac) please do me a large favor and send me updates to this
information. Thanks.
(note: M-Set is short hand for Mandelbrot Set)
For MSDOS:
DEEPZOOM: a high-precision M-Set program for displaying highly zoomed
fractals
http://spanky.triumf.ca/pub/fractals/programs/ibmpc/depzm13.zip
Fractal WitchCraft: a very fast fractal design program
ftp://garbo.uwasa.fi/pc/demo/fw1-08.zip
ftp://ftp.cdrom.com/pub/garbo/garbo_pc/show/fw1-08.zip
CAL: generates more than 15 types of fractals including Lyapunov, IFS,
user-defined, logistic, and Quaternion Julia
ftp://ftp.coast.net/SimTel/msdos/graphics/frcal040.zip
Fractal Discovery Laboratory: designed for use in a science museum or
school setting. The Lab has five sections: Art Gallery,
Microscope, Movies, Tools, and Library
Sampler available from Compuserve GRAPHDEV Lib 4 in DISCOV.ZIP,
or send high-density disk and self-addressed, stamped envelope
to: Earl F. Glynn, 10808 West 105th Street, Overland Park,
Kansas 66214-3057.
WL-Plot 2.59 : plots functions including bifurcations and recursive
relations
ftp://archives.math.utk.edu/software/msdos/graphing/wlplt/wlplt
259.zip
From ftp://ftp.coast.net/SimTel/msdos/graphics/
forb01a.zip: Displays orbits of M-Set mapping. C/E/VGA
fract30.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
frain192.zip: FRACTINT v19.2 EGA/VGA/XGA fractal generator
frasr192.zip: C & ASM src for FRACTINT v19.2
frcal040.zip: Fractal drawing program: 15 formulae available
frcaldmo.zip: 800x600x256 demo images for FRCAL040.ZIP
vlotkatc uses VESA 640x480x16 Million colour mode to generate
Volterra-Lotka images.
http://spanky.triumf.ca/pub/fractals/programs/ibmpc/vlotkatc.zi
p
http://spanky.triumf.ca/pub/fractals/programs/ibmpc/vlotkatc.do
c
ftp://spanky.triumf.ca/pub/fractals/programs/ibmpc/vlotkatc.zip
ftp://spanky.triumf.ca/pub/fractals/programs/ibmpc/vlotkatc.doc
Fast FPU Fractal Fun 2.0 (FFFF2.0) is the first M-Set generator
working in hicolor gfx modes thus using up to 32768 different
colors on screen by Daniele Paccaloni requires 386DX+ and VESA
support
http://spanky.triumf.ca/pub/fractals/programs/IBMPC/FFFF20.ZIP
ftp://spanky.triumf.ca/pub/fractals/programs/IBMPC/FFFF20.ZIP
3DFract generates 3-D fractals including Sierpinski cheese and 3-D
snowflake
http://www.cstp.umkc.edu/users/bhugh/home.html
FracTrue 2.00 - Hi/TrueColor Generator including a formular parser.
286+ VGA by Bernd Hemmerling
http://www.cs.tu-berlin.de/~hemmerli/
HOP based on the HOPALONG fractal type. Math coprocessor (386DX and
above) and SuperVGA required. shareware ($30) Places to
download HOPZIP.EXE from:
Compuserve GRAPHDEV forum, lib 4
The Well under ibmpc/graphics
http://ourworld.compuserve.com/homepages/mpeters/hop.htm
ftp://ftp.uni-heidelberg.de/pub/msdos/graphics/
http://spanky.triumf.ca/pub/fractals/programs/ibmpc/
ftp://spanky.triumf.ca/pub/fractals/programs/ibmpc/
ZsManJul 1.0 (requires 386DX+) by Zsolt Zsoldos
http://www.chem.leeds.ac.uk/ICAMS/people/zsolt/zsmanjul.html
Fractal Movie a real-time 3D IFS fractal movie renderer (running on
the 486DX+)
http://home.pacific.net.sg/~yqchen/
FracZoom shareware by Niels Ulrik Reinwald 386DX+
http://www.geocities.com/siliconvalley/4602/index.html
RMandel 1.2 80-bit floating point M-Set animation generator by Marvin
R. Lipford
ftp://ftp.cnam.fr/pub/Fractals/anim/FRACSOFT/rmandel.zip
TruMand 1.0 by Mike Freeman 486DX+ True-colour M-Set generator
http://spanky.triumf.ca/pub/fractals/programs/ibmpc/TRMAND10.ZI
P
FAE - Fractal Animation Engine shareware by Brian Towles
http://spanky.triumf.ca/pub/fractals/programs/ibmpc/FAE210B.ZIP
For Windows:
dy-syst: Explores Newton's method, Mandelbrot and Julia sets
ftp://cssun.mathcs.emory.edu/pub/riddle/
bmand 1.1 shareware by Christopher Bare M-Set program
http://www.gi.net/MSDOS_A/PM-1995/95-01/95-01-24/0012.html
For Amiga:
(all entries marked "ff###" are directories where the inividual
archives of the Fish Disk set available at
ftp://ftp.funet.fi/pub/amiga/fish/ and other sites)
General Mandelbrot generators with many features: Mandelbrot (ff030),
Mandel (ff218), Mandelbrot (ff239), TurboMandel (ff302), MandelBltiz
(ff387), SMan (ff447), MandelMountains (ff383, in 3-D), MandelPAUG
(ff452, MandFXP movies), MandAnim (ff461, anims), ApfelKiste (ff566,
very fast), MandelSquare (ff588, anims)
Mandelbrot and Julia sets generators: MandelVroom (ff215), Fractals
(ff371, also Newton-R and other sets)
With different algorithmic approaches (shown): FastGro (ff188, DLA),
IceFrac (ff303, DLA), DEM (ff303, DEM), CPM (ff303, CPM in 3-D),
FractalLab (ff391, any equation)
Iterated Function System generators (make ferns, etc): FracGen (ff188,
uses "seeds"), FCS (ff465), IFSgen (ff554), IFSLab (ff696, "Collage
Theorem"")
Unique fractal types: Cloud (ff216, cloud surfaces), Fractal (ff052,
terrain), IMandelVroom (strange attractor contours?), Landscape
(ff554, scenery), Scenery (ff155, scenery), Plasma (ff573, plasma
clouds)
Fractal generators: PolyFractals (ff015), FFEX (ff549)
Lyapunov fractals
ftp://ftp.luth.se/pub/aminet/gfx/misc/Lyapunovia15.lha
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.
Software for computing fractal dimension:
Fractal Dimension Calculator is a Macintosh program which uses the
box-counting method to compute the fractal dimension of planar
graphical objects.
http://wuarchive.wustl.edu/edu/math/software/mac/fractals/FDC/
http://wuarchive.wustl.edu/packages/architec/Fractals/FDC2D.sea.hqx
http://wuarchive.wustl.edu/packages/architec/Fractals/FDC3D.sea.hqx
FD3: estimates capacity, information, and correlation dimension from a
list of points. It computes log cell sizes, counts, log counts, log of
Shannon statistics based on counts, log of correlations based on
counts, two-point estimates of the dimensions at all scales examined,
and over-all least-square estimates of the dimensions.
ftp://inls.ucsd.edu/pub/cal-state-stan
ftp://inls.ucsd.edu/pub/inls-ucsd
for an enhanced Grassberger-Procaccia algorithm for correlation
dimension.
A MS-DOS version of FP3 is available by request to
gentry@altair.csustan.edu.
Java applets
Chaos!
http://www.vt.edu:10021/B/bwn/Chaos.html
Take's Online
http://www.geocities.com/Hollywood/3618/java.html
Fractal Lab
http://www.wmin.ac.uk/~storyh/fractal/frac.html
Mandelbrot Set Escape Iterations
http://www.voidstar.org/java/escape.html
The Mandelbrot Set
http://www.mindspring.com/~chroma/mandelbrot.html
Paton J. Lewis: Graphics Projects
http://www.cs.brown.edu/people/pjl/mandelbrot.html
Mark's Java Julia Set Generator
http://liberty.uc.wlu.edu/~mmcclure/java/Julia/
Fractals by Sun Microsystems
http://java.sun.com/java.sun.com/applets/applets/Fractal/example1.html
The Mandelbrot set
http://www.franceway.com/java/fractale/mandel_b.htm
Mandelbrot Java Applet
http://www.mit.edu:8001/people/mkgray/java/Mandel.html
Ken Shirriff Java language pages
http://www.sunlabs.com/~shirriff/java/
_________________________________________________________________
Subject: FTP questions
Q23a: How does anonymous ftp work?
A23a: 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 lyapunov.ucsd.edu", or whatever system you wish to
access. You are prompted for your name and you reply "anonymous". You
are prompted for your password and you reply with your email address.
You then use "ls" to list the files, "cd" to change directories, "get"
to get files, an "quit" to exit. For example, you could say "cd /pub",
"ls", "get README", and "quit"; this would get you the file "README".
See the man page ftp(1) or ask someone at your site for more
information.
In this FAQ, anonymous ftp addresses are given in the form
ftp://name.of.machine:/pub/path [1.2.3.4]. 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 1.2.3.4". The part after the name: "/pub/path"
is the file or directory to access once you are connected to the
remote machine.
Q23b: What if I can't use ftp to access files?
A23b: If you don't have access to ftp because you are on a UUCP,
Fidonet, BITNET network there is an e-mail gateway at
ftpmail@decwrl.dec.com that can retrieve the files for you. To get
instructions on how to use the ftp gateway send a message to
ftpmail@decwrl.dec.com 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, be careful not exceed it.
_________________________________________________________________
Subject: Archived pictures
Q24a: Where are fractal pictures archived? NEW
A24a: Fractal images (GIFs, etc.) used to be posted to
alt.fractals.pictures; this newsgroup has been replaced by
alt.binaries.pictures.fractals. Pictures from 1990 and 1991 are
available via anonymous ftp at
ftp://csus.edu/pub/alt.fractals.pictures
Many Mandelbrot set images are available via
ftp://ftp.ira.uka.de/pub/graphic/fractals
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/
From Paris, France one of the largest collections (>= 460MB) is Frank
Roussel's at http://www.cnam.fr/fractals.html. These images are also
available via FTP at ftp://ftp.cnam.fr/pub/Fractals. Fractal
animations in MPG and FLI format are in
ftp://ftp.cnam.fr/pub/Fractals/anim or
http://www.cnam.fr/fractals/anim.html. In Bordeaux (France) there is a
mirror of this site,
http://www.bdx1.u-bordeaux.fr/MAPBX/roussel/fractals.html. Another
collection of fractal images is archived at
ftp.maths.tcd.ie/pub/images/Computer
A collection of interesting smoke- and flame-like jpeg iterated
function system images is available on the WWW at
http://www.cs.cmu.edu/afs/cs.cmu.edu/user/spot/web/images.html. Some
images are also available from:
ftp://hopeless.mess.cs.cmu.edu/spot/film/
Other tutorials, resources, and galleries of images are:
Cliff Pickover
http://sprott.physics.wisc.edu/pickover/home.htm
Fractal Gallery (J. C. Sprott)
http://sprott.physics.wisc.edu/fractals.htm
Fractal Microscope
http://www.ncsa.uiuc.edu/Edu/Fractal/Fractal_Home.html
"Contours of the Mind"
http://online.anu.edu.au/ITA/ACAT/contours/contours.html
Computer Graphics Gallery
http://www.maths.tcd.ie/pub/images/images.html
The San Francisco Fractal Factory.
http://www.awa.com/sfff/sfff.html
Spanky Fractal Datbase (Noel Giffin)
http://spanky.triumf.ca/www/spanky.html
Fractal Gallery (Frank Rousell)
http://www.cnam.fr/fractals.html
Fractal Animations Gallery (Frank Rousell)
http://www.cnam.fr/fractals/anim.html
Yahoo Index of Fractal Art
http://www.yahoo.com/Art/Computer_Generated/Fractals/
Geometry Centre at University of Minnesota
http://www.geom.umn.edu/pix/archive/subjects/fractals.html
Fractal from Ojai (Art Baker)
http://www.fishnet.net/~ayb/
Skal's 3D-fractal collection (Pascal Massimino)
http://acacia.ens.fr:8080/home/massimin/quat/f_gal.ang.html
3d Fractals (Stewart Dickson) via Mathart.com
http://www.wri.com/~mathart/portfolio/SPD_Frac_portfolio.html
Softsource
http://www.softsource.com/softsource/fractal.html
Favourite Fractals (Ryan Grant)
http://www.ncsa.uiuc.edu/SDG/People/rgrant/fav_pics.html
A.F.P. Fractal FTP Archive
ftp://csus.edu/pub/alt.fractals.pictures
Eric Schol
http://hydra.cs.utwente.nl/~schol/video.html
Mandelbrot and Julia Sets (David E. Joyce)
http://aleph0.clarku.edu/~djoyce/home.html
Newton's method
http://aleph0.clarku.edu/~djoyce/newton/newton.html
Gratuitous Fractals (evans@ctrvax.vanderbilt.edu)
http://www.vanderbilt.edu/VUCC/Misc/Art1/fractals.html
Xmorphia
http://www.ccsf.caltech.edu/ismap/image.html
Fractal Prairie Page (George Krumins)
http://www.prairienet.org/astro/fractal.html
Fractal Gallery (Paul Derbyshire)
http://chat.carleton.ca/~pderbysh/fractgal.html
David Finton's homepage
http://www.d.umn.edu/~dfinton/
Algorithmic Image Gallery (Giuseppe Zito)
http://www.ba.infn.it/gallery
Octonion Fractals built using hyper-hyper-complex numbers by Onar Em
http://www.stud.his.no/~onar/Octonion.html
B' Plasma Cloud (animated gif)
http://www.az.com/~rsears/fractp1.html
John Bailey's fractal images
http://www.servtech.com/public/jmb184/images
Q24b: How do I view fractal pictures from
alt.binaries.pictures.fractals?
A24b: A detailed explanation is given in the "alt.binaries.pictures
FAQ" (see "pictures-FAQ"). This is posted to the pictures newsgroups
and is available by ftp:
ftp://rtfm.mit.edu:/pub/usenet/news.answers/pictures-faq/
[18.181.0.24].
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 your
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.
_________________________________________________________________
Subject: Where can I obtain fractal papers?
Q25: Where can I obtain fractal papers?
A25: 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.
A collection of short papers on fractal formulas, drawing methods, and
transforms is available by ftp://ftp.coe.montana.edu:/pub/fractals
(this site hasn't been working lately).
The WWW site http://inls.ucsd.edu/y/Complex/ has some fractal papers.
The site life.csu.edu.au has a collection of fractal programs, papers,
information related to complex systems, and gopher and World Wide Web
connections.
The ftp path is:
ftp://life.csu.edu.au/pub/complex/. Look in fractals, tutorial,
and anu92.
via WWW:
http://life.csu.edu.au/complex/.
_________________________________________________________________
Subject: How can I join the FRAC-L fractal discussion?
Q26: How can I join the FRAC-L fractal discussion?
A26: 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
listproc@archives.math.utk.edu containing the sole line of text:
SUBSCRIBE FRAC-L Your_first_name Your_last_name
(substituting your actual first and last names, of course).
To unsubscribe from frac-l, send LISTPROC (NOT frac-l) the message:
UNSUBSCRIBE FRAC-L
Messages may be posted to frac-l by sending email to:
frac-l@archives.math.utk.edu
If there is any difficulty contact the listowner: Ermel Stepp
(stepp@marshall.edu).
_________________________________________________________________
Subject: Complexity
Q27: What is complexity?
A27: Emerging paradigms of thought encompassing fractals, chaos,
nonlinear science, dynamic systems, self-organization, artificial
life, neural networks, and similar systems comprise the science of
complexity. Several helpful online resources on complexity are:
Institute for Research on Complexity
http://www.marshall.edu/~stepp/vri/irc/irc.html
The site life.csu.edu.au has a collection of fractal programs, papers,
information related to complex systems, and gopher and World Wide Web
connections.
LIFE via WWW
http://life.csu.edu.au/complex/
Complex Systems (UPENN)
http://www.seas.upenn.edu/~ale/cplxsys.html
Center for Complex Systems Research (UIUC)
http://www.ccsr.uiuc.edu/
Complexity International Journal
http://www.csu.edu.au/ci/ci.html
Nonlinear Science Preprints
ftp://xyz.lanl.gov/nlin-sys
Nonlinear Science Preprints via email:
To subscribe to public bulletin board to receive announcements of the
availability of preprints from Los Alamos National Laboratory, send
email to nlin-sys@xyz.lanl.gov containing the sole line of text:
subscribe your-real-name
_________________________________________________________________
Subject: References
Q28a: What are some general references on fractals, chaos, and
complexity? NEW
A28a: Some references are:
M. Barnsley, Fractals Everywhere, Academic Press Inc., 1988, 1993.
ISBN 0-12-079062-9. This is an excellent text book on fractals. This
is probably the best book for learning about the math underpinning
fractals. It is also a good source for new fractal types.
M. Barnsley, The Desktop Fractal Design System Versions 1 and 2. 1992,
1988. Academic Press. Available from Iterated Systems.
M. Barnsley and P H Lyman, Fractal Image Compression. 1993. AK Peters
Limited. Available from Iterated Systems.
M. Barnsley and L. Anson, The Fractal Transform, Jones and Bartlett,
April, 1993. ISBN 0-86720-218-1. This book is a sequel to Fractals
Everywhere. Without assuming a great deal of technical knowledge, the
authors explain the workings of the Fractal Transform(tm). The Fractal
Transform is the compression tool for storing high-quality images in a
minimal amount of space on a computer. Barnsley uses examples and
algorithms to explain how to transform a stored pixel image into its
fractal representation.
R. Devaney and L. Keen, eds., Chaos and Fractals: The Mathematics
Behind the Computer Graphics, American Mathematical Society,
Providence, RI, 1989. This book contains detailed mathematical
descriptions of chaos, the Mandelbrot set, etc.
R. L. Devaney, An Introduction to Chaotic Dynamical Systems, Addison-
Wesley, 1989. ISBN 0-201-13046-7. This book introduces many of the
basic concepts of modern dynamical systems theory and leads the reader
to the point of current research in several areas. It goes into great
detail on the exact structure of the logistic equation and other 1-D
maps. The book is fairly mathematical using calculus and topology.
R. L. Devaney, Chaos, Fractals, and Dynamics, Addison-Wesley, 1990.
ISBN 0-201-23288-X. This is a very readable book. It introduces chaos
fractals and dynamics using a combination of hands-on computer
experimentation and precalculus math. Numerous full-color and black
and white images convey the beauty of these mathematical ideas.
R. Devaney, A First Course in Chaotic Dynamical Systems, Theory and
Experiment, Addison Wesley, 1992. A nice undergraduate introduction to
chaos and fractals.
A. K. Dewdney, (1989, February). Mathematical Recreations. Scientific
American, pp. 108-111.
G. A. Edgar, Measure Topology and Fractal Geometry, Springer-Verlag
Inc., 1990. ISBN 0-387-97272-2. This book provides the math necessary
for the study of fractal geometry. It includes the background material
on metric topology and measure theory and also covers topological and
fractal dimension, including the Hausdorff dimension.
K. Falconer, Fractal Geometry: Mathematical Foundations and
Applications, Wiley, New York, 1990.
J. Feder, Fractals, Plenum Press, New York, 1988. This book is
recommended as an introduction. It introduces fractals from
geometrical ideas, covers a wide variety of topics, and covers things
such as time series and R/S analysis that aren't usually considered.
Y. Fisher (Ed), Fractal Image Compression: Theory and Application.
Springer Verlag, 1995.
L. Gardini(Editor), Chaotic Dynamics in Two-Dimensional Noninvertive
Maps. World Scientific 1996, ISBN: 9810216475
J. Gleick, Chaos: Making a New Science, Penguin, New York, 1987.
B. Hao, ed., Chaos, World Scientific, Singapore, 1984. This is an
excellent collection of papers on chaos containing some of the most
significant reports on chaos such as "Deterministic Nonperiodic Flow"
by E.N. Lorenz.
I. Hargittai and C. Pickover. Spiral Symmetry 1992 World Scientific
Publishing, River Edge, New Jersey 07661. ISBN 981-02-0615-1. Topics:
Spirals in nature, art, and mathematics. Fractal spirals, plant
spirals, artist's spirals, the spiral in myth and literature... Loads
of images.
H. Jurgens, H. O Peitgen, & D. Saupe. (1990, August). The Language of
Fractals. Scientific American, pp. 60-67.
H. Jurgens, H. O. Peitgen, H.O., & D. Saupe. (1992). Chaos and
Fractals: New Frontiers of Science. New York: Springer-Verlag.
S. Levy, Artificial life : the quest for a new creation, Pantheon
Books, New York, 1992. This book takes off where Gleick left off. It
looks at many of the same people and what they are doing post-Gleick.
B. Mandelbrot, The Fractal Geometry of Nature, W. H. FreeMan, New
York. ISBN 0-7167-1186-9. In this book Mandelbrot attempts to show
that reality is fractal-like. He also has pictures of many different
fractals.
E. R. Mac Cormac(Ed), M. Stamenov(Ed), Fractals of Brain, Fractals of
Mind : In Searchg of a Symmetry Bond (Advances in Consciousness
Research, No 7), John Benjamins, ISBN: 1556191871, Subjects include:
Neural networks (Neurobiology), Mathematical models, Fractals, and
Consciousness
G.V. Middleton, (ed.), 1991: Nonlinear Dynamics, Chaos and Fractals
(w/ application to geological systems) Geol. Assoc. Canada, Short
Course Notes Vol. 9, 235 p. This volume contains a disk with some
examples (also as pascal source code) ($25 CDN)
T.F. Nonnenmacher, G.A Losa, E.R Weibel (ed.) Fractals in Biology and
Medicine Birkhaeuser Verlag
L. Nottale, Fractal Space-Time and Microphysics, Towards a Theory of
Scale Relativity, World Scientific (1993).
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 on the back cover
H. O. Peitgen and P. H. Richter, The Beauty of Fractals,
Springer-Verlag, New York, 1986. ISBN 0-387-15851-0. This book has
lots of nice pictures. There is also an appendix giving the
coordinates and constants for the color plates and many of the other
pictures.
H. Peitgen and D. Saupe, eds., The Science of Fractal Images,
Springer-Verlag, New York, 1988. ISBN 0-387-96608-0. This book
contains many color and black and white photographs, high level math,
and several pseudocoded algorithms.
H. Peitgen, H. Juergens and D. Saupe, Fractals for the Classroom,
Springer-Verlag, New York, 1992. These two volumes are aimed at
advanced secondary school students (but are appropriate for others
too), have lots of examples, explain the math well, and give BASIC
programs.
H. Peitgen, H. Juergens and D. Saupe, Chaos and Fractals: New
Frontiers of Science, Springer-Verlag, New York, 1992.
C. Pickover, Computers, Pattern, Chaos, and Beauty: Graphics from an
Unseen World, St. Martin's Press, New York, 1990. This book contains a
bunch of interesting explorations of different fractals.
C. Pickover, Keys to Infinity, (1995) John Wiley: NY. ISBN
0-471-11857-5.
C. Pickover, (1995) Chaos in Wonderland: Visual Adventures in a
Fractal World. St. Martin's Press: New York. ISBN 0-312-10743-9.
(Devoted to the Lyapunov exponent.)
C. Pickover, Computers and the Imagination (Subtitled: Visual
Adventures from Beyond the Edge) (1993) St. Martin's Press: New York.
C. Pickover. The Pattern Book: Fractals, Art, and Nature (1995) World
Scientific. ISBN 981-02-1426-X Some of the patterns are ultramodern,
while others are centuries old. Many of the patterns are drawn from
the universe of mathematics.
C. Pickover, Visualizing Biological Information (1995) World
Scientific: Singapore, New Jersey, London, Hong Kong.
on the use of computer graphics, fractals, and musical techniques to
find patterns in DNA and amino acid sequences.
J. Pritchard, The Chaos Cookbook: A Practical Programming Guide,
Butterworth-Heinemann, Oxford, 1992. ISBN 0-7506-0304-6. It contains
type in and go listings in BASIC and Pascal. It also eases you into
some of the mathematics of fractals and chaos in the context of
graphical experimentation. So it's more than just a
type-and-see-pictures book, but rather a lab tutorial, especially good
for those with a weak or rusty (or even nonexistent) calculus
background.
P. Prusinkiewicz and A. Lindenmayer, The Algorithmic Beauty of Plants,
Springer-Verlag, NY, 1990. ISBN 0-387-97297-8. A very good book on
L-systems, which can be used to model plants in a very realistic
fashion. The book contains many pictures.
Edward R. Scheinerman, Invitation to Dynamical Systems, Prentice-Hall,
1996, ISBN 0-13-185000-8, xvii + 373 pages
M. Schroeder, Fractals, Chaos, and Power Laws: Minutes from an
Infinite Paradise, W. H. Freeman, New York, 1991. This book contains a
clearly written explanation of fractal geometry with lots of puns and
word play.
J. Sprott, Strange Attractors: Creating Patterns in Chaos, M&T Books
(subsidary of Henry Holt and Co.), New York. ISBN 1-55851-298-5. This
book describes a new method for generating beautiful fractal patterns
by iterating simple maps and ordinary differential equations. It
contains over 350 examples of such patterns, each producing a
corresponding piece of fractal music. It also describes methods for
visualizing objects in three and higher dimensions and explains how to
produce 3-D stereoscopic images using the included red/blue glasses.
The accompanying 3.5" IBM-PC disk contain source code in BASIC, C,
C++, Visual BASIC for Windows, and QuickBASIC for Macintosh as well as
a ready-to-run IBM-PC executable version of the program. Available for
$39.95 + $3.00 shipping from M&T Books (1-800-628-9658).
D. Stein, ed., Proceedings of the Santa Fe Institute's Complex Systems
Summer School, Addison-Wesley, Redwood City, CA, 1988. See especially
the first article by David Campbell: "Introduction to nonlinear
phenomena".
R. Stevens, Fractal Programming in C, M&T Publishing, 1989 ISBN
1-55851-038-9. This is a good book for a beginner who wants to write a
fractal program. Half the book is on fractal curves like the Hilbert
curve and the von Koch snow flake. The other half covers the
Mandelbrot, Julia, Newton, and IFS fractals.
I. Stewart, Does God Play Dice?: the Mathematics of Chaos, B.
Blackwell, New York, 1989.
Y. Takahashi, Algorithms, Fractals, and Dynamics, Plenum Pub Corp,
(May) 1996, ISBN: 0306451271 Subjects: Differentiable dynamical syste,
Congresses, Fractals, Algorithms, Differentiable Dynamical Systems,
Algorithms (Computer Programming)
T. Wegner and B. Tyler, Fractal Creations, 2nd ed. The Waite Group,
1993. ISBN 1-878739-34-4 This is the book describing the Fractint
program.
Q28b: What are some relevant journals?
A28b: Some relevant journals are:
"Chaos and Graphics" section in the quarterly journal Computers and
Graphics. This contains recent work in fractals from the graphics
perspective, and usually contains several exciting new ideas.
"Mathematical Recreations" section by I. Stewart in Scientific
American.
Fractal Report. Reeves Telecommunication Labs.
West Towan House, Porthtowan, TRURO, Cornwall TR4 8AX, U.K.
WWW: http://ourworld.compuserve.com/homepages/JohndeR/fractalr
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: 100112.1700@compuserve.com
Fractals, An interdisciplinary Journal On The Complex Geometry of
Nature
This is a new journal published by World Scientific. B.B Mandelbrot is
the Honorary Editor and T. Vicsek, M.F. Shlesinger, M.M Matsushita are
the Managing Editors). The aim of this first international journal on
fractals is to bring together the most recent developments in the
research of fractals so that a fruitful interaction of the various
approaches and scientific views on the complex spatial and temporal
behavior could take place.
Q28c: What are some other Internet references?
A28c: Some other Internet references:
Web references to nonlinear dynamics
Dynamical Systems (G. Zito)
http://alephwww.cern.ch/~zito/chep94sl/sd.html
Scanning huge number of events (G. Zito)
http://alephwww.cern.ch/~zito/chep94sl/chep94sl.html
The Who Is Who Handbook of Nonlinear Dynamics
http://www.nonlin.tu-muenchen.de/chaos/Dokumente/WiW/wiw.html
_________________________________________________________________
Multifractals
Q29: What are multifractals? NEW
A29: It is not easy to give a succinct definition of multifractals.
Following Feder (1988) one may distinguish a measure (of probability,
or some physical quantity) from its geometric support -- which might
or might not have fractal geometry. Then if the measure has different
fractal dimension on different parts of the support, the measure is a
multifractal.
Hastings and Sugihara (1993) distinguish multifractals from
multiscaling fractals -- which have different fractal dimensions at
different scales (e.g. show a break in slope in a dividers plot, or
some other power law). I believe different authors use different names
for this phenomenon, which is often confused with true multifractal
behaviour.
______________________________________________________________________
Subject: Notices
Q30: Are there any special notices? NEW
From: dfinton@ub.d.umn.edu (David Finton)
1. Well, I've been doing some computation of fractals, and thought,
"You know, it would be really cool if there was another fractal
art contest." So I thought I'd coordinate the next contest (giving
Tim Wegner a break :). Here are the contest rules I propose:
2. The files must be in PAR file format. If it turns out you can only
post a GIF or JPEG, post it on a binaries newsgroup, and post the
location of the image on here. Do not post binary images like GIFs
or JPEGs on this newsgroup!
3. Mandelbrot images only (from the equation z <- z² + c). Apologies
to all the talented formula fractal artists out there, but I
wanted to narrow down the parameters of the contest to make it
easier to judge. :)
4. Deep Zoom. Yes that's right. I'd like to see some deep zoom
fractals out there, and if anybody has found something interesting
or amazing hidden in the depths of the Mandelbrot Set, please post
them!
Remember, originality counts. It would be nice to see something I
would have never found rather than to see a generic image that
everyone and their grandmother could have found. I hope I get a lot of
responses out of this. Thanks!
- Dave
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
one-page summary.
Chaos Data Analyzer requires an IBM PC or compatible with at least
512K of memory. A math coprocessor is recommended (but not required)
to speed some of the calculations. The program is available on 5.25 or
3.5" disk and includes a 62-page User's Manual. Chaos Data Analyzer is
peer-reviewed software published by Physics Academic Software, a
cooperative Project of the American Institute of Physics, the American
Physical Society, And the American Association of Physics Teachers.
Chaos Data Analyzer and other related programs are available from The
Academic Software Library, North Carolina State University, Box 8202,
Raleigh, NC 27695-8202, Tel: (800) 955-TASL or (919) 515-7447 or Fax:
(919) 515-2682. The price is $99.95. Add $3.50 for shipping in U.S. or
$12.50 for foreign airmail. All TASL programs come with a 30-day,
money-back guarantee.
_________________________________________________________________
Subject: Acknowledgements
Q31: Who has contributed to the Fractal FAQ?
A31: 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, special thanks go to:
Alex Antunes, Simon Arthur, John Beale, Steve Bondeson, Erik Boman,
Jacques Carette, John Corbit, Predrag Cvitanovic, Paul Derbyshire,
John de Rivaz, Abhijit Deshmukh, Tony Dixon, Jürgen Dollinger, Robert
Drake, Detlev Droege, Gerald Edgar, Gordon Erlebacher, Yuval Fisher,
Duncan Foster, David Fowler, Murray Frank, Jean-loup Gailly, Noel
Giffin, Frode Gill, Earl Glynn, Lamont Granquist, John Holder, Jon
Horner, Luis Hernandez- Urëa, Jay Hill, Arto Hoikkala, Carl Hommel,
Robert Hood, Larry Husch, Oleg Ivanov, Simon Juden, J. Kai-Mikael,
Leon Katz, Matt Kennel, Robert Klep, Dave Kliman, Tal Kubo, Jon Leech,
Otmar Lendl, 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, Rollo Silver, J. C. Sprott, Ken Shirriff, Gerolf
Starke, Bruce Stewart, Dwight Stolte, Tommy Vaske, Tim Wegner, Andrea
Whitlock, Erick Wong, Wayne Young, Giuseppe Zito, and others.
Special thanks to Matthew J. Bernhardt (mjb@acsu.buffalo.edu) for
collecting many of the chaos definitions.
If I have missed you, I am very sorry, let me know and I will add you
to the list. Without the help of these contributors, the sci.fractals
FAQ would be not be possible.
_________________________________________________________________
Subject: Copyright
Q32: Copyright?
A32: This document, "sci.fractals FAQ", is Copyright 1995-1996 by
Michael C. Taylor. All Rights Reserved.
Previous versions:
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. Standing permission is granted for non-profit
reproduction and distribution of this issue of the sci.fractals FAQ as
a complete document. This does not mean automatic permission for usage
in CD-ROM collections or commerical educational products. If you would
like to include sci.fractals FAQ in a commerical product, in whole or
in part, contact Michael Taylor. If you would like to send a review
sample of a program, or books, feel free send them to the editor:
* Michael Taylor
* P.O. Box 36
* Centreville (Kings)
* Nova Scotia, B0P 1J0
* CANADA
email:
* aa459@chebucto.ns.ca
* mctaylor@mailserv.mta.ca (until August 1996)