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- From: mctaylor@mta.ca (Michael C Taylor (CSD))
- Newsgroups: sci.fractals,sci.answers,news.answers
- Subject: sci.fractals FAQ
- Followup-To: poster
- Date: 9 Mar 1998 00:16:03 GMT
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- Summary: Frequently Asked Questions about Fractals
- Keywords: fractals Mandelbrot Julia chaos IFS
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- Archive-name: sci/fractals-faq
- Posting-Frequency: monthly
- Last-modified: March 8, 1998
- Version: v5n3
- URL: http://www.mta.ca/~mctaylor/sci.fractals-faq/
- Copyright: Copyright 1997-1998 by Michael C. Taylor and Jean-Pierre Louvet
- Maintainer: Michael C. Taylor and Jean-Pierre Louvet
-
- sci.fractals FAQ (Frequently Asked Questions)
-
-
-
- _________________________________________________________________
-
- _Volume_ 5 _Number_ 3
- _Date_ March 8, 1998
-
- _________________________________________________________________
-
- _Copyright_ 1997-1998 by Michael C. Taylor and Jean-Pierre Louvet. All
- Rights Reserved.
-
- Introduction
-
- This FAQ is posted monthly to sci.fractals, a Usenet newsgroup about
- fractals; mathematics and software. This document is aimed at being a
- reference about fractals, including answers to commonly asked
- questions, archive listings of fractal software, images, and papers
- that can be accessed via the Internet using FTP, gopher, or
- World-Wide-Web (WWW), and a bibliography for further readings.
-
- The FAQ does not give a textbook approach to learning about fractals,
- but a summary of information from which you can learn more about and
- explore fractals.
-
- This FAQ is posted monthly to the Usenet newsgroups: sci.fractals
- ("Objects of non-integral dimension and other chaos"), sci.answers,
- and news.answers. Like most FAQs it can be obtained freely with a WWW
- browser (such as Mosaic or Netscape), or by anonymous FTP from
- ftp://rtfm.mit.edu/pub/usenet/news.answers/sci/fractals-faq (USA). It
- is also available from
- ftp://ftp.Germany.EU.net/pub/newsarchive/news.answers/sci/fractals-faq
- .gz (Europe),
- http://graffiti.cribx1.u-bordeaux.fr/MAPBX/louvet/sci.fractals-faq/faq
- .html (France) and http://www.mta.ca/~mctaylor/sci.fractals-faq/
- (Canada).
-
- Those without FTP or WWW access can obtain the FAQ via email, by
- sending a message to mail-server@rtfm.mit.edu with the _message_:
-
- send usenet/news.answers/sci/fractals-faq
-
- _________________________________________________________________
-
- Suggestions, Comments, Mistakes
-
- Please send suggestions and corrections about the sci.fractals FAQ to
- fractal-faq@mta.ca. Without your contributions, the FAQ for
- sci.fractals will not grow in its wealth. _"For the readers, by the
- readers."_ Rather than calling me a fool behind my back, if you find a
- mistake, whether spelling or factual, please send me a note. That way
- readers of future versions of the FAQ will not be misled. Also if you
- have problems with the appearance of the hypertext version. There
- should not be any Netscape only markup tags contained in the hypertext
- verion, but I have not followed strict HTML 3.2 specifications. If the
- appearance is "incorrect" let me know what problems you experience.
-
- Why the different name?
-
- The old Fractal FAQ about fractals _has not been updated for over two
- years_ and has not been posted by Dr. Ermel Stepp, in as long. So this
- is a new FAQ based on the previous FAQ's information and the readers
- of primarily sci.fractals with contributions from the FRAC-L and
- Fractal-Art mailing lists. Thus it is now called the _sci.fractals
- FAQ_.
-
- ______________________________________________________________________
-
-
-
-
-
- Table of contents
- The questions which are answered include:
-
-
-
- Q0: I am new to the 'Net. What should I know about being online?
-
- Q1: I want to learn about fractals. What should I read first? New
-
- Q2: What is a fractal? What are some examples of fractals?
-
- Q3a: What is chaos?
-
- Q3b: Are fractals and chaos synonymous?
-
- Q3c: Are there references to fractals used as financial models?
-
- Q4a: What is fractal dimension? How is it calculated?
-
- Q4b: What is topological dimension?
-
- Q5: What is a strange attractor?
-
- Q6a: What is the Mandelbrot set?
-
- Q6b: How is the Mandelbrot set actually computed?
-
- Q6c: Why do you start with z = 0?
-
- Q6d: What are the bounds of the Mandelbrot set? When does it diverge?
-
- Q6e: How can I speed up Mandelbrot set generation?
-
- Q6f: What is the area of the Mandelbrot set?
-
- Q6g: What can you say about the structure of the Mandelbrot set?
-
- Q6h: Is the Mandelbrot set connected?
-
- Q6i: What is the Mandelbrot Encyclopedia?
-
- Q6j: What is the dimension of the Mandelbrot Set?
-
- Q6k: What are the seahorse and the elephant valleys?
-
- Q6l: What is the relation between Pi and the Mandelbrot Set?
-
- Q7a: What is the difference between the Mandelbrot set and a Julia
- set?
-
- Q7b: What is the connection between the Mandelbrot set and Julia sets?
-
- Q7c: How is a Julia set actually computed?
-
- Q7d: What are some Julia set facts?
-
- Q8a: How does complex arithmetic work?
-
- Q8b: How does quaternion arithmetic work?
-
- Q9: What is the logistic equation?
-
- Q10: What is Feigenbaum's constant?
-
- Q11a: What is an iterated function system (IFS)?
-
- Q11b: What is the state of fractal compression?
-
- Q12a: How can you make a chaotic oscillator?
-
- Q12b: What are laboratory demonstrations of chaos?
-
- Q13: What are L-systems?
-
- Q14: What are sources of fractal music?
-
- Q15: How are fractal mountains generated?
-
- Q16: What are plasma clouds?
-
- Q17a: Where are the popular periodically-forced Lyapunov fractals
- described?
-
- Q17b: What are Lyapunov exponents?
-
- Q17c: How can Lyapunov exponents be calculated?
-
- Q18: Where can I get fractal T-shirts and posters?
-
- Q19: How can I take photos of fractals?
-
- Q20a: What are the rendering methods commonly used for 256-colour
- fractals?
-
- Q20b: How does rendering differ for true-colour fractals??
-
- Q21: How can 3-D fractals be generated?
-
- Q22a: What is Fractint?
-
- Q22b: How does Fractint achieve its speed?
-
- Q23: Where can I obtain software packages to generate fractals? New
-
- Q24a: How does anonymous ftp work?
-
- Q24b: What if I can't use ftp to access files?
-
- Q25a: Where are fractal pictures archived? New
-
- Q25b: How do I view fractal pictures from
- alt.binaries.pictures.fractals?
-
- Q26: Where can I obtain fractal papers?
-
- Q27: How can I join fractal mailing lists? New
-
- Q28: What is complexity?
-
- Q29a: What are some general references on fractals and chaos?
-
- Q29b: What are some relevant journals?
-
- Q29c: What are some other Internet references?
-
- Q30: What is a multifractal?
-
- Q31a: What is aliasing? New
-
- Q31b: What does aliasing have to do with fractals? New
-
- Q31c: How Do I "Anti-Alias" Fractals? New
-
- Q32: Ideas for science fair projects? New
-
- Q33: Are there any special notices?
-
- Q34: Who has contributed to the Fractal FAQ? New
-
- Q35: Copyright? New
-
- ____________________________________________________
-
-
-
-
- Subject: USENET and Netiquette
-
- _Q0_: I am new to sci.fratals. What should I know about being online?
-
- _A0_: There are a couple of common mistakes people make, posting ads,
- posting large binaries (images or programs), and posting off-topic.
-
- _Do Not Post Images to sci.fractals._ If you follow this rule people
- will be your friend. There is a special place for you to post your
- images, _alt.binaries.pictures.fractals_. The other group
- (alt.fractals.pictures) is considered obsolete and may not be carried
- to as many people as _alt.binaries.pictures.fractals._ In fact there
- is/was a CancelBot which will delete any binary posts it finds in
- sci.fractals (and most other non-binaries newsgroup) so nearly no one
- will see it.
-
- _Post only about fractals_, this includes fractal mathematics,
- software to generate fractals, where to get fractal posters and
- T-shirts, and fractals as art. Do not bother posting about news events
- not directly related to fractals, or about which OS / hardware /
- language is better. These lead to flame wars.
-
- _Do not post advertisements._ I should not have to mention this, but
- people get excited. If you have some _fractal_ software (or posters)
- available as shareware or shrink-wrap commercial, post your _brief_
- announcement _once_. After than, you should limit yourself to notices
- of upgrades and responding _via e-mail_ to people looking for fractal
- software.
-
- If you are new to Usenet and/or being online, read the guidelines and
- Frequently Asked Questions (FAQ) in news.announce.newusers. They are
- available from:
-
- Welcome to news.newusers.questions
- ftp://rtfm.mit.edu/pub/usenet/news.answers/news-newusers-intro
- ftp://garbo.uwasa.fi/pc/doc-net/usenews.zip
-
- A Primer on How to Work With the Usenet Community
- ftp://rtfm.mit.edu/pub/usenet/news.answers/usenet/primer/part1
-
- Frequently Asked Questions about Usenet
- ftp://rtfm.mit.edu/pub/usenet/news.answers/usenet/faq/part1
-
- Rules for posting to Usenet
- ftp://rtfm.mit.edu/pub/usenet/news.answers/usenet/posting-rules
- /part1
-
- Emily Postnews Answers Your Questions on Netiquette
- ftp://rtfm.mit.edu/pub/usenet/news.answers/usenet/emily-postnew
- s/part1
-
- Hints on writing style for Usenet
- ftp://rtfm.mit.edu/pub/usenet/news.answers/usenet/writing-style
- /part1
-
- What is Usenet?
- ftp://rtfm.mit.edu/pub/usenet/news.answers/usenet/what-is/part1
-
- Subject: Learning about fractals
-
- _Q1_: I want to learn about fractals. What should I read/view first?
-
- _A1_: _Chaos: Making a New Science_, by James Gleick, is a good book
- to get a general overview and history that does not require an
- extensive math background. _Fractals Everywhere,_ by Michael Barnsley,
- and _Measure Topology and Fractal Geometry_, by G. A. Edgar, are
- textbooks that describe mathematically what fractals are and how to
- generate them, but they requires a college level mathematics
- background. _Chaos, Fractals, and Dynamics_, by R. L. Devaney, is also
- a good start. There is a longer book list at the end of the FAQ (see
- "What are some general references?").
-
- Also, there are networked resources available, such as :
-
- Exploring Fractals and Chaos
- http://www.lib.rmit.edu.au/fractals/exploring.html
-
- Fractal Microscope
- http://www.ncsa.uiuc.edu/Edu/Fractal/Fractal_Home.html
-
- Dynamical Systems and Technology Project: a introduction for
- high-school students
- http://math.bu.edu/DYSYS/dysys.html
-
- An Introduction to Fractals (Written by Paul Bourke)
- http://www.mhri.edu.au/~pdb/fractals/fracintro/
-
- Fractals and Scale (by David G. Green)
- http://life.csu.edu.au/complex/tutorials/tutorial3.html
-
- What are fractals? (by Neal Kettler)
- http://www.vis.colostate.edu/~user1209/fractals/fracinfo.html
-
- Fract-ED a fractal tutorial for beginners, targeted for high
- school/tech school students.
- http://www.ealnet.com/ealsoft/fracted.htm
-
- Mandelbrot Questions & Answers (without any scary details) by Paul
- Derbyshire
- http://chat.carleton.ca/~pderbysh/mandlfaq.html
-
- Godric's fractal gallery. A brief introduction to Fractals clear and
- well illustrated explanations
- http://www.gozen.demon.co.uk/godric/fracgall.html
-
- Lystad Fractal Info complex numbers and fractals
- http://www.iglobal.net/lystad/lystad-fractal-info.html
-
- Fractal eXtreme: fractal theory theoritical informations
- http://www.cygnus-software.com/theory/theory.htm
-
- Frode Gill Fractal pages mathematical and programming data about
- classical fractals and quaternions
- http://www.krs.hia.no/~fgill/fractal.html
-
- Fractals: a history
- http://graffiti.cribx1.u-bordeaux.fr/MAPBX/louvet/history.html
-
- Basic informations about fractals
- http://graffiti.cribx1.u-bordeaux.fr/MAPBX/louvet/jpl1a.html
-
- Fantastic Fractals a very comprehensive site with tutorials for
- beginners and more advanced readers, workshops etc.
- http://library.advanced.org/12740/cgi-bin/welcome.cgi
-
- Chaos, Fractals, Dimension: mathematics in the age of the computer by
- Glenn Elert. A huge (>100 pages double-spaced) essay on
- chaos, fractals, and non-linear dynamics. It requires a
- moderate math background, though is not aimed at the
- mathematician.
- http://www.columbia.edu/~gae4/chaos/
-
- Mathsnet this site has several pages devoted to fractals and complex
- numbers.
-
-
- http://www.anglia.co.uk/education/mathsnet/
-
- Fractals in Your Future by Ronald Lewis <ronlewis@sympatico.ca>
- http://www.eureka.ca/resources/fiyf/fiyf.html
-
- Subject: What is a fractal?
-
- _Q2_: What is a fractal? What are some examples of fractals?
-
- _A2_: A fractal is a rough or fragmented geometric shape that can be
- subdivided in parts, each of which is (at least approximately) a
- reduced-size copy of the whole. Fractals are _generally_ self-similar
- and independent of scale.
-
- There are many mathematical structures that are fractals; e.g.
- Sierpinski triangle, Koch snowflake, Peano curve, Mandelbrot set, and
- Lorenz attractor. Fractals also describe many real-world objects, such
- as clouds, mountains, turbulence, coastlines, roots, branches of
- trees, blood vesels, and lungs of animals, that do not correspond to
- simple geometric shapes.
-
- Benoit B. Mandelbrot gives a mathematical definition of a fractal as a
- set of which the Hausdorff Besicovich dimension strictly exceeds the
- topological dimension. However, he is not satisfied with this
- definition as it excludes sets one would consider fractals.
-
- According to Mandelbrot, who invented the word: "I coined _fractal_
- from the Latin adjective _fractus_. The corresponding Latin verb
- _frangere_ means "to break:" to create irregular fragments. It is
- therefore sensible - and how appropriate for our needs! - that, in
- addition to "fragmented" (as in _fraction_ or _refraction_), _fractus_
- should also mean "irregular," both meanings being preserved in
- _fragment_." (The Fractal Geometry of Nature, page 4.)
-
- Subject: Chaos
-
- _Q3a_: What is chaos?
-
- _A3a_: Chaos is apparently unpredictable behavior arising in a
- deterministic system because of great sensitivity to initial
- conditions. Chaos arises in a dynamical system if two arbitrarily
- close starting points diverge exponentially, so that their future
- behavior is eventually unpredictable.
-
- Weather is considered chaotic since arbitrarily small variations in
- initial conditions can result in radically different weather later.
- This may limit the possibilities of long-term weather forecasting.
- (The canonical example is the possibility of a butterfly's sneeze
- affecting the weather enough to cause a hurricane weeks later.)
-
- Devaney defines a function as chaotic if it has sensitive dependence
- on initial conditions, it is topologically transitive, and periodic
- points are dense. In other words, it is unpredictable, indecomposable,
- and yet contains regularity.
-
- Allgood and Yorke define chaos as a trajectory that is exponentially
- unstable and neither periodic or asymptotically periodic. That is, it
- oscillates irregularly without settling down.
-
- sci.fractals may not be the best place for chaos/non-linear dynamics
- questions, sci.nonlinear newsgroup should be much better.
-
- _Q3b_: Are fractals and chaos synonymous?
-
- _A3b_: No. Many people do confuse the two domains because books or
- papers about chaos speak of the two concepts or are illustrated with
- fractals.
- _Fractals_ and _deterministic chaos_ are mathematical tools to
- modelise different kinds of natural phenomena or objects. _The
- keywords in chaos_ are impredictability, sensitivity to initial
- conditions in spite of the deterministic set of equations describing
- the phenomenon.
-
- On the other hand, _the keywords to fractals are self-similarity,
- invariance of scale_. Many fractals are in no way chaotic (Sirpinski
- triangle, Koch curve...).
-
- However, starting from very differents point of view, the two domains
- have many things in common : many chaotic phenomena exhibit fractals
- structures (in their strange attractors for example... fractal
- structure is also obvious in chaotics phenomena due to successive
- bifurcations ; see for example the logistic equation Q9 )
-
- The following resources may be helpful to understand chaos:
-
- sci.nonlinear FAQ (UK)
- http://www.fen.bris.ac.uk/engmaths/research/nonlinear/faq.html
-
- sci.nonlinear FAQ (US)
- http://amath.colorado.edu/appm/faculty/jdm/faq.html
-
- Exploring Chaos and Fractals
- http://www.lib.rmit.edu.au/fractals/exploring.html
-
- Chaos and Complexity Homepage (M. Bourdour)
- http://www.cc.duth.gr/~mboudour/nonlin.html
-
- The Institute for Nonlinear Science
- http://inls.ucsd.edu/
-
- _Q3c_: Are there references to fractals used as financial models?
-
- _A3c_: Most references are related to chaos being used as a model for
- financial forecasting.
-
- One reference that is about fractal models is, Fractal Market Analysis
- - Applying Chaos Theory to Investment & Economics by Edgar Peters.
-
- Some recommended Chaos-related texts in financial forecasting.
-
- Medio: Chaotic Dynmics - Theory and Applications to Economics
- Cambridge University Press, 1993, ISBN 0-521-48461-8
-
- Vaga: Profiting from Chaos - Using Chaos Theory for Market Timing,
- Stock Selection and Option Valuation
- McGraw-Hill Inc, 1994, ISBN 0-07-066786-1
-
- Subject: Fractal dimension
-
- _Q4a_ : What is fractal dimension? How is it calculated?
-
- _A4a_: A common type of fractal dimension is the Hausdorff-Besicovich
- Dimension, but there are several different ways of computing fractal
- dimension.
-
- Roughly, fractal dimension can be calculated by taking the limit of
- the quotient of the log change in object size and the log change in
- measurement scale, as the measurement scale approaches zero. The
- differences come in what is exactly meant by "object size" and what is
- meant by "measurement scale" and how to get an average number out of
- many different parts of a geometrical object. Fractal dimensions
- quantify the static _geometry_ of an object.
-
- For example, consider a straight line. Now blow up the line by a
- factor of two. The line is now twice as long as before. Log 2 / Log 2
- = 1, corresponding to dimension 1. Consider a square. Now blow up the
- square by a factor of two. The square is now 4 times as large as
- before (i.e. 4 original squares can be placed on the original square).
- Log 4 / log 2 = 2, corresponding to dimension 2 for the square.
- Consider a snowflake curve formed by repeatedly replacing ___ with
- _/\_, where each of the 4 new lines is 1/3 the length of the old line.
- Blowing up the snowflake curve by a factor of 3 results in a snowflake
- curve 4 times as large (one of the old snowflake curves can be placed
- on each of the 4 segments _/\_). Log 4 / log 3 = 1.261... Since the
- dimension 1.261 is larger than the dimension 1 of the lines making up
- the curve, the snowflake curve is a fractal.
-
- For more information on fractal dimension and scale, via the WWW
-
- Fractals and Scale (by David G. Green)
- http://life.csu.edu.au/complex/tutorials/tutorial3.html
-
- Fractal dimension references:
-
- 1. J. P. Eckmann and D. Ruelle, _Reviews of Modern Physics_ 57, 3
- (1985), pp. 617-656.
- 2. K. J. Falconer, _The Geometry of Fractal Sets_, Cambridge Univ.
- Press, 1985.
- 3. T. S. Parker and L. O. Chua, _Practical Numerical Algorithms for
- Chaotic Systems_, Springer Verlag, 1989.
- 4. H. Peitgen and D. Saupe, eds., _The Science of Fractal Images_,
- Springer-Verlag Inc., New York, 1988. ISBN 0-387-96608-0.
- This book contains many color and black and white photographs,
- high level math, and several pseudocoded algorithms.
- 5. G. Procaccia, _Physica D_ 9 (1983), pp. 189-208.
- 6. J. Theiler, _Physical Review A_ 41 (1990), pp. 3038-3051.
-
- References on how to estimate fractal dimension:
-
- 1. S. Jaggi, D. A. Quattrochi and N. S. Lam, Implementation and
- operation of three fractal measurement algorithms for analysis of
- remote-sensing data., _Computers & Geosciences _19, 6 (July 1993),
- pp. 745-767.
- 2. E. Peters, _Chaos and Order in the Capital Markets _, New York,
- 1991. ISBN 0-471-53372-6
- Discusses methods of computing fractal dimension. Includes several
- short programs for nonlinear analysis.
- 3. J. Theiler, Estimating Fractal Dimension, _Journal of the Optical
- Society of America A-Optics and Image Science_ 7, 6 (June 1990),
- pp. 1055-1073.
-
- There are some programs available to compute fractal dimension. They
- are listed in a section below (see Q22 "Fractal software").
-
- Reference on the Hausdorff-Besicovitch dimension
-
- A clear and concise (2 page) write-up of the definition of the
- Hausdorff-Besicovitch dimension in MS-Word 6.0 format is available in
- zip format.
-
- hausdorff.zip (~26KB)
- http://www.newciv.org/jhs/hausdorff.zip
-
- _Q4b_ : What is topological dimension?
-
- _A4b_: Topological dimension is the "normal" idea of dimension; a
- point has topological dimension 0, a line has topological dimension 1,
- a surface has topological dimension 2, etc.
-
- For a rigorous definition:
- A set has topological dimension 0 if every point has arbitrarily small
- neighborhoods whose boundaries do not intersect the set.
-
- A set S has topological dimension k if each point in S has arbitrarily
- small neighborhoods whose boundaries meet S in a set of dimension k-1,
- and k is the least nonnegative integer for which this holds.
-
- Subject: Strange attractors
-
- _Q5_: What is a strange attractor?
-
- _A5_: A strange attractor is the limit set of a chaotic trajectory. A
- strange attractor is an attractor that is topologically distinct from
- a periodic orbit or a limit cycle. A strange attractor can be
- considered a fractal attractor. An example of a strange attractor is
- the Henon attractor.
-
- Consider a volume in phase space defined by all the initial conditions
- a system may have. For a dissipative system, this volume will shrink
- as the system evolves in time (Liouville's Theorem). If the system is
- sensitive to initial conditions, the trajectories of the points
- defining initial conditions will move apart in some directions, closer
- in others, but there will be a net shrinkage in volume. Ultimately,
- all points will lie along a fine line of zero volume. This is the
- strange attractor. All initial points in phase space which ultimately
- land on the attractor form a Basin of Attraction. A strange attractor
- results if a system is sensitive to initial conditions and is not
- conservative.
-
- Note: While all chaotic attractors are strange, not all strange
- attractors are chaotic.
-
- Reference:
-
- 1. Grebogi, et al., Strange Attractors that are not Chaotic, _Physica
- D_ 13 (1984), pp. 261-268.
-
- Subject: The Mandelbrot set
-
- _Q6a_ : What is the Mandelbrot set?
-
- _A6a_: The Mandelbrot set is the set of all complex _c_ such that
- iterating _z_ -> _z^2_ + _c_ does not go to infinity (starting with _z_
- = 0).
-
- Other images and resources are:
-
- Frank Rousell's hyperindex of clickable/retrievable Mandelbrot images
- http://www.cnam.fr/fractals/mandel.html
-
- Neal Kettler's Interactive Mandelbrot
- http://www.vis.colostate.edu/~user1209/fractals/explorer/
-
- Panagiotis J. Christias' Mandelbrot Explorer
- http://www.softlab.ntua.gr/mandel/mandel.html
-
- 2D & 3D Mandelbrot fractal explorer (set up by Robert Keller)
- http://reality.sgi.com/employees/rck/hydra/
-
- Mandelbrot viewer written in Java (by Simon Arthur)
- http://www.mindspring.com/~chroma/mandelbrot.html
-
- Mandelbrot Questions & Answers (without any scary details) by Paul
- Derbyshire
- http://chat.carleton.ca/~pderbysh/mandlfaq.html
-
- Quick Guide to the Mandelbrot Set (includes a tourist map) by Paul
- Derbyshire
- http://chat.carleton.ca/~pderbysh/manguide.html
-
- The Mandelbrot Set by Eric Carr
- http://www.cs.odu.edu/~carr/fractals/mandelbr.html
-
- Java program to view the Mandelbrot Set by Ken Shirriff
- http://www.sunlabs.com/~shirriff/java/
-
- Mu-Ency The Encyclopedia of the Mandelbrot Set by Robert Munafo
- http://home.earthlink.net/~mrob/muency.html
-
- _Q6b_ : How is the Mandelbrot set actually computed?
-
- _A6b_: The basic algorithm is: For each pixel c, start with z = 0.
- Repeat z = z^2 + c up to N times, exiting if the magnitude of z gets
- large. If you finish the loop, the point is probably inside the
- Mandelbrot set. If you exit, the point is outside and can be colored
- according to how many iterations were completed. You can exit if
- |z| > 2, since if z gets this big it will go to infinity. The maximum
- number of iterations, N, can be selected as desired, for instance 100.
- Larger N will give sharper detail but take longer.
-
- Frode Gill has some information about generating the Mandelbrot Set at
- http://www.krs.hia.no/~fgill/mandel.html.
-
- _Q6c_ : Why do you start with z = 0?
-
- _A6c_: Zero is the critical point of z = z^2 + c, that is, a point
- where d/dz (z^2 + c) = 0. If you replace z^2 + c with a different
- function, the starting value will have to be modified. E.g. for z ->
- z^2 + z, the critical point is given by 2z + 1 = 0, so start with
- z = -0.5. In some cases, there may be multiple critical values, so
- they all should be tested.
-
- Critical points are important because by a result of Fatou: every
- attracting cycle for a polynomial or rational function attracts at
- least one critical point. Thus, testing the critical point shows if
- there is any stable attractive cycle. See also:
-
- 1. M. Frame and J. Robertson, A Generalized Mandelbrot Set and the
- Role of Critical Points, _Computers and Graphics_ 16, 1 (1992),
- pp. 35-40.
-
- Note that you can precompute the first Mandelbrot iteration by
- starting with z = c instead of z = 0, since 0^2 + c = c.
-
- _Q6d_: What are the bounds of the Mandelbrot set? When does it
- diverge?
-
- _A6d_: The Mandelbrot set lies within |c| <= 2. If |z| exceeds 2, the
- z sequence diverges.
- Proof: if |z| > 2, then |z^2 + c| >= |z^2| - |c| > 2|z| - |c|. If
- |z| >= |c|, then 2|z| - |c| > |z|. So, if |z| > 2 and |z| >= c, then
- |z^2 + c| > |z|, so the sequence is increasing. (It takes a bit more
- work to prove it is unbounded and diverges.) Also, note that |z| = c,
- so if |c| > 2, the sequence diverges.
-
- _Q6e_ : How can I speed up Mandelbrot set generation?
-
- _A6e_: See the information on speed below (see "Fractint"). Also see:
-
- 1. R. Rojas, A Tutorial on Efficient Computer Graphic Representations
- of the Mandelbrot Set, _Computers and Graphics_ 15, 1 (1991), pp.
- 91-100.
-
- _Q6f_: What is the area of the Mandelbrot set?
-
- _A6f_: Ewing and Schober computed an area estimate using 240,000 terms
- of the Laurent series. The result is 1.7274... However, the Laurent
- series converges very slowly, so this is a poor estimate. A project to
- measure the area via counting pixels on a very dense grid shows an
- area around 1.5066. (Contact rpm%mrob.uucp@spdcc.com for more
- information.) Hill and Fisher used distance estimation techniques to
- rigorously bound the area and found the area is between 1.503 and
- 1.5701. Jay Hill's latest results using Root Solving and Component
- Series Evaluation shows the area is at least 1.506302 and less than
- 1.5613027. See Fractal Horizons edited by Cliff Pickover and Hill's
- home page for details about his work.
-
- References:
-
- 1. J. H. Ewing and G. Schober, The Area of the Mandelbrot Set,
- _Numer. Math._ 61 (1992), pp. 59-72.
- 2. Y. Fisher and J. Hill, Bounding the Area of the Mandelbrot Set,
- _Numerische Mathematik,_. (Submitted for publication). Available
- via
-
- World Wide Web (in Postscript format)
- http://inls.ucsd.edu/y/Complex/area.ps.Z.
-
- 3. Jay Hill's Home page which includes his latest updates.
-
- Jay's Hill Home Page via the World Wide Web.
- http://www.geocities.com/CapeCanaveral/Lab/3825/
-
- _Q6g_: What can you say about the structure of the Mandelbrot set?
-
- _A6g_: Most of what you could want to know is in Branner's article in
- _Chaos and Fractals: The Mathematics Behind the Computer Graphics_.
-
- Note that the Mandelbrot set in general is _not_ strictly
- self-similar; the tiny copies of the Mandelbrot set are all slightly
- different, mainly because of the thin threads connecting them to the
- main body of the Mandelbrot set. However, the Mandelbrot set is
- quasi-self-similar. However, the Mandelbrot set is self-similar under
- magnification in neighborhoods of Misiurewicz points (e.g.
- -.1011 + .9563i). The Mandelbrot set is conjectured to be self-similar
- around generalized Feigenbaum points (e.g. -1.401155 or
- -.1528 + 1.0397i), in the sense of converging to a limit set.
-
- References:
-
- 1. T. Lei, Similarity between the Mandelbrot set and Julia Sets,
- _Communications in Mathematical Physics_ 134 (1990), pp. 587-617.
- 2. J. Milnor, Self-Similarity and Hairiness in the Mandelbrot Set, in
- _Computers in Geometry and Topology_, M. Tangora (editor), Dekker,
- New York, pp. 211-257.
-
- The "external angles" of the Mandelbrot set (see Douady and Hubbard or
- brief sketch in "Beauty of Fractals") induce a Fibonacci partition
- onto it.
-
- The boundary of the Mandelbrot set and the Julia set of a generic c in
- M have Hausdorff dimension 2 and have topological dimension 1. The
- proof is based on the study of the bifurcation of parabolic periodic
- points. (Since the boundary has empty interior, the topological
- dimension is less than 2, and thus is 1.)
-
- Reference:
-
- 1. M. Shishikura, The Hausdorff Dimension of the Boundary of the
- Mandelbrot Set and Julia Sets, The paper is available from
- anonymous ftp: ftp://math.sunysb.edu/preprints/ims91-7.ps.Z
-
- _Q6h_: Is the Mandelbrot set connected?
-
- _A6h_: The Mandelbrot set is simply connected. This follows from a
- theorem of Douady and Hubbard that there is a conformal isomorphism
- from the complement of the Mandelbrot set to the complement of the
- unit disk. (In other words, all equipotential curves are simple closed
- curves.) It is conjectured that the Mandelbrot set is locally
- connected, and thus pathwise connected, but this is currently
- unproved.
-
- Connectedness definitions:
- Connected: X is connected if there are no proper closed subsets A and
- B of X such that A union B = X, but A intersect B is empty. I.e. X is
- connected if it is a single piece.
-
- Simply connected: X is simply connected if it is connected and every
- closed curve in X can be deformed in X to some constant closed curve.
- I.e. X is simply connected if it has no holes.
-
- Locally connected: X is locally connected if for every point p in X,
- for every open set U containing p, there is an open set V containing p
- and contained in the connected component of p in U. I.e. X is locally
- connected if every connected component of every open subset is open in
- X. Arcwise (or path) connected: X is arcwise connected if every two
- points in X are joined by an arc in X.
-
- (The definitions are from _Encyclopedic Dictionary of Mathematics_.)
-
- Reference:
- Douady, A. and Hubbard, J., "Comptes Rendus" (Paris) 294, pp.123-126,
- 1982.
-
- _Q6i_: What is the Mandelbrot Encyclopedia?
-
- _A6i_: The Mandelbrot Encyclopedia is a web page by Robert Munafo
- <rpm%mrob.uucp@spdcc.com> about the Mandelbrot Set. It is available
- via WWW at <http://home.earthlink.net/~mrob/muency.html>.
-
- _Q6j_: What is the dimension of the Mandelbrot Set?
-
- _A6j_: The Mandelbrot Set has a dimension of 2. The Mandelbrot Set
- contains and is contained in a disk. A disk has a dimension of 2, thus
- so does the Mandelbrot Set.
-
- The Koch snowflake (Hausdorff dimension 1.2619...) does not satisfy
- this condition because it is a thin boundary curve, thus containing no
- disk. If you add the region inside the curve then it does have
- dimension of 2.
-
- The boundary of the Mandelbrot set and the Julia set of a generic c in
- M have Hausdorff dimension 2 and have topological dimension 1. The
- proof is based on the study of the bifurcation of parabolic periodic
- points. (Since the boundary has empty interior, the topological
- dimension is less than 2, and thus is 1.) See reference above
-
- _Q6k_: What are the seahorse and the elephant valleys?
-
- _A6k_: The Mandelbrot set being the most famous fractal, its various
- regions are well known and many of them have popular names evoking
- graphic details found by zooming into them. The seahorse valley is the
- limit border of the main cardioid at the negative side of the x axis
- (near to x=-0.75, y=0.0). You can see here convoluted and complex buds
- looking more or less like seahorses. The elephant valley is near the
- symetry plane on the positive side of the x axis (x=0.25, y=0.0).
- Spirals protuding from the border evoke trunks of elephants. By
- zooming in these regions many interesting structures can be seen.
-
- A nice guide (by Paul Derbyshire) to explore the various regions of
- the Mandelbrot set can be found at :
-
- http://chat.carleton.ca/~pderbysh/manguide.htlm
-
- Subject: Julia sets
-
- _Q7a_: What is the difference between the Mandelbrot set and a Julia
- set?
-
- _A7a_: The Mandelbrot set iterates z^2 + c with z starting at 0 and
- varying c. The Julia set iterates z^2 + c for fixed c and varying
- starting z values. That is, the Mandelbrot set is in parameter space
- (c-plane) while the Julia set is in dynamical or variable space
- (z-plane).
-
- _Q7b_: What is the connection between the Mandelbrot set and Julia
- sets?
-
- _A7b_: Each point c in the Mandelbrot set specifies the geometric
- structure of the corresponding Julia set. If c is in the Mandelbrot
- set, the Julia set will be connected. If c is not in the Mandelbrot
- set, the Julia set will be a Cantor dust.
-
- _Q7c_: How is a Julia set actually computed?
-
- _A7c_: The Julia set can be computed by iteration similar to the
- Mandelbrot computation. The only difference is that the c value is
- fixed and the initial z value varies.
-
- Alternatively, points on the boundary of the Julia set can be computed
- quickly by using inverse iterations. This technique is particularly
- useful when the Julia set is a Cantor Set. In inverse iteration, the
- equation z1 = z0^2 + c is reversed to give an equation for z0: z0 =
- ▒sqrt(z1 - c). By applying this equation repeatedly, the resulting
- points quickly converge to the Julia set boundary. (At each step,
- either the positive or negative root is randomly selected.) This is a
- nonlinear iterated function system.
-
- In pseudocode:
-
- z = 1 (or any value)
- loop
- if (random number < .5) then
- z = sqrt(z - c)
- else
- z = -sqrt(z - c)
- endif
- plot z
- end loop
-
- _Q7d_: What are some Julia set facts?
-
- _A7d_: The Julia set of any rational map of degree greater than one is
- perfect (hence in particular uncountable and nonempty), completely
- invariant, equal to the Julia set of any iterate of the function, and
- also is the boundary of the basin of attraction of every attractor for
- the map.
-
- Julia set references:
-
- 1. A. F. Beardon, _Iteration of Rational Functions : Complex Analytic
- Dynamical Systems_, Springer-Verlag, New York, 1991.
- 2. P. Blanchard, Complex Analytic Dynamics on the Riemann Sphere,
- _Bull. of the Amer. Math. Soc_ 11, 1 (July 1984), pp. 85-141.
-
- This article is a detailed discussion of the mathematics of iterated
- complex functions. It covers most things about Julia sets of rational
- polynomial functions.
-
- Subject: Complex arithmetic and quaternion arithmetic
-
- _Q8a_: How does complex arithmetic work?
-
- _A8a_: It works mostly like regular algebra with a couple additional
- formulas:
- (note: a, b are reals, _x_, _y_ are complex, _i_ is the square root of
- -1)
-
- Powers of _i_:
- _i_^2 = -1
-
- Addition:
- (a+_i_*b)+(c+_i_*d) = (a+c)+_i_*(b+d)
-
- Multiplication:
- (a+_i_*b)*(c+_i_*d) = a*c-b*d + _i_*(a*d+b*c)
-
- Division:
- (a+_i_*b) / (c+_i_*d) = (a+_i_*b)*(c-_i_*d) / (c^2+d^2)
-
- Exponentiation:
- exp(a+_i_*b) = exp(a)*(cos(b)+_i_*sin(b))
-
- Sine:
- sin(_x_) = (exp(_i_*_x_) - exp(-_i_*_x_)) / (2*_i_)
-
- Cosine:
- cos(_x_) = (exp(_i_*_x_) + exp(-_i_*_x_)) / 2
-
- Magnitude:
- |a+_i_*b| = sqrt(a^2+b^2)
-
- Log:
- log(a+_i_*b) = log(|a+_i_*b|)+_i_*arctan(b / a) (Note: log is
- multivalued.)
-
- Log (polar coordinates):
- log(r e^(_i_*a)) = log(r)+_i_*a
-
- Complex powers:
- _x_^y = exp(y*log(x))
-
- de Moivre's theorem:
- _x_^n = r^n [cos(n*a) + _i_*sin(n*a)] (where n is an integer)
-
- More details can be found in any complex analysis book.
-
- _Q8b_: How does quaternion arithmetic work?
-
- _A8b_: quaternions have 4 components (a + _i_b + _j_c + _k_d) compared
- to the two of complex numbers. Operations such as addition and
- multiplication can be performed on quaternions, but multiplication is
- not commutative.
-
- Quaternions satisfy the rules
-
- * i^2 = j^2 = k^2 = -1
- * ij = -ji = k
- * jk = -kj = i,
- * ki = -ik = j
-
- See:
-
- Frode Gill's quaternions page
- http://www.krs.hia.no/~fgill/quatern.html
-
- Subject: Logistic equation
-
- _Q9_: What is the logistic equation?
-
- _A9_: It models animal populations. The equation is x -> c x (1 - x),
- where x is the population (between 0 and 1) and c is a growth
- constant. Iteration of this equation yields the period doubling route
- to chaos. For c between 1 and 3, the population will settle to a fixed
- value. At 3, the period doubles to 2; one year the population is very
- high, causing a low population the next year, causing a high
- population the following year. At 3.45, the period doubles again to 4,
- meaning the population has a four year cycle. The period keeps
- doubling, faster and faster, at 3.54, 3.564, 3.569, and so forth. At
- 3.57, chaos occurs; the population never settles to a fixed period.
- For most c values between 3.57 and 4, the population is chaotic, but
- there are also periodic regions. For any fixed period, there is some c
- value that will yield that period. See _An Introduction to Chaotic
- Dynamical Systems_, by R. L. Devaney, for more information.
-
- Subject: Feigenbaum's constant
-
- _Q10_: What is Feigenbaum's constant?
-
- _A10_: In a period doubling cascade, such as the logistic equation,
- consider the parameter values where period-doubling events occur (e.g.
- r[1]=3, r[2]=3.45, r[3]=3.54, r[4]=3.564...). Look at the ratio of
- distances between consecutive doubling parameter values; let delta[n]
- = (r[n+1]-r[n])/(r[n+2]-r[n+1]). Then the limit as n goes to infinity
- is Feigenbaum's (delta) constant.
-
- Based on computations by F. Christiansen, P. Cvitanovic and H.H. Rugh,
- it has the value 4.6692016091029906718532038... _Note_: several books
- have published incorrect values starting 4.6692016_6_...; the last
- repeated 6 is a _typographical error_.
-
- The interpretation of the delta constant is as you approach chaos,
- each periodic region is smaller than the previous by a factor
- approaching 4.669...
-
- Feigenbaum's constant is important because it is the same for any
- function or system that follows the period-doubling route to chaos and
- has a one-hump quadratic maximum. For cubic, quartic, etc. there are
- different Feigenbaum constants.
-
- Feigenbaum's alpha constant is not as well known; it has the value
- 2.50290787509589282228390287272909. This constant is the scaling
- factor between x values at bifurcations. Feigenbaum says,
- "Asymptotically, the separation of adjacent elements of period-doubled
- attractors is reduced by a constant value [alpha] from one doubling to
- the next". If d[a] is the algebraic distance between nearest elements
- of the attractor cycle of period 2^a, then d[a]/d[a+1] converges to
- -alpha.
-
- References:
-
- 1. K. Briggs, How to calculate the Feigenbaum constants on your PC,
- _Aust. Math. Soc. Gazette_ 16 (1989), p. 89.
- 2. K. Briggs, A precise calculation of the Feigenbaum constants,
- _Mathematics of Computation_ 57 (1991), pp. 435-439.
- 3. K. Briggs, G. R. W. Quispel and C. Thompson, Feigenvalues for
- Mandelsets, _J. Phys. A_ 24 (1991), pp. 3363-3368.
- 4. F. Christiansen, P. Cvitanovic and H.H. Rugh, "The spectrum of the
- period-doubling operator in terms of cycles", _J. Phys A_ 23, L713
- (1990).
- 5. M. Feigenbaum, The Universal Metric Properties of Nonlinear
- Transformations, _J. Stat. Phys_ 21 (1979), p. 69.
- 6. M. Feigenbaum, Universal Behaviour in Nonlinear Systems, _Los
- Alamos Sci_ 1 (1980), pp. 1-4. Reprinted in _Universality in
- Chaos_, compiled by P. Cvitanovic.
-
- Feigenbaum Constants
- http://www.mathsoft.com/asolve/constant/fgnbaum/fgnbaum.html
-
- Subject: Iterated function systems and compression
-
- _Q11a_: What is an iterated function system (IFS)?
-
- _A11a_: If a fractal is self-similar, you can specify mappings that
- map the whole onto the parts. Iteration of these mappings will result
- in convergence to the fractal attractor. An IFS consists of a
- collection of these (usually affine) mappings. If a fractal can be
- described by a small number of mappings, the IFS is a very compact
- description of the fractal. An iterated function system is By taking a
- point and repeatedly applying these mappings you end up with a
- collection of points on the fractal. In other words, instead of a
- single mapping x -> F(x), there is a collection of (usually affine)
- mappings, and random selection chooses which mapping is used.
-
- For instance, the Sierpinski triangle can be decomposed into three
- self-similar subtriangles. The three contractive mappings from the
- full triangle onto the subtriangles forms an IFS. These mappings will
- be of the form "shrink by half and move to the top, left, or right".
-
- Iterated function systems can be used to make things such as fractal
- ferns and trees and are also used in fractal image compression.
- _Fractals Everywhere_ by Barnsley is mostly about iterated function
- systems.
-
- The simplest algorithm to display an IFS is to pick a starting point,
- randomly select one of the mappings, apply it to generate a new point,
- plot the new point, and repeat with the new point. The displayed
- points will rapidly converge to the attractor of the IFS.
-
- Interactive IFS Playground (Otmar Lendl)
- http://www.cosy.sbg.ac.at/rec/ifs/
-
- Frank Rousell's hyperindex of IFS images
- http://www.cnam.fr/fractals/ifs.html
-
- _Q11b_: What is the state of fractal compression?
-
- _A11b_: Fractal compression is quite controversial, with some people
- claiming it doesn't work well, and others claiming it works
- wonderfully. The basic idea behind fractal image compression is to
- express the image as an iterated function system (IFS). The image can
- then be displayed quickly and zooming will generate infinite levels of
- (synthetic) fractal detail. The problem is how to efficiently generate
- the IFS from the image. Barnsley, who invented fractal image
- compression, has a patent on fractal compression techniques
- (4,941,193). Barnsley's company, Iterated Systems Inc
- (http://www.iterated.com/), has a line of products including a Windows
- viewer, compressor, magnifier program, and hardware assist board.
-
- Fractal compression is covered in detail in the comp.compression FAQ
- file (See "compression-FAQ").
- ftp://rtfm.mit.edu/pub/usenet/comp.compression .
-
- One of the best online references for Fractal Compress is Yuval
- Fisher's Fractal Image Encoding page
- (http://inls.ucsd.edu/y/Fractals/) at the Institute for Nonlinear
- Science, University for California, San Diego. It includes references
- to papers, other WWW sites, software, and books about Fractal
- Compression.
-
- Three major research projects include:
-
- Waterloo Montreal Verona Fractal Research Initiative
- http://links.uwaterloo.ca/
-
- Groupe FRACTALES
- http://www-syntim.inria.fr/fractales/
-
- Bath Scalable Video Software Mk 2
- http://dmsun4.bath.ac.uk/bsv-mk2/
-
- Several books describing fractal image compression are:
-
- 1. M. Barnsley, _Fractals Everywhere_, Academic Press Inc., 1988.
- ISBN 0-12-079062-9. This is an excellent text book on fractals.
- This is probably the best book for learning about the math
- underpinning fractals. It is also a good source for new fractal
- types.
- 2. M. Barnsley and L. Anson, _The Fractal Transform_, Jones and
- Bartlett, April, 1993. ISBN 0-86720-218-1. Without assuming a
- great deal of technical knowledge, the authors explain the
- workings of the Fractal Transform(TM).
- 3. M. Barnsley and L. Hurd, _Fractal Image Compression_, Jones and
- Bartlett. ISBN 0-86720-457-5. This book explores the science of
- the fractal transform in depth. The authors begin with a
- foundation in information theory and present the technical
- background for fractal image compression. In so doing, they
- explain the detailed workings of the fractal transform. Algorithms
- are illustrated using source code in C.
- 4. Y. Fisher (Ed), _Fractal Image Compression: Theory and
- Application_. Springer Verlag, 1995.
- 5. Y. Fisher (Ed), _Fractal Image Encoding and Analysis: A NATO ASI
- Series Book_, Springer Verlag, New York, 1996 contains the
- proceedings of the Fractal Image Encoding and Analysis Advanced
- Study Institute held in Trondheim, Norway July 8-17, 1995. The
- book is currently being produced.
-
- Some introductary articles about fractal compression:
-
- 1. The October 1993 issue of Byte discussed fractal compression. You
- can ftp sample code:
- ftp://ftp.uu.net/published/byte/93oct/fractal.exe .
- 2. A Better Way to Compress Images," M.F. Barnsley and A.D. Sloan,
- BYTE, pp. 215-223, January 1988.
- 3. "Fractal Image Compression," M.F. Barnsley, Notices of the
- American Mathematical Society, pp. 657-662, June 1996.
- (http://www.ams.org/publications/notices/199606/barnsley.html)
- 4. A. E. Jacquin, Image Coding Based on a Fractal Theory of Iterated
- Contractive Image Transformation, _IEEE Transactions on Image
- Processing_, January 1992.
- 5. A "Hitchhiker's Guide to Fractal Compression" For Beginners by
- E.R. Vrscay
- ftp://links.uwaterloo.ca/pub/Fractals/Papers/Waterloo/vr95.ps.gz
-
- Andreas Kassler wrote a Fractal Image Compression with WINDOWS package
- for a Fractal Compression thesis. It is available at
- http://www-vs.informatik.uni-ulm.de/Mitarbeiter/Kassler/papers.htm
-
- Other references:
-
- Fractal Compression Bibliography
- http://www.dip.ee.uct.ac.za/imageproc/compression/fractal/fract
- al.bib.html
-
- Fractal Video Compression
- http://inls.ucsd.edu/y/Fractals/Video/fracvideo.html
-
- Many fractal image compression papers are available from
- ftp://ftp.informatik.uni-freiburg.de/documents/papers/fractal
-
- A review of the literature is in Guide.ps.gz.
- ftp://ftp.informatik.uni-freiburg.de/documents/papers/fractal/R
- EADME
-
- Subject: Chaotic demonstrations
-
- _Q12a_: How can you make a chaotic oscillator?
-
- _A12a_: Two references are:
-
- 1. T. S. Parker and L. O. Chua, Chaos: a tutorial for engineers,
- _Proceedings IEEE_ 75 (1987), pp. 982-1008.
- 2. _New Scientist_, June 30, 1990, p. 37.
-
- _Q12b_: What are laboratory demonstrations of chaos?
-
- _A12b_: Robert Shaw at UC Santa Cruz experimented with chaos in
- dripping taps. This is described in:
-
- 1. J. P. Crutchfield, Chaos, _Scientific American_ 255, 6 (Dec.
- 1986), pp. 38-49.
- 2. I. Stewart, _Does God Play Dice?: the Mathematics of Chaos_, B.
- Blackwell, New York, 1989.
-
- Two references to other laboratory demonstrations are:
-
- 1. K. Briggs, Simple Experiments in Chaotic Dynamics, _American
- Journal of Physics_ 55, 12 (Dec 1987), pp. 1083-1089.
- 2. J. L. Snider, Simple Demonstration of Coupled Oscillations,
- _American Journal of Physics_ 56, 3 (Mar 1988), p. 200.
-
- See sci.nonlinear FAQ and the sci.nonlinear newsgroup for further
- information.
-
- Subject: L-Systems
-
- _Q13_: What are L-systems?
-
- _A13_: A L-system or Lindenmayer system is a formal grammar for
- generating strings. (That is, it is a collection of rules such as
- replace X with XYX.) By recursively applying the rules of the L-system
- to an initial string, a string with fractal structure can be created.
- Interpreting this string as a set of graphical commands allows the
- fractal to be displayed. L-systems are very useful for generating
- realistic plant structures.
-
- Some references are:
-
- 1. P. Prusinkiewicz and J. Hanan, _Lindenmayer Systems, Fractals, and
- Plants_, Springer-Verlag, New York, 1989.
- 2. P. Prusinkiewicz and A. Lindenmayer, _The Algorithmic Beauty of
- Plants_, Springer-Verlag, NY, 1990. ISBN 0-387-97297-8. A very
- good book on L-systems, which can be used to model plants in a
- very realistic fashion. The book contains many pictures.
-
- _________________________________________________________________
-
- More information can be obtained via the WWW at:
-
- L-Systems Tutorial by David Green
- http://life.csu.edu.au/complex/tutorials/tutorial2.html
- http://www.csu.edu.au/complex_systems/tutorial2.html
-
- Graphics Archive at the Center for the Computation and Visualization
- of Geometric Structures contains various fractals created from
- L-Systems.
- http://www.geom.umn.edu/graphics/pix/General_Interest/Fractals/
-
- Subject: Fractal music
-
- _Q14_: What are sources of fractal music?
-
- _A14_: One fractal recording is "The Devil's Staircase: Composers and
- Chaos" on the Soundprint label. A second is "Curves and Jars" by Barry
- Lewis. You can contact MPS Music & Video for further information:
- Rosegarth, Hetton Road, Houghton-le-Spring, DH5 8JN, England or online
- at CDeMUSIC (http://www.emf.org/focus_lewisbarry.html).
-
- Does anyone know of others? Mail me at fractal-faq@mta.ca.
-
- Some references, many from an unpublished article by Stephanie Mason,
- are:
-
- 1. R. Bidlack, Chaotic Systems as Simple (But Complex) Compositional
- Algorithms, _Computer Music Journal_, Fall 1992.
- 2. C. Dodge, A Musical Fractal, _Computer Music Journal_ 12, 13 (Fall
- 1988), p. 10.
- 3. K. J. Hsu and A. Hsu, Fractal Geometry of Music, _Proceedings of
- the National Academy of Science, USA_ 87 (1990), pp. 938-941.
- 4. K. J. Hsu and A. Hsu, Self-similatrity of the '1/f noise' called
- music., _Proceedings of the National Academy of Science USA_ 88
- (1991), pp. 3507-3509.
- 5. C. Pickover, _Mazes for the Mind: Computers and the Unexpected_,
- St. Martin's Press, New York, 1992.
- 6. P. Prusinkiewicz, Score Generation with L-Systems, _International
- Computer Music Conference 86 Proceedings, _1986, pp. 455-457.
- 7. _Byte_ 11, 6 (June 1986), pp. 185-196.
-
- Online resources include:
-
- Well Tempered Fractal v3.0 by Robert Greenhouse
- http://www-ks.rus.uni-stuttgart.de/people/schulz/fmusic/wtf/
-
- A fractal music C++ package is available at
- http://hamp.hampshire.edu/~gpzF93/inSanity.html
-
- The Fractal Music Project (Claus-Dieter Schulz)
- http://www-ks.rus.uni-stuttgart.de/people/schulz/fmusic
-
- Chua's Oscillator: Applications of Chaos to Sound and Music
- http://www.ccsr.uiuc.edu/People/gmk/Projects/ChuaSoundMusic/Chu
- aSoundMusic.html
-
- Fractal Music Lab
- http://members.aol.com/strohbeen/fml.html
-
- Fractal Music - Phil Thompson
- http://easyweb.easynet.co.uk/~cenobyte/
-
- fractal music in MIDI format by Jose Oscar Marques
- http://midiworld.com/jmarques.htm
-
- Don Archer's fractal art and music contains several pieces of fractal
- music in MIDI format.
- http://www.dorsai.org/~arch/
-
- LMUSe, a DOS program that generates MIDI music and files from 3D
- L-systems.
- http://www.interport.net/~dsharp/lmuse.html
-
- There is now a Fractal Music mailing list. It's purposes are:
-
- 1. To inform people about news, updates, changes on the Fractal Music
- Projects WWW pages.
- 2. To encourage discussion between people working in that area.
-
- The Fractal Music Mailinglist: 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 sub-triangles, and perturbing the center point. This process is then
- repeated on the sub-triangles. This results in a 2-d table of heights,
- which can then be rendered as a 3-d image. This is referred to as
- midpoint displacement. Two references are:
-
- 1. M. Ausloos, _Proc. R. Soc. Lond. A_ 400 (1985), pp. 331-350.
- 2. H.O. Peitgen, D. Saupe, _The Science of Fractal Images_,
- Springer-Velag, 1988
-
- Available online is an implementation of fractal Brownian motion (fBm)
- such as described in _The Science of Fractal Images_. Lucasfilm became
- famous for its fractal landscape sequences in _Star Trek II: The Wrath
- of Khan_ the primary one being the _Genesis_ planet transformation.
- Pixar and Digital Productions are have produced fractal landscapes for
- Hollywood.
-
- Fractal landscape information available online:
-
- EECS News: Fall 1994: Building Fractal Planets by Ken Musgrave
- http://www.seas.gwu.edu/faculty/musgrave/article.html
-
- Gforge and Landscapes (John Beale)
- http://www.best.com/~beale/
-
- Java fractal landscapes :
-
- Fractal landscapes (applet and sources) by Chris Thornborrow
- http://www-europe.sgi.com/Fun/free/java/chris-thornborrow/index
- .html
-
- Subject: Plasma clouds
-
- _Q16_: What are plasma clouds?
-
- _A16_: They are a Fractint fractal and are similar to fractal
- mountains. Instead of a 2-d table of heights, the result is a 2-d
- table of intensities. They are formed by repeatedly subdividing
- squares.
-
- Robert Cahalan has fractal information about Earth's Clouds including
- how they differ from plasma clouds.
-
- Fractal Clouds Reference by Robert F. Cahalan
- (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
-
- The Center for the Computation and Visualization of Geometric
- Structures also has some fractal clouds.
- http://www.geom.umn.edu/graphics/pix/General_Interest/Fractals/
-
- Two articles about the fractal nature of Earth's clouds:
-
- 1. "Fractal statistics of cloud fields," R. F. Cahalan and J. H.
- Joseph, _Mon. Wea.Rev._ 117, 261-272, 1989
- 2. "The albedo of fractal stratocumulus clouds," R. F. Cahalan, W.
- Ridgway, W. J. Wiscombe, T. L. Bell and J. B. Snider, _J. Atmos.
- Sci._ 51, 2434-2455, 1994
-
- Subject: Lyapunov fractals
-
- _Q17a_: Where are the popular periodically-forced Lyapunov fractals
- described?
-
- _A17a_: See:
-
- 1. A. K. Dewdney, Leaping into Lyapunov Space, _Scientific American_,
- Sept. 1991, pp. 178-180.
- 2. M. Markus and B. Hess, Lyapunov Exponents of the Logistic Map with
- Periodic Forcing, _Computers and Graphics_ 13, 4 (1989), pp.
- 553-558.
- 3. M. Markus, Chaos in Maps with Continuous and Discontinuous Maxima,
- _Computers in Physics_, Sep/Oct 1990, pp. 481-493.
-
- _Q17b_: What are Lyapunov exponents?
-
- _A17b_: Lyapunov exponents quantify the amount of linear stability or
- instability of an attractor, or an asymptotically long orbit of a
- dynamical system. There are as many Lyapunov exponents as there are
- dimensions in the state space of the system, but the largest is
- usually the most important.
-
- Given two initial conditions for a chaotic system, a and b, which are
- close together, the average values obtained in successive iterations
- for a and b will differ by an exponentially increasing amount. In
- other words, the two sets of numbers drift apart exponentially. If
- this is written e^(n*(lambda) for _n_ iterations, then e^(lambda) is
- the factor by which the distance between closely related points
- becomes stretched or contracted in one iteration. Lambda is the
- Lyapunov exponent. At least one Lyapunov exponent must be positive in
- a chaotic system. A simple derivation is available in:
-
- 1. H. G. Schuster, _Deterministic Chaos: An Introduction_, Physics
- Verlag, 1984.
-
- _Q17c_: How can Lyapunov exponents be calculated?
-
- _A17c_: For the common periodic forcing pictures, the Lyapunov
- exponent is:
-
- lambda = limit as N -> infinity of 1/N times sum from n=1 to N of
- log2(abs(dx sub n+1 over dx sub n))
-
- In other words, at each point in the sequence, the derivative of the
- iterated equation is evaluated. The Lyapunov exponent is the average
- value of the log of the derivative. If the value is negative, the
- iteration is stable. Note that summing the logs corresponds to
- multiplying the derivatives; if the product of the derivatives has
- magnitude < 1, points will get pulled closer together as they go
- through the iteration.
-
- MS-DOS and Unix programs for estimating Lyapunov exponents from short
- time series are available by ftp: ftp://inls.ucsd.edu/pub/ncsu/
-
- Computing Lyapunov exponents in general is more difficult. Some
- references are:
-
- 1. H. D. I. Abarbanel, R. Brown and M. B. Kennel, Lyapunov Exponents
- in Chaotic Systems: Their importance and their evaluation using
- observed data, _International Journal of Modern Physics B_ 56, 9
- (1991), pp. 1347-1375.
- 2. A. K. Dewdney, Leaping into Lyapunov Space, _Scientific American_,
- Sept. 1991, pp. 178-180.
- 3. M. Frank and T. Stenges, _Journal of Economic Surveys_ 2 (1988),
- pp. 103- 133.
- 4. T. S. Parker and L. O. Chua, _Practical Numerical Algorithms for
- Chaotic Systems_, Springer Verlag, 1989.
-
- Subject: Fractal items
-
- _Q18_: Where can I get fractal T-shirts, posters and other items?
-
- _A18_: One source is Art Matrix, P.O. box 880, Ithaca, New York,
- 14851, 1-800-PAX-DUTY.
-
- Another source is Media Magic; they sell many fractal posters,
- calendars, videos, software, t-shirts, ties, and a huge variety of
- books on fractals, chaos, graphics, etc. Media Magic is at PO Box 598
- Nicasio, CA 94946, 415-662-2426.
-
- A third source is Ultimate Image; they sell fractal t- shirts,
- posters, gift cards, and stickers. Ultimate Image is at PO Box 7464,
- Nashua, NH 03060-7464.
-
- Yet another source is Dave Kliman (516) 625-2504 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/gift.html. For foreign requests
- (outside USA) include two IRCs (international reply coupons) to cover
- the weight.
-
- ReFractal Design (http://www.refractal.com/) sells jewelry based on
- fractals.
-
- Lifesmith Classic Fractals (http://www.lifesmith.com/) claims to be
- the largest fractal art studio in USA. You can contact Jeff Berkowitz
- at Fractalier@aol.com.
-
- There is a form of broccoli called Romanesco which is actually
- cauli-brocs, cross between cauliflowers and broccoli. It has a fractal
- like form. It was created in Italy about eight years ago and available
- in many stores in Europe.
-
- Subject: How can I take photos of fractals?
-
- _Q19_: How can I take photos of fractals?
-
- _A19_: Noel Giffin gets good results with the following setup: Use 100
- ISO (ASA) Kodak Gold for prints or 64 ISO (ASA) for slides. Use a long
- lens (100mm) to flatten out the field of view and minimize screen
- curvature. Use f/4 stop. Shutter speed must be longer than frame rate
- to get a complete image; 1/4 seconds works well. Use a tripod and
- cable release or timer to get a stable picture. The room should be
- completely blackened, with no light, to prevent glare and to prevent
- the monitor from showing up in the picture.
-
- You can also obtain high quality images by sending your Targa or GIF
- images to a commercial graphics imaging shop. They can provide much
- higher resolution images. Prices are about $10 for a 35mm slide or
- negative and about $50 for a high quality 4x5 negative.
-
- Subject: Colour Rendering Techniques
-
- _Q20a_: What are the rendering methods commonly used for 256-colour
- fractals?
-
- _A20a_: The simplest form of rendering uses escape times. Pixels are
- coloured according to the number of iterations it takes for a pixel to
- _blow-up_ or escape the loop. Different criteria may be chosen to
- speed a pixel to its blow-up point and therefore change the rendering
- of a fractal. These include the biomorph method and epsilon-cross
- method, both developed by Clifford Pickover. Similar to the
- escape-time methods are Fractint's _real_, _imag_ and _summ_ options.
- These add the real and/or imaginary values of a points Z-potential (at
- the blow-up time) to the escape time. Normally, escape-time fractals
- exhibit a flat 2-D appearance with _banding_ quite apparent at the
- lowest escape times. The addition of z-potential to the escape times
- tends to reduce banding and simulate 3-D effects in the outer bands.
-
- Other traditional rendering methods for 256-colour fractals include
- continuous potential, external decomposition and level-set methods
- like Fractint's Bof60 and Bof61. Here the colour of a point is based
- on its Z-potential and/or exit angle. The potential may be obtained
- for when it is at its lowest or at its last value, or some other
- criteria. The potential is scaled then applied to the palette used.
- Scaling may be linear or logarithmic, as for example palettes are
- defined in Fractint. Orbit-trap fractals make extensive use of level
- curves, which are based on z-potentials scaled linearly. Decomposition
- uses exit angles to define colours. Exit angles are derived from the
- polar notation of a point's complex value. Akin to decomposition is
- Paul Carlson's atan method (which uses an average of the last two
- angles) and the _atan_ (single angle) method in Fractint. All of these
- methods can be used to simulated 3-D effects because of the continuous
- shadings possible.
-
- _Q20b_: How does rendering differ for true-colour fractals?
-
- _A20b_: The problem with true-colour rendering is that computers use a
- 3D approach to simulating 16 million colours. The basic components for
- addressing true colour are red, green and blue (256 shades each.)
- There is no logical way to determine an one-dimensional index which
- can be used to address all the RGB colours available in true colour.
- Palettes can be simulated in true colour but are limited to about
- 65000 colours (256x256). Even so, this is enough to eliminate most
- banding found in 256-colour fractals due to limited colour spread.
-
- Because of the flexability in choosing colours from an expanded
- "palette", the best rendering methods will use a combination of level
- curves and exit angles. While escape times can be fractionalized using
- interpolated iteration, the result is still very flat. One promising
- addition to true-colour rendering is acheived by accumulating data
- about a point as it is iterated. The data is then used as an offset to
- the colour normally calculated by other methods. Depending on the
- algorithm used, the "filter" (sic: Stephen C. Ferguson) can intensify,
- fragment or add interesting details to a picture.
-
- Subject: 3-D fractals
-
- _Q21_: How can 3-D fractals be generated?
-
- _A21_: A common source for 3-D fractals is to compute Julia sets with
- quaternions instead of complex numbers. The resulting Julia set is
- four dimensional. By taking a slice through the 4-D Julia set (e.g. by
- fixing one of the coordinates), a 3-D object is obtained. This object
- can then be displayed using computer graphics techniques such as ray
- tracing.
-
- Frank Rousell's hyperindex of 3D images
- http://www.cnam.fr/fractals/mandel3D.html
-
- 4D Quaternions by Tom Holroyd
- http://bambi.ccs.fau.edu/~tomh/fractals/fractals.html
-
- The papers to read on this are:
-
- 1. J. Hart, D. Sandin and L. Kauffman, Ray Tracing Deterministic 3-D
- Fractals, _SIGGRAPH_, 1989, pp. 289-296.
- 2. A. Norton, Generation and Display of Geometric Fractals in 3-D,
- _SIGGRAPH_, 1982, pp. 61-67.
- 3. A. Norton, Julia Sets in the Quaternions, _Computers and
- Graphics_, 13, 2 (1989), pp. 267-278.
-
- Two papers on cubic polynomials, which can be used to generate 4-D
- fractals:
-
- 1. B. Branner and J. Hubbard, The iteration of cubic polynomials,
- part I., _Acta Math_ 66 (1988), pp. 143-206.
- 2. J. Milnor, Remarks on iterated cubic maps, This paper is available
- from ftp://math.sunysb.edu/preprints/ims90-6.ps.Z. Published in
- 1991 SIGGRAPH Course Notes #14: Fractal Modeling in 3D Computer
- Graphics and Imaging.
-
- Instead of quaternions, you can of course use hypercomplex number such
- as in "FractInt", or other functions. For instance, you could use a
- map with more than one parameter, which would generate a
- higher-dimensional fractal.
-
- Another way of generating 3-D fractals is to use 3-D iterated function
- systems (IFS). These are analogous to 2-D IFS, except they generate
- points in a 3-D space.
-
- A third way of generating 3-D fractals is to take a 2-D fractal such
- as the Mandelbrot set, and convert the pixel values to heights to
- generate a 3-D "Mandelbrot mountain". This 3-D object can then be
- rendered with normal computer graphics techniques.
-
- POV-Ray 3.0, a freely available ray tracing package, has added 4-D
- fractal support. It takes a 3-D slice of a 4-D Julia set based on an
- arbitrary 3-D "plane" done at any angle. For more information see the
- POV Ray web site at http://www.povray.org/ .
-
- Subject: Fractint
-
- _Q22a_: What is Fractint?
-
- _A22a_: Fractint is a very popular freeware (not public domain)
- fractal generator. There are DOS, MS-Windows, OS/2, Amiga, and
- Unix/X-Windows versions. The DOS version is the original version, and
- is the most up-to-date.
-
- _Please note_: sci.fractals is not a product support newsgroup for
- Fractint. Bugs in Fractint/Xfractint should usually go to the authors
- rather than being posted.
-
- Fractint is on many ftp sites. For example:
-
- A Guide to getting FractInt by Noel at Spanky (Canada)
- http://spanky.triumf.ca/www/fractint/getting.html
-
- DOS
-
- 19.6 executable via FTP and WWW from SimTel & mirrors world-wide
- http://www.coast.net/cgi-bin/coast/dwn?msdos/graphics/frain196.
- zip
-
- 19.6 source via FTP and WWW from SimTel & mirrors world-wide
- http://www.coast.net/cgi-bin/coast/dwn?msdos/graphics/frasr196.
- zip
-
- 19.6 executable via FTP from Canada
- ftp://fractal.mta.ca/pub/spanky/programs/ibmpc/frain196.zip
-
- 19.6 source via FTP from Canada
- ftp://fractal.mta.ca/pub/spanky/programs/ibmpc/frasr196.zip
-
- (The suffix _196_ will change as new versions are released.)
-
- Fractint is available on Compuserve: GO GRAPHDEV and look for
- FRAINT.EXE and FRASRC.EXE in LIB 4.
-
- Windows
-
- MS-Window FractInt 18.21 via FTP and WWW from SimTel & mirrors
- world-wide
- http://www.coast.net/cgi-bin/coast/dwn?win3/graphics/winf1821.z
- ip
-
- MS-Window FractInt 18.21 via FTP from Canada
- ftp://fractal.mta.ca/pub/spanky/programs/ibmpc/windows/winf1821
- .zip
-
- MS-Windows FractInt 18.21 source via FTP and WWW from SimTel & mirrors
- world-wide
- http://www.coast.net/cgi-bin/coast/dwn?win3/graphics/wins1821.z
- ip
-
- MS-Windows FractInt 18.21 source via FTP from Canada
- ftp://fractal.mta.ca/pub/spanky/programs/ibmpc/windows/wins1821
- .zip
-
- OS/2
-
- Available on Compuserve in its GRAPHDEV forum. The files are PM*.ZIP.
- These files are also available on many sites, for example
- http://oak.oakland.edu/pub/os2/graphics/
-
- Unix
-
- The Unix version of FractInt, called _XFractInt_ requires X-Windows.
- The current version 3.04 is based on FractInt 19.6.
-
- 3.04 source Western Canada
- http://spanky.triumf.ca/pub/fractals/programs/unix/xfract304.tg
- z
-
- 3.04 source Atlantic Canada
- http://fractal.mta.ca/spanky/programs/unix/xfract304.tgz
-
- XFractInt is also available in LIB 4 of Compuserve's GO GRAPHDEV forum
- in XFRACT.ZIP.
-
- _Xmfract_ by Darryl House is a port of FractInt to a X/Motif
- multi-window interface. The current version is 1.4 which is compatible
- with FractInt 18.2.
-
- README
- http://fractal.mta.ca/pub/fractals/programs/unix/xmfract_1-4.re
- adme
-
- xmfract_1-4_tar.gz
- http://fractal.mta.ca/pub/fractals/programs/unix/xmfract_1-4_ta
- r.gz
-
- Macintosh
-
- There is _NO_ Macintosh version of Fractint, although there may be
- several people working on a port. It is possible to run Fractint on
- the Macintosh if you use a PC emulator such as Insignia Software's
- SoftAT.
-
- Amiga
-
- There is an Amiga version also available:
-
- FracInt 3.2
- http://spanky.triumf.ca/pub/fractals/programs/AMIGA/
-
- FracXtra
-
- There is a collection of map, parameter, etc. files for
- FractInt, called FracXtra. It is available at
-
- FracXtra Home Page by Dan Goldwater
- http://fatmac.ee.cornell.edu/~goldwada/fracxtra.html
-
- FracXtra via FTP and WWW from SimTel & mirrors world-wide
- http://www.coast.net/cgi-bin/coast/dwn?msdos/graphics/fra
- cxtr6.zip
-
- FracXtra via FTP
- ftp://fractal.mta.ca/pub/spanky/programs/ibmpc/fracxtr6.z
- ip
-
- _Q22b_: How does Fractint achieve its speed?
-
- _A22b_: Fractint's speed (such as it is) is due to a
- combination of:
-
- 1. Reducing computation by Periodicity checking and guessing
- solid areas (especially the "lake" area).
- 2. Using hand-coded assembler in many places.
- 3. Using fixed point math rather than floating point where
- possible (huge improvement for non-coprocessor machine, small
- for 486's, moot for Pentium processors).
- 4. Exploiting symmetry of the fractal.
- 5. Detecting nearly repeating orbits, avoid useless iteration
- (e.g. repeatedly iterating 02+0 etc. etc.).
- 6. Obtaining both sin and cos from one 387 math coprocessor
- instruction.
- 7. Using good direct memory graphics writing in 256-color modes.
-
- The first three are probably the most important. Some of these
- introduce errors, usually quite acceptable.
-
- Subject: Fractal software
-
- _Q23_: Where can I obtain software packages to generate fractals?
-
- _A23_:
-
- * Amiga
- * Java
- * Macintosh
- * MS-DOS
- * MS-Windows
- * SunView
- * UNIX
- * X-Windows
-
- * Software to calculate fractal dimension
-
- For Amiga:
-
- (all entries marked "ff###" are directories where the inividual
- archives of the Fred Fish Disk set available at
- ftp://ftp.funet.fi/pub/amiga/fish/ and other sites)
-
- General Mandelbrot generators with many features: Mandelbrot (ff030),
- Mandel (ff218), Mandelbrot (ff239), TurboMandel (ff302), MandelBltiz
- (ff387), SMan (ff447), MandelMountains (ff383, in 3-D), MandelPAUG
- (ff452, MandFXP movies), MandAnim (ff461, anims), ApfelKiste (ff566,
- very fast), MandelSquare (ff588, anims)
-
- Mandelbrot and Julia sets generators: MandelVroom (ff215), Fractals
- (ff371, also Newton-R and other sets)
-
- With different algorithmic approaches (shown): FastGro (ff188, DLA),
- IceFrac (ff303, DLA), DEM (ff303, DEM), CPM (ff303, CPM in 3-D),
- FractalLab (ff391, any equation)
-
- Iterated Function System generators (make ferns, etc): FracGen (ff188,
- uses "seeds"), FCS (ff465), IFSgen (ff554), IFSLab (ff696, "Collage
- Theorem")
-
- Unique fractal types: Cloud (ff216, cloud surfaces), Fractal (ff052,
- terrain), IMandelVroom (strange attractor contours?), Landscape
- (ff554, scenery), Scenery (ff155, scenery), Plasma (ff573, plasma
- clouds)
-
- Fractal generators: PolyFractals (ff015), FFEX (ff549)
-
- Fractint for Amiga
- http://spanky.triumf.ca/pub/fractals/programs/AMIGA/
-
- Lyapunov fractals
- http://www.itsnet.com/~bug/fractals/Lyapunovia.html
-
- XaoS, by Jan Hubicka, fast portable real-time interactive fractal
- zoomer. 256 workbench displays only.
- http://www.paru.cas.cz/~hubicka/XaoS/
-
- Commercial packages: Fractal Pro 5.0, Scenery Animator 2.0, Vista
- Professional, Fractuality (reviewed in April '93 Amiga User
- International). MathVISION 2.4. Generates Julia, Mandelbrot, and
- others. Includes software for image processing, complex arithmetic,
- data display, general equation evaluation. Available for $223 from
- Seven Seas Software, Box 1451, Port Townsend WA 98368.
-
- Java applets
-
- Chaos!
- http://www.vt.edu:10021/B/bwn/Chaos.html
-
- Fractal Lab
- http://www.wmin.ac.uk/~storyh/fractal/frac.html
-
- The Mandelbrot Set
- http://www.mindspring.com/~chroma/m andelbrot.html
-
- The Mandelbrot set (Paton J. Lewis)
- http://numinous.com/_private/people/pjl/graphics/mandelbrot/man
- delbrot.html
-
- Mark's Java Julia Set Generator
- http://www.stolaf.edu/people/mcclure/java/Julia/
-
- Fractals by Sun Microsystems
- http://java.sun.com/jav
- a.sun.com/applets/applets/Fractal/example1.html
-
- The Mandelbrot set
- http://www.franceway.com/java/fractale /mandel_b.htm
-
- Mandelbrot Java Applet
- http://www.mit.edu:8001/people/m kgray/java/Mandel.html
-
- Ken Shirriff Java language pages
- http://www.sunlabs.com/~shirriff/java/
-
- example of the plasma method of fractal terrain by Carl Burke,
- <cburke@mitre.org>
- http://www.geocities.com/Area51/6902/t_sd_app.html
-
- Mandelbrot generator in Javascript by Frode Gill.
- http://www.krs.hia.no/~fgill/javascript/mandscr.htm
-
- Fracula Java Applet. A java applet to glide into the Mandelbrot set
- (best with Pentium and MSIE 3.0). Vince Ruddy
- <vruddy1@san.rr.com>
- http://www.geocities.com/SiliconValley/Pines/5788/index.html
-
- Chaos and Fractals. Many java applets by Stephen Oswin
- <stephen.oswin@ukmail.org>
- www.ukmail.org/~oswin/
-
- IFS Fractals using javascript (Richard L. Bowman
- <rbowman@bridgewater.edu>)
- http://www.bridgewater.edu/departments/physics/ISAW/FracMain.ht
- ml
-
- A lot of Java applets
- http://java.developer.com/pages/tmp-Gamelan.mm.graphics.fractal
- s.html
-
- ChaosLab. A nice fully java site with several interactive applets
- showing different types of Mandelbrot, Julia, and strange
- attractors. By Cameron Mckechnie <chaoslab@actrix.gen.nz>
- http://www.actrix.gen.nz/users/chaoslab/chaoslab.html
-
- Fractal landscapes (applet and sources) by Chris Thornborrow
- http://www-europe.sgi.com/Fun/free/java/chris-thornborrow/index
- .html
-
- Forest Echo Farm Fractal Fern
- http://www.forestecho.com/ferns.html
-
- Fractal java generator by Patrick Charles
- http://www.csn.org/~pcharles/classes/FractalApp.html
-
- 3 interactive java applets by Robert L. Devaney <bob@math.bu.edu>
- http://math.bu.edu/DYSYS/applets/index.html
-
- Interactive java applets by Philip Baker <phil@pjbsware.demon.co.uk>
- http://www.pjbsware.demon.co.uk/java/index.htm
-
- Chaos and order by Eric Leese
- http://www.geocities.com/CapeCanaveral/Hangar/7959/
-
- MB applet by Russ <RBinNJ@worldnet.att.net>
- http://home.att.net/~RBinNJ/mbapplet.htm
-
- Stand alone application
-
- Filmer by Julian Haight. Filmer is a front-end program for Fractint
- that generates amazing fractal animation. Fractint is a program
- for calculating still fractal images (you need Fractint
- installed to use Filmer). Filmer uses Fractint parameter (.par)
- files to specify the coordinates and other parameters of a
- fractal. It then calculates the intermediate frames and calls
- Fractint to make a continuous animation. Filmer also has many
- options for pallete rotation and generation.
- http://www.julianhaight.com/filmer/
- Javaquat by Garr Lystad. Can also be run as an applet from Lystad's
- page.
- http://www.iglobal.net/lystad/fractal-top.html
-
- For Macs:
-
- For PowerMacs
- (and PowerPC-based Macintosh compatible computers)
-
- Fractal Domains v. 1.2
-
- * Fractal generator for PowerMacs only, by Dennis C. De Mars
- (formerly FracPPC)
- * Generates the Mandelbrot set and associated Julia sets, allows the
- user to edit the color map, 24-bit colour
- + http://members.aol.com/ddemars/fracppc.html
-
- MandelBrowser 2.0
-
- * by the author of Mandella, 24-bit colour
- +
- ftp://mirrors.aol.com/pub/mac/graphics/fractal/mandelbrowser2.
- 0.sit.hqx
-
- _________________________________________________________________
-
- For 68K Macs
-
- Mandella 8.7
-
- * generation of many different types of fractals, allow editing of
- the color map, and other display & calculation options. Some
- features not available on PowerMacs.
- +
- ftp://mirrors.aol.com/pub/mac/graphics/fractal/mandella8.7.cpt
- .hqx
-
- Mandelzot 4.0.1
-
- * generation of many different types of fractals, allow editing of
- the color map, and other display & calculation options. Some
- features not available on PowerMacs.
- +
- ftp://mirrors.aol.com/pub/mac/graphics/fractal/mandelzot4.01.c
- pt.hqx
-
- SuperMandelZoom 1.0.6
-
- * useful to those rare individuals who are still using a Mac Plus/SE
- class machine
- +
- ftp://mirrors.aol.com/pub/mac/graphics/fractal/supermandelzoom
- 1.06.cpt.hqx
-
- _________________________________________________________________
-
- Miscellaneous programs
-
- * _FDC and FDC 3D_ - Fractal Dimension Calculators
- + http://www.mhri.edu.au/~pdb/software/
- * _Lsystem, 3D-L-System, IFS, FracHill_
- + http://www.mhri.edu.au/~pdb/fractals/
- * _Color Fractal Generator_ 2.12
- +
- ftp://mirrors.aol.com/pub/mac/graphics/fractal/colorfractalgen
- 2.12.sit.hqx
- * _MandelNet_ (uses several Macs on an AppleTalk network to
- calculate the Mandebrot set!)
- +
- ftp://mirrors.aol.com/pub/mac/graphics/fractal/mandelnet1.2.si
- t.hqx
- * _Julia's Nightmare_ - original and cool program, as you drag the
- mouse about the complex plane, the corresponding Julia set is
- generated in real time!
- +
- ftp://mirrors.aol.com/pub/mac/graphics/fractal/juliasnightmare
- .sit.hqx
- * _Lyapunov_ 1.0.1
- +
- ftp://mirrors.aol.com/pub/mac/graphics/fractal/lyapunov1.01.cp
- t.hqx
- * _Fract_ 1.0 - A fractal-drawing program that uses the IFS
- algorithm. Change parameters to get different self-similar
- patterns.
- +
- ftp://mirrors.aol.com/pub/mac/graphics/fractal/fract1.0.cpt.hq
- x
- * _XaoS_ 2.1 - fast portable real-time interactive fractal zoomer
- + http://www.paru.cas.cz/~hubicka/XaoS/
-
- _________________________________________________________________
-
- Commerical
-
- There are also commercial programs: _IFS Explorer_ and _Fractal Clip
- Art_ (published by Koyn Software (314) 878-9125), _Kai's Fractal
- Explorer_ (part of the Kai's Power Tools package)
-
- For MSDOS:
-
- DEEPZOOM: a high-precision Mandelbrot Set program for displaying
- highly zoomed fractals
- http://spanky.triumf.ca/pub/fractals/programs/ibmpc/depzm13.zip
-
- Fractal WitchCraft: a very fast fractal design program
- ftp://garbo.uwasa.fi/pc/demo/fw1-08.zip
- ftp://ftp.cdrom.com/pub/garbo/garbo_pc/show/fw1-08.zip
-
- Fractal Discovery Laboratory: designed for use in a science museum or
- school setting. The Lab has five sections: Art Gallery,
- Microscope, Movies, Tools, and Library
- Sampler available from Compuserve GRAPHDEV Lib 4 in DISCOV.ZIP,
- or send high-density disk and self-addressed, stamped envelope
- to: Earl F. Glynn, 10808 West 105th Street, Overland Park,
- Kansas 66214-3057.
-
- WL-Plot 2.59 : plots functions including bifurcations and recursive
- relations
- ftp://archives.math.utk.edu/software/msdos/graphing/wlplt/wlplt
- 259.zip
-
- From http://www.simtel.net/pub/simtelnet/msdos/graphics/
- forb01a.zip: Displays orbits of Mandelbrot Set mapping. C/E/VGA
-
- fract3.zip: Mandelbrot/Julia set 2D/3D EGA/VGA Fractal Gen
- fractfly.zip: Create Fractal flythroughs with FRACTINT
- fdesi313.zip: Program to visually design IFS fractals
- frain196.zip: FRACTINT v19.6 EGA/VGA/XGA fractal generator
- frasr196.zip: C & ASM src for FRACTINT v19.6
- frcal040.zip: CAL: more than 15 types of fractals including
- Lyapunov, IFS, user-defined, logistic, and Quaternion Julia
-
- Vlotkatc uses VESA 640x480x16 Million colour mode to generate
- Volterra-Lotka images.
- http://spanky.triumf.ca/pub/fractals/programs/ibmpc/vlotkatc.zi
- p
- http://spanky.triumf.ca/pub/fractals/programs/ibmpc/vlotkatc.do
- c
-
- Fast FPU Fractal Fun 2.0 (FFFF2.0) is the first Mandelbrot Set
- generator working in hicolor gfx modes thus using up to 32768
- different colors on screen by Daniele Paccaloni requires 386DX+
- and VESA support
- http://spanky.triumf.ca/pub/fractals/programs/IBMPC/FFFF20.ZIP
-
- 3DFract generates 3-D fractals including Sierpinski cheese and 3-D
- snowflake
- http://www.cstp.umkc.edu/users/bhugh/home.html
-
- FracTrue 2.10 - Hi/TrueColor Generator including a formular parser.
- 286+ VGA by Bernd Hemmerling
-
- LyapTrue 2.10 Lyapunov generator
-
- ChaosTrue 2.00 - 18 types
-
- Atractor 1.00 256 colour
- http://www.cs.tu-berlin.de/~hemmerli/fractal.html
-
- HOP based on the HOPALONG fractal type. Math coprocessor (386DX and
- above) and SuperVGA required. shareware ($30) Places to
- download HOPZIP.EXE from:
- Compuserve GRAPHDEV forum, lib 4
- The Well under ibmpc/graphics
- http://ourworld.compuserve.com/homepages/mpeters/hop.htm
- ftp://ftp.uni-heidelberg.de/pub/msdos/graphics/
- http://spanky.triumf.ca/pub/fractals/programs/ibmpc/
-
- ZsManJul 1.0 (requires 386DX+) by Zsolt Zsoldos
- http://www.chem.leeds.ac.uk/ICAMS/people/zsolt/zsmanjul.html
-
- FractMovie 1.62 a real-time 2D/3D IFS fractal movie renderer (requires
- 486DX+) with GIF save
- http://pub.vse.cz/pub/msdos/SAC/pc/graph/frcmv162.zip
-
- FracZoom Explorer and FracZoom Navigator by Niels Ulrik Reinwald
- 386DX+
- http://www.softorange.com/software.html
-
- RMandel 1.2 80-bit floating point Mandelbrot Set animation generator
- by Marvin R. Lipford
- ftp://fractal.mta.ca/pub/cnam/anim/FRACSOFT/rmandel.zip
-
- M24, the new version of TruMand by Mike Freeman 486DX+ True-colour
- Mandelbrot Set generator
- http://www.capcollege.bc.ca/~mfreeman/mand.html
-
- FAE - Fractal Animation Engine shareware by Brian Towles
- http://spanky.triumf.ca/pub/fractals/programs/ibmpc/FAE210B.ZIP
-
- XaoS 2.2 fast portable realtime interactive fractal zoomer/morpher for
- MS-DOS (and others) by Jan Hubicka <hubicka@limax.paru.cas.cz>
- 11 fractal formulas, "Autopilot", solid guessing, zoom up to
- 64051194700380384 times
- http://www.paru.cas.cz/~hubicka/XaoS/
-
- Ultra Fractal. A DOS program with graphic interface, 256 colors or
- truecolor. Very fast, many formulas. Shareware (Frederik
- Slijkerman <slijkerman@compuserve.com>)
- http://ourworld.compuserve.com/homepages/slijkerman/
-
- Fractal worldmap generator. A simple program to generate fractal
- pseudo geographic maps, by John Olsson <d91johol@isy.liu.se>,
- DOS adaptation by Martijn Faassen <faassen@phil.ruu.nl>
- http://www.lysator.liu.se/~johol/fwmg/fwmg.html
-
- Quat - A 3D-Fractal-Generator (Quaternions).
- http://wwwcip.rus.uni-stuttgart.de/~phy11733/quat_e.html
-
- For MS-Windows:
-
- dy-syst: Explores Newton's method, Mandelbrot and Julia sets
- ftp://cssun.mathcs.emory.edu/pub/riddle/
-
- bmand 1.1 shareware by Christopher Bare Mandelbrot program
- http://www.ualberta.ca/~jdawe/mandelbrot/bmand11.zip
-
- Quaternion-generator generates Julia-set Quaternions by Frode Gill
- http://www.krs.hia.no/~fgill/fractal.html
-
- Quat - A 3D-Fractal-Generator (Quaternions).
- http://wwwcip.rus.uni-stuttgart.de/~phy11733/quat_e.html
-
- A Fractal Experience 32 for Windows 95/NT by David Wright
- <wgwright@mnsinc.com>
- http://www.mnsinc.com/wgwright/fracexp/
-
- Iterate 32 for Windows 95/NT written in VisualBasic. Generates IFS,
- includes 10 built-in attractors, plots via chaos algorithm or
- MRCM (multiple reduction copy machine), includes MS-Word
- document about IFS and fractal compression in easy to
- understand terms. Freeware by Jeff Colvin <kd4syw@usit.net>
- http://hamnetcenter.com/jeffc/fractal.html
-
- IFS Explorer for Windows 95/NT, a companion to Iterate 32, allows
- users to explore IFS by changing the IFS parameters. Requires
- 800x600 screen. Freeware by Jeff Colvin <kd4syw@usit.net>
- http://hamnetcenter.com/jeffc/fractal.html
-
- DFRAC 1.4 by John Ratcliff a Windows 95 DirectDraw Mandelbrot explorer
- with movie feature. Requires DirectDraw, FPU, and
- monitor/graphics card capable of 800x600 graphic mode.
- Freeware.
- http://www.inlink.com/~jratclif/john.htm
-
- QS W95 Fractals generates several fractals types in 24-bit colour
- includind Volterra-Lotka, enhanced sine, "Escher-like tiling"
- of Julia Set, magnetism formulae, and "self-squared dragons".
- Supports FractInt MAP files, saves 24-bit Targa or 8-bit GIF,
- several colour options. Freeware by Michael Sargent
- <msargent@zoo.uvm.edu>.
- http://www.uvm.edu/~msargent/
-
- Other fractal programs by Michael Sargent.
- http://www.uvm.edu/~msargent/fractals.htm
-
- Fractal eXtreme for 32-bit Windows 1.01c. A fast interactive fractal
- explorer of Mandelbrot, Julia Set, and Mandelbrot to various
- powers, Newton, "Hidden Mandelbrot", and Auto Quadratic.
- Movies, curve-based palette editor, deep zoom (>2000 digits
- precision for some types), Auto-Explore. Shareware, with
- ability to register online, by Cygnus Software.
- http://www.cygnus-software.com/
-
- Iterations, Flarium24 and Inkblot Kaos Original programs : Now
- Iterations is true color as are Flarium 24 and Inkblot Kaos.
- For W95 or NT. Freeware by Stephen C. Ferguson
- (<itriazon@gte.net>)
- http://home1.gte.net/itriazon/
-
- JuliaSaver : a W95 screen saver that does real-time fractals, by
- Damien M. Jones (<dmj@emi.net>)
- http://www.icd.com/tsd/juliasaver/
-
- Mndlzoom W95 or Nt program which iterate the Mandelbrot set within the
- coprocessor stack : very fast, 19-digits significance (Philip
- A. Seeger <PASeeger@aol.com>)
- http://members.aol.com/paseeger/
-
- Frang : a real-time zooming Mandelbrot set generator. Needs DirectX
- (can be downloaded from the same URL or from Microsoft).
- Shareware (Michael Baldwin <baldwin@servtech.com>)
- http://www.servtech.com/public/baldwin/frang/frang.html
-
- Fractal Orbits; A nice implementation of Bubble, Ring, Stalk methods
- by Phil Pickard <plptrigon@enterprise.net >. Very easy to use.
- W95, NT.
- ftp://ftp-hs.iuta.u-bordeaux.fr/fractorb/
-
- Fractal Commander and Fractal Elite (formerly Zplot) Very
- comprehensive programs which gather several powerful methods
- (original or found in other programs). Now only 32 bits version
- is supported. You can download a free simplified version
- (Fractal Agent) at
- http://www.simtel.net/pub/simtelnet/win95/math/fa331.zip.
- Registered users will receive the full version and a true color
- one. Shareware by Terry W. Gintz <twgg@ix.netcom.com>.
- http://www.geocities.com/SoHo/Lofts/5601/gallery.htm
-
- Set surfer. A nice small program. Draws a variety of fractals of
- Mandelbrot or Julia types. Freeware by Jason Letbetter
- <redbeard@flash.net>.
- http://www.flash.net/~redbeard/
-
- Kai Power Tools 2 and 3 include Fractal Explorer.
- MetaCreations will mail a replacement CD to early KPT 3.0
- owners which didn't include Fractal Explorer.
-
- Fantastic Fractals. This program can draw several sorts of fractals
- (IFS, L-system, Julia...). Well designed for IFS.
- http://library.advanced.org/12740/
-
- Screen savers
-
- Free screen savers : By Philip Baker (<phil@pjbsware.demon.co.uk>)
- http://www.pjbsware.demon.co.uk/snsvdsp.htm
-
- JuliaSaver : a W95 screen saver that does real-time fractals, by
- Damien M. Jones (<dmj@emi.net>)
- http://www.icd.com/tsd/juliasaver/
-
- IFS screen saver: a Windows 3 screen saver, by Bill Decker
- (<wdecker@acm.org>)
- http://www.geocities.com/SoHo/Studios/1450/
-
- Fractint Screen Saver: a Windows 95 - NT screen saver, by Thore
- Berntsen ; needs the DOS program Fractint (<thbernt@online.no>)
-
- http://home.sol.no/~thbernt/fintsave.htm
-
- Seractal Screen Saver: Windows 3 and Windows 95 time limited versions
- (shareware) (<info@seraline.com)>
- http://www.seraline.com/seractal.htm
-
- the Orb series by 'O' from RuneTEK. For MS-Windows 95/NT only.
- http://www.hypermart.net/runetek/
-
- For SunView:
-
- Mandtool: generates Mandelbrot Set
- http://fractal.mta.ca/spanky/programs/mandtool/m_tar.z
- ftp://spanky.triumf.ca/fractals/programs/mandtool/M_TAR.Z
-
- For Unix/C:
-
- lsys: L-systems as PostScript (in C++)
- ftp://ftp.cs.unc.edu/pub/users/leech/lsys.tar.gz
-
- lyapunov: PGM Lyapunov exponent images
- ftp://ftp.uu.net/usenet/comp.sources.misc/volume23/lyapunov/
-
- SPD: fractal mountain, tree, recursive tetrahedron
- ftp://ftp.povray.org/pub/povray/spd/
-
- Fractal Studio: Mandelbrot set; handles distributed computing
- ftp://archive.cs.umbc.edu/pub/peter/fractal-studio
-
- fanal: analysis of fractal dimension for Linux by Jⁿrgen Dollinger
- ftp://ftp.uni-stuttgart.de/pub/systems/linux/local/math/fanal-0
- 1b.tar.gz
-
- XaoS, by Jan Hubicka, fast portable real-time interactive fractal
- zoomer. supports X11 (8,15,16,24,31-bit colour, StaticGray,
- StaticColor), Curses, Linux/SVGAlib
- http://www.paru.cas.cz/~hubicka/XaoS/
-
- For X windows :
-
- xmntns xlmntn: fractal mountains
- ftp://ftp.uu.net/usenet/comp.sources.x/volume8/xmntns
-
- xfroot: fractal root window
- X11 distribution
-
- xmartin: Martin hopalong root window
- X11 distribution
-
- xmandel: Mandelbrot/Julia sets
- X11 distribution
-
- lyap: Lyapunov exponent images
- ftp://ftp.uu.net/usenet/comp.sources.x/volume17/lyapunov-xlib
-
- spider: Uses Thurston's algorithm, Kobe algorithm, external angles
- http://inls.ucsd.edu/y/Complex/spider.tar.Z
-
- xfractal_explorer: fractal drawing program
- ftp://ftp.x.org/contrib/applications/xfractal_explorer-v1.0.tar
- .gz
-
- Xmountains: A fractal landscape generator
- ftp://ftp.epcc.ed.ac.uk/pub/personal/spb/xmountains
-
- xfractint: the Unix version of Fractint : look at XFRACTxxx (xxx being
- the version number)
- http://spanky.triumf.ca/www/fractint/getting.html
-
- xmfract v1.4: Needs Motif 1.2+, based on FractInt
- http://hpftp.cict.fr/hppd/hpux/X11/Misc/xmfract-1.4/
-
- Fast Julia Set and Mandelbrot for X-Windows by Zsolt Zsoldos
- http://www.chem.leeds.ac.uk/ICAMS/people/zsolt/mandel.html
-
- XaoS realtime fractal zoomer for X11 or SVGAlibs by Jan Hubicka
- <hubicka@limax.paru.cas.cz>
- http://www.paru.cas.cz/~hubicka/XaoS/
-
- AlmondBread-0.2. Fast algorithm ; simultaneous orbit iteration ;
- Fractint-compatible GIF and MAP files ; Tcl/Tk user interface
- (Michael R. Ganss <rms@cs.tu-berlin.de>)
- http://www.cs.tu-berlin.de/~rms/AlmondBread/
-
- Quat - A 3D-Fractal-Generator (Quaternions).
- http://wwwcip.rus.uni-stuttgart.de/~phy11733/quat_e.html
-
- XFracky 2.5 by Henrik Wann Jensen <hwj@gk.dtu.dk> based on Tcl/Tk
- http://www.gk.dtu.dk/~hwj/
- http://sunsite.unc.edu/pub/Linux/X11/apps/math/fractals/
-
- Distributed X systems:
-
- MandelSpawn: Mandelbrot/Julia on a network
- ftp://ftp.x.org/R5contrib/mandelspawn-0.07.tar.Z
- ftp://ftp.funet.fi/pub/X11/R5contrib/mandelspawn-0.07.tar.Z
-
- gnumandel: Mandelbrot on a network
- ftp://ftp.elte.hu/pub/software/unix/gnu/gnumandel.tar.Z
-
- Software for computing fractal dimension:
-
- _Fractal Dimension Calculator_ is a Macintosh program which uses the
- box-counting method to compute the fractal dimension of planar
- graphical objects.
-
- http://wuarchive.wustl.edu/edu/math/software/mac/fractals/FDC/
-
- http://wuarchive.wustl.edu/packages/architec/Fractals/FDC2D.sea.hqx
-
- http://wuarchive.wustl.edu/packages/architec/Fractals/FDC3D.sea.hqx
-
- _FD3_: estimates capacity, information, and correlation dimension from
- a list of points. It computes log cell sizes, counts, log counts, log
- of Shannon statistics based on counts, log of correlations based on
- counts, two-point estimates of the dimensions at all scales examined,
- and over-all least-square estimates of the dimensions.
-
- ftp://inls.ucsd.edu/pub/cal-state-stan
- for an enhanced Grassberger-Procaccia algorithm for correlation
- dimension.
-
- A MS-DOS version of FP3 is available by request to
- gentry@altair.csustan.edu.
-
- Subject: FTP questions
-
- _Q24a_: How does anonymous ftp work?
-
- _A24a_: Anonymous ftp is a method of making files available to anyone
- on the Internet. In brief, if you are on a system with ftp (e.g.
- Unix), you type "ftp fractal.mta.ca", or whatever system you wish to
- access. You are prompted for your name and you reply "anonymous". You
- are prompted for your password and you reply with your email address.
- You then use "ls" to list the files, "cd" to change directories, "get"
- to get files, an "quit" to exit. For example, you could say "cd /pub",
- "ls", "get README", and "quit"; this would get you the file "README".
- See the man page ftp(1) or ask someone at your site for more
- information.
-
- In this FAQ, anonymous ftp addresses are given in the URL form
- ftp://name.of.machine/pub/path [138.73.1.18]. The first part is the
- protocol, FTP, rather than say "gopher", the second part
- "name.of.machine" is the machine you must ftp to. If your machine
- cannot determine the host from the name, you can try the numeric
- Internet address: "ftp 138.73.1.18". The part after the name:
- "/pub/path" is the file or directory to access once you are connected
- to the remote machine.
-
- _Q24b_: What if I can't use ftp to access files?
-
- _A24b_: If you don't have access to ftp because you are on a UUCP,
- Fidonet, BITNET network there is an e-mail gateway at
- 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, often 1MB or less, be careful not exceed
- it.
-
- Subject: Archived pictures
-
- _Q25a_: Where are fractal pictures archived?
-
- News groups
-
- _A25a_: Fractal images (GIFs, JPGs...) are posted to
- alt.binaries.pictures.fractals (also known as abpf); this newsgroup
- has replaced alt.fractals.pictures. However, several
- alt.binaries.pictures groups being badly reputed,
- alt.fractals.pictures seems to have some new activity.
-
- The fractals posted in alt.binaries.pictures.fractals are recorded daily at
-
- http://www.xmission.com/~legalize/fractals/index.html
- http://galaxy.uci.agh.edu.pl/pictures//alt.binaries.pictures.fractals/
- last.html
- http://www.cs.uni-magdeburg.de/pictures/Usenet/fractals/summary/
-
- The following lists are scanty and will evolve soon.
-
- Other archives and university sites (images, tutorials...)
-
- Many Mandelbrot set images are available via
- ftp://ftp.ira.uka.de/pub/graphic/fractals
- Pictures from 1990 and 1991 are available via anonymous ftp at
- ftp://csus.edu/pub/alt.fractals.pictures
- Fractal images including some recent alt.binaries.pictures.fractals
- images are archived at ftp://spanky.triumf.ca/fractals
- This can also be accessed via WWW at http://spanky.triumf.ca/ or
- http://fractal.mta.ca/spanky/
- From Paris, France one of the largest collections (>= 820MB) is Frank
- Roussel's at http://www.cnam.fr/fractals.html
- Fractal animations in MPEG and FLI format are in
- http://www.cnam.fr/fractals/anim.html
- In Bordeaux (France) there is a mirror of this site,
- http://graffiti.cribx1.u-bordeaux.fr/MAPBX/roussel/fractals.htm
- l
- and a Canadian mirror at http://fractal.mta.ca/cnam/
- Another collection of fractal images is archived at
- ftp://ftp.maths.tcd.ie/pub/images/Computer
- Fractal Microscope
- http://www.ncsa.uiuc.edu/Edu/Fractal/Fractal_Home.html
- "Contours of the Mind"
- http://online.anu.edu.au/ITA/ACAT/contours/contours.html
- Spanky Fractal Datbase (Noel Giffin)
- http://spanky.triumf.ca/www/spanky.html
- Yahoo Index of Fractal Art
- http://www.yahoo.com/Arts/Visual_Arts/Computer_Generated/Fracta
- ls/
- Geometry Centre at University of Minnesota
- http://www.geom.umn.edu/graphics/pix/General_Interest/Fractals/
- Computer Graphics Gallery
- http://www.maths.tcd.ie/pub/images/images.html
-
- Many fractal creators have personal web pages showing images, tutorials...
-
- Flame Index A collection of interesting smoke- and flame-like jpeg
- iterated function system images
- http://www.cs.cmu.edu/~spot/flame.htm
- Some images are also available from:
- ftp://hopeless.mess.cs.cmu.edu/spot/film/
-
- Cliff Pickover
- http://sprott.physics.wisc.edu/pickover/home.htm
-
- Fractal Gallery (J. C. Sprott) Personal images and a thousand of
- fractals collected in abpf
- http://sprott.physics.wisc.edu/fractals.htm
-
- Fractal from Ojai (Art Baker)
- http://www.bhs.com/ffo/
-
- Skal's 3D-fractal collection (Pascal Massimino)
- http://www.eleves.ens.fr:8080/home/massimin/quat/f_gal.ang.html
-
- 3d Fractals (Stewart Dickson) via Mathart.com
- http://www.wri.com/~mathart/portfolio/SPD_Frac_portfolio.html
-
- Dirk's 3D-Fractal-Homepage
- http://wwwcip.rus.uni-stuttgart.de/~phy11733/index_e.html
-
- Softsource
- http://www.softsource.com/softsource/fractal.html
-
- Favourite Fractals (Ryan Grant)
- http://www.ncsa.uiuc.edu/SDG/People/rgrant/fav_pics.html
-
- Eric Schol
- http://snt.student.utwente.nl/~schol/gallery/
-
- Mandelbrot and Julia Sets (David E. Joyce)
- http://aleph0.clarku.edu/~djoyce/home.html
-
- Newton's method
- http://aleph0.clarku.edu/~djoyce/newton/newton.html
-
- Gratuitous Fractals (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 fractal homepage
- http://www.d.umn.edu/~dfinton/fractals/
-
- Algorithmic Image Gallery (Giuseppe Zito)
- http://www.ba.infn.it/gallery
-
- Octonion Fractals built using hyper-hyper-complex numbers by Onar Em
- http://www.stud.his.no/~onar/Octonion.html
-
- B' Plasma Cloud (animated gif)
- http://www.az.com/~rsears/fractp1.html
-
- John Bailey's fractal images (<john_bayley@wb.xerox.com>)
- http://www.frontiernet.net/~jmb184/interests/fractals/
-
- Fractal Art Parade (Douglas "D" Cootey <D@itsnet.com>)
- http://www.itsnet.com/~bug/fractals.html
-
- The Fractory (John/Alex <kulesza@math.gmu.edu>)
- http://tqd.advanced.org/3288/
-
- FracPPC gallery (Dennis C. De Mars <demars@netcom.com>)
- http://members.aol.com/ddemars/gallery.html
-
- http://galifrey.triode.net.au/ (Frances Griffin
- <kgriffin@triode.net.au>)
- http://galifrey.triode.net.au/
-
- J.P. Louvet's Fractal Album
- http://graffiti.cribx1.u-bordeaux.fr/MAPBX/louvet/jpl0a.html )
- (Jean-Pierre Louvet <louvet@iuta.u-bordeaux.fr> French and
- English versions)
-
- Carlson's Fractal Gallery
- http://sprott.physics.wisc.edu/carlson.htm (Paul Carlson
- <pjcarlsn@ix.netcom.com>)
-
- Fractals by Paul Carlson
- http://fractal.mta.ca/fractals/carlson/ (an other Paul
- Carlson's Gallery)
-
- Daves's Graphics Page
- http://www.unpronounceable.com/graphics/ (David J. Grossman
- <graphics AT unpronounceable DOT com> replace the AT with '@'
- and DOT with '.' I apologize that I must take this drastic step
- to foil the spammers)
-
- Gumbycat's cyberhome
- http://www.geocities.com/~gumbycat/index.html (Linda Allison
- <gumby-cat@ix.netcom.com> Delete the dash ("-") in gumbycat to
- send e-mail. It's only purpose is to act as a spam deterent!)
-
- Sylvie Gallet Gallery
- http://spanky.triumf.ca/www/fractint/SYLVIE/GALLET.HTML
-
- Sylvie Gallet's Fractal Gallery New pages
- http://ourworld.compuserve.com/homepages/Sylvie_Gallet/homepage
- .htm (Sylvie Gallet <sylvie_gallet@compuserve.com>)
-
- Howard Herscovitch's Home Page
- http://home.echo-on.net/~hnhersco/
-
- Fractalus Home. Fractals by Damien M. Jones
- http://www.geocities.com/SoHo/Lofts/2605/ (Damien M. Jones
- <dmj@emi.net>)
-
- Fractopia Home page. Bill Rossi
- http://members.aol.com/billatny/fractopi.htm (Bill Rossi
- <billatny@aol.com>)
-
- Doug's Gallery. Doug Owen
- http://www.zenweb.com/rayn/doug/ (Doug Owen
- <dougowen@mindspring.com>)
-
- TWG's Gallery. Terry W. Gintz
- http://www.zenweb.com/rayn/twg/ (Terry W. Gintz
- <twgg@ix.netcom.com>)
-
- Fractal Gallery
- http://members.aol.com/MKing77043/index.htm (Mark King
- <MKing77043@aol.com>)
-
- Julian's fractal page
- http://members.aol.com/julianpa/index.htm (Julian Adamaitis
- <julianpa@aol.com>)
-
- Don Archer's fractal art
- http://www.ingress.com/~arch/ (Don Archer <arch@dorsai.org>)
-
- The 4D Julibrot Homepage
- http://www.shop.de/priv/hp/3133/fr_4d.htm (Benno Schmid
- <bm459885@muenchen.org>)
-
- The Fractal of the Day
- http://home.att.net/~Paul.N.Lee/FotD/FotD.html Each day Jim Muth
- (<jamth@mindspring.com>) post a new fractal !
-
- The Beauty of Chaos
- http://i30www.ira.uka.de/~ukrueger/fractals/ A journey in the
- Mandelbrot set (Uwe Krⁿger <uwe.krueger@sap-ag.de>)
-
- The Brian E. Jones Computer Art Gallery
- http://ourworld.compuserve.com/homepages/Brian_E_Jones/ (Brian E.
- Jones <bej2001@netmcr.com>)
-
- Phractal Phantasies
- http://www.globalserve.net/~jval/intro.htm (Margaret
- <mval@globalserve.net> and Jack <jval@globalserve.net> Valero)
-
- Glimpses of a fugitive Universe
- http://www.artvark.com/artvark/ (Rollo Silver <rollo@artvark.com>)
-
- Earl's Computer Art Gallery
- http://computerart.org/
-
- Jacco's Homepage (Jaap Burger <Jacco.Burger@kabelfoon.nl>)
- http://wwwserv.caiw.nl/~jaccobu/index.htm
-
- MOCA: the Museum Of Computer Art The fractal art of Sylvie Gallet, and
- several other artists (Bob Dodson, MOCA curator <bgdodson@ncn.com> ;
- Don Archer, MOCA director)
- http://www.dorsai.org/~moca/
-
- Les St Clair's Fractal Home Page (Les St Clair
- <les_stclair@compuserve.com>)
- http://ourworld.compuserve.com/homepages/Les_StClair/
-
- Numerous links to fractal galleries and other fractal subjects can be found
- at
-
- Spanky fractal database
- http://spanky.triumf.ca/www/welcome1.html
-
- Fractal Images / Immagini frattali su Internet
- http://www.ba.infn.it/www/fractal.html
-
- Chaffey High School's Fractal Image Gallery Links
- http://www.chaffey.org/fractals/galleries.html
-
- Fantastic Fractals. Reference Desk
- http://library.advanced.org/12740/cgi-bin/linking.cgi?browser=m
- sie&language=enu
-
- The Infinite Fractal Loop
-
- The Infinite Fractal Loop was initiated by Douglas Cootey ; it is now
- managed by Damien M. Jones. It is a link between a number of personal
- fractal galleries. The home page of the subscribers display the logo
- of the Infinite Fractal Loop. By clicking on selected areas of this
- logo the server of the loop will call an other site of this loop and
- from this new page, you can go to an other gallery... There are nearly
- 40 members in the loop.
-
- You can have more information and subscribe at
- http://www.emi.net/~dmj/ifl/
-
- _Q25b_: How do I view fractal pictures from
- alt.binaries.pictures.fractals?
-
- _A25b_: A detailed explanation is given in the "alt.binaries.pictures
- FAQ" (see "pictures-FAQ"). This is posted to the pictures newsgroups
- and is available by ftp:
- ftp://rtfm.mit.edu:/pub/usenet/news.answers/pictures-faq/.
-
- In brief, there is a series of things you have to do before viewing
- these posted images. It will depend a little on the system you are
- working with, but there is much in common. Some newsreaders have
- features to automatically extract and decode images ready to display
- ("e" in trn) but if you don't you can use the following manual method.
-
- Manual method
-
- 1. Save/append all posted parts sequentially to one file.
- 2. Edit this file and delete all text segments except what is between
- the BEGIN-CUT and END-CUT portions. This means that BEGIN-CUT and
- END-CUT lines will disappear as well. There will be a section to
- remove for each file segment as well as the final END-CUT line.
- What is left in the file after editing will be bizarre garbage
- starting with begin 660 imagename.GIF and then about 6000 lines
- all starting with the letter "M" followed by a final "end" line.
- This is called a uuencoded file.
- 3. You must uudecode the uuencoded file. There should be an
- appropriate utility at your site; "uudecode filename " should work
- under Unix. Ask a system person or knowledgeable programming type.
- It will decode the file and produce another file called
- imagename.GIF. This is the image file.
- 4. You must use another utility to view these GIF images. It must be
- capable of displaying color graphic images in GIF format. (If you
- get a JPG or JPEG format file, you may have to convert it to a GIF
- file with yet another utility.) In the XWindows environment, you
- may be able to use "xv", "xview", or "xloadimage" to view GIF
- files. If you aren't using X, then you'll either have to find a
- comparable utility for your system or transfer your file to some
- other system. You can use a file transfer utility such as Kermit
- to transfer the binary file to an IBM-PC.
-
- Automated method
-
- Most of the news readers for Windows or Macintosh, as well as web
- browsers such as Netscape or MSIE will automate the decoding for you.
- This may not be true of all web browsers.
-
- Subject: Where can I obtain papers about fractals?
-
- _Q26_: Where can I obtain papers about fractals?
-
- _A26_: There are several Internet sites with fractal papers: There is
- an ftp archive site for preprints and programs on nonlinear dynamics
- and related subjects at: ftp://inls.ucsd.edu/pub.
-
- There are also articles on dynamics, including the IMS preprint
- series, available from ftp://math.sunysb.edu/preprints.
-
- The WWW site http://inls.ucsd.edu/y/Complex/ has some fractal papers.
-
- The site life.csu.edu.au has a collection of fractal programs, papers,
- information related to complex systems, and gopher and World Wide Web
- connections.
-
- The ftp path is:
- ftp://life.csu.edu.au/pub/complex/ (Look in fractals and chaos)
-
- via WWW:
- http://life.csu.edu.au/complex/
-
- R. Vojak has some papers and preprints available from his home page at
- UniversitΘ Paris IX Dauphine.
-
- R. Vojak's home page
- http://www.ceremade.dauphine.fr/~vojak/
-
- Subject: How can I join fractal mailing lists?
-
- _Q27_: How can I join fractal mailing lists?
-
- _A27_: There are now 4 mailing lists devoted to fractals.
-
- FRAC-L
-
- Fractal-Art
-
- Fractint
-
- Fractal Programmers
-
- The FRAC-L mailing list
-
- FRAC-L is a mailing list "Forum on Fractals, Chaos, and Complexity".
- The purpose of frac-l is to be a globally networked forum for
- discourse and collaboration on fractals, chaos, and complexity in
- multiple disciplines, professions, and arts.
-
- To subscribe to frac-l an email message to
- listproc@archives.math.utk.edu containing the sole line of text:
- SUBSCRIBE FRAC-L [email address optional]
-
- To unsubscribe from frac-l, send LISTPROC (_NOT frac-l_) the message:
- UNSUBSCRIBE FRAC-L
-
- Messages may be posted to frac-l by sending email to:
- frac-l@archives.math.utk.edu
-
- Ermel Stepp founded this list; the current listowner is Larry Husch
- and you should contact him (husch@math.utk.edu) if there are any
- difficulties.
-
- The Frac-L archives (http://archives.math.utk.edu/hypermail/frac-l/)
- go back to Fri 09 Jun 1995.
-
- The Fractal-Art Discussion List
-
- This mailing list is open to all individuals and organizations
- interested in all aspects of Fractal Art. This would include fractal
- and digital artists, fractal software developers, gallery owners,
- museum curators, art marketers and brokers, printers, art collectors,
- and simply anybody who just plain likes to look at fractal images.
- This should include just about everybody!
-
- Administrator: Jon Noring noring@netcom.com
-
- To subscribe Fractal-Art send an email message to majordomo@aros.net
- containing the sole line of text:
-
- subscribe fractal-art
-
- Messages may be posted to the fractal-art mailing list by sending
- email to: fractal-art@aros.net
-
- An innovative member of Fractal-Art has created the Unofficial Links
- from Fractal-Art Email Digest
- (http://www.ee.calpoly.edu/~jcline/fractalart-links.htm) which
- collects all the URLs posted to the Fractal-Art mailing list and makes
- them into a web page. Created by Jonathan Cline.
-
- The Fractint mailing list
-
- This mailing list is for the discussion of fractals, fractal art,
- fractal algorithms, fractal software, and fractal programming.
- Specific discussion related to the freeware MS-DOS program Fractint
- and it's ports to other platforms is welcome, but discussion need not
- be Fractint related. Technical discussion is welcome, but so are
- beginner's questions, so don't be shy. This is a good place to share
- Fractint tips, tricks, and techniques, or to wax poetic about other
- fractal software.
-
- To subscribe you can send a mail to majordomo@xmission.com with the
- following command in the body of your email message:
-
- subscribe fractint
-
- Messages may be posted to the fractint mailing list by sending email
- to: Fractint@xmission.com
-
- You can contact the fractint list administrator, Tim Wegner, by
- sending e-mail to: twegner@phoenix.com
-
- The Fractal Programmers mailing list
-
- Subcription/unsubscription/info requests should always be sent to the
- -request address of the mailinglist. This would be:
- <fracprogrammers-list-request@terindell.com>. To subscribe to the
- mailinglist, simply send a message with the word "subscribe" in the
- _Subject:_ field to <fracprogrammers-list-request@terindell.com>.
-
- As in: To: fracprogrammers-list-request@terindell.com
- Subject: subscribe
-
- To unsubscribe from the mailinglist, simply send a message with the
- word "unsubscribe" in the _Subject:_ field to
- <fracprogrammers-list-request@terindell.com>.
-
- Subject: Complexity
-
- _Q28_: What is complexity?
-
- _A28_: Emerging paradigms of thought encompassing fractals, chaos,
- nonlinear science, dynamic systems, self-organization, artificial
- life, neural networks, and similar systems comprise the science of
- complexity. Several helpful online resources on complexity are:
-
- Institute for Research on Complexity
- http://webpages.marshall.edu/~stepp/vri/irc/irc.html
-
- The site life.csu.edu.au has a collection of fractal programs, papers,
- information related to complex systems, and gopher and World Wide Web
- connections.
-
- LIFE via WWW
- http://life.csu.edu.au/complex/
-
- Center for Complex Systems Research (UIUC)
- http://www.ccsr.uiuc.edu/
-
- Complexity International Journal
- http://www.csu.edu.au/ci/ci.html
-
- Nonlinear Science Preprints
- http://xxx.lanl.gov/archive/nlin-sys
-
- Nonlinear Science Preprints via email:
-
- To subscribe to public bulletin board to receive announcements of the
- availability of preprints from Los Alamos National Laboratory, send
- email to nlin-sys@xxx.lanl.gov containing the sole line of text:
- subscribe your-real-name
-
- The Complexity and Management Mailing List. For more information see
- the web archive at http://HOME.EASE.LSOFT.COM/archives/complex-m.html
- or their lexicon of terms at http://lissack.com/lexicon/lexicon.html.
-
- To subscribe:
- http://home.ease.lsoft.com/scripts/wa.exe?SUBED1=complex-m or send a
- message to list@lissack.com with the message "subscribe complex-m" in
- the _body_.
-
- To send a message to the list, send them to COMPLEX@lissack.com or to
- COMPLEX-M@HOME.EASE.LSOFT.COM.
-
- Subject: References
-
- _Q29a_: What are some general references on fractals, chaos, and
- complexity?
-
- _A29a_: Some references are:
-
- M. Barnsley, _Fractals Everywhere_, Academic Press Inc., 1988, 1993.
- ISBN 0-12-079062-9. This is an excellent text book on fractals. This
- is probably the best book for learning about the math underpinning
- fractals. It is also a good source for new fractal types.
-
- M. Barnsley, _The Desktop Fractal Design System_ Versions 1 and 2.
- 1992, 1988. Academic Press. Available from Iterated Systems.
-
- M. Barnsley and P H Lyman, _Fractal Image Compression_. 1993. AK
- Peters Limited. Available from Iterated Systems.
-
- M. Barnsley and L. Anson, _The Fractal Transform_, Jones and Bartlett,
- April, 1993. ISBN 0-86720-218-1. This book is a sequel to _Fractals
- Everywhere_. Without assuming a great deal of technical knowledge, the
- authors explain the workings of the Fractal Transform(tm). The Fractal
- Transform is the compression tool for storing high-quality images in a
- minimal amount of space on a computer. Barnsley uses examples and
- algorithms to explain how to transform a stored pixel image into its
- fractal representation.
-
- R. Devaney and L. Keen, eds., _Chaos and Fractals: The Mathematics
- Behind the Computer Graphics_, American Mathematical Society,
- Providence, RI, 1989. This book contains detailed mathematical
- descriptions of chaos, the Mandelbrot set, etc.
-
- R. L. Devaney, _An Introduction to Chaotic Dynamical Systems_,
- Addison- Wesley, 1989. ISBN 0-201-13046-7. This book introduces many
- of the basic concepts of modern dynamical systems theory and leads the
- reader to the point of current research in several areas. It goes into
- great detail on the exact structure of the logistic equation and other
- 1-D maps. The book is fairly mathematical using calculus and topology.
-
- R. L. Devaney, _Chaos, Fractals, and Dynamics_, Addison-Wesley, 1990.
- ISBN 0-201-23288-X. This is a very readable book. It introduces chaos
- fractals and dynamics using a combination of hands-on computer
- experimentation and precalculus math. Numerous full-color and black
- and white images convey the beauty of these mathematical ideas.
-
- R. Devaney, _A First Course in Chaotic Dynamical Systems, Theory and
- Experiment_, Addison Wesley, 1992. A nice undergraduate introduction
- to chaos and fractals.
-
- A. K. Dewdney, (1989, February). Mathematical Recreations. _Scientific
- American_, pp. 108-111.
-
- G. A. Edgar, _Measure Topology and Fractal Geometry_, Springer-Verlag
- Inc., 1990. ISBN 0-387-97272-2. This book provides the math necessary
- for the study of fractal geometry. It includes the background material
- on metric topology and measure theory and also covers topological and
- fractal dimension, including the Hausdorff dimension.
-
- K. Falconer, _Fractal Geometry: Mathematical Foundations and
- Applications_, Wiley, New York, 1990.
-
- J. Feder, _Fractals_, Plenum Press, New York, 1988. This book is
- recommended as an introduction. It introduces fractals from
- geometrical ideas, covers a wide variety of topics, and covers things
- such as time series and R/S analysis that aren't usually considered.
-
- Y. Fisher (ed), _Fractal Image Compression: Theory and Application_.
- Springer Verlag, 1995.
-
- L. Gardini (ed), _Chaotic Dynamics in Two-Dimensional Noninvertive
- Maps_. World Scientific 1996, ISBN: 9810216475
-
- J. Gleick, _Chaos: Making a New Science_, Penguin, New York, 1987.
-
- B. Hao, ed., _Chaos_, World Scientific, Singapore, 1984. This is an
- excellent collection of papers on chaos containing some of the most
- significant reports on chaos such as "Deterministic Nonperiodic Flow"
- by E.N. Lorenz.
-
- I. Hargittai and C. Pickover. _Spiral Symmetry_ 1992 World Scientific
- Publishing, River Edge, New Jersey 07661. ISBN 981-02-0615-1. Topics:
- Spirals in nature, art, and mathematics. Fractal spirals, plant
- spirals, artist's spirals, the spiral in myth and literature... Loads
- of images.
-
- H. Jⁿrgens, H. O Peitgen, & D. Saupe. 1990 August, The Language of
- Fractals. _Scientific American_, pp. 60-67.
-
- H. Jⁿrgens, H. O. Peitgen, H.O., & D. Saupe, 1992, _Chaos and
- Fractals: New Frontiers of Science_. New York: Springer-Verlag.
-
- S. Levy, _Artificial life : the quest for a new creation_, Pantheon
- Books, New York, 1992. This book takes off where Gleick left off. It
- looks at many of the same people and what they are doing post-Gleick.
-
- B. Mandelbrot, _The Fractal Geometry of Nature_, W. H. FreeMan, New
- York. ISBN 0-7167-1186-9. In this book Mandelbrot attempts to show
- that reality is fractal-like. He also has pictures of many different
- fractals.
-
- B. Mandelbrot, _Les objets fractals_, Flammarion, Paris. ISBN
- 2-08-211188-1. The French Mandelbrot's book, where the word _fractal_
- has been used for the first time.
-
- J.L. McCauley, _Chaos, dynamics, and fractals : an algorithmic
- approach to deterministic chaos_, Cambridge University Press, 1993.
-
- E. R. MacCormac (ed), M. Stamenov (ed), _Fractals of Brain, Fractals
- of Mind : In Search of a Symmetry Bond (Advances in Consciousness
- Research, No 7)_, John Benjamins, ISBN: 1556191871, Subjects include:
- Neural networks (Neurobiology), Mathematical models, Fractals, and
- Consciousness
-
- G.V. Middleton, (ed), _1991: Nonlinear Dynamics, Chaos and Fractals
- (w/ application to geological systems)_ Geol. Assoc. Canada, Short
- Course Notes Vol. 9, 235 p. This volume contains a disk with some
- examples (also as pascal source code) ($25 CDN)
-
- T.F. Nonnenmacher, G.A Losa, E.R Weibel (ed.) _Fractals in Biology and
- Medicine_ ISBN 0817629890, Springer Verlag, 1994
-
- L. Nottale, _Fractal Space-Time and Microphysics, Towards a Theory of
- Scale Relativity_, World Scientific (1993).
-
- E. Ott, _Chaos in dynamical systems_, Cambridge University Press,
- 1993.
-
- E. Ott, T. Sauer, J.A. Yorke (ed.) _Coping with chaos : analysis of
- chaotic data and the exploitation of chaotic systems_, New York, J.
- Wiley, 1994.
-
- D. Peak and M. Frame, _Chaos Under Control: The Art and Science of
- Complexity_, W.H. Freeman and Company, New York 1994, ISBN
- 0-7167-2429-4 "The book is written at the perfect level to help a
- beginner gain a solid understanding of both basic and subtler appects
- of chaos and dynamical systems." - a review from the back cover
-
- H. O. Peitgen and P. H. Richter, _The Beauty of Fractals_,
- Springer-Verlag, New York, 1986. ISBN 0-387-15851-0. This book has
- lots of nice pictures. There is also an appendix giving the
- coordinates and constants for the color plates and many of the other
- pictures.
-
- H. Peitgen and D. Saupe, eds., _The Science of Fractal Images_,
- Springer-Verlag, New York, 1988. ISBN 0-387-96608-0. This book
- contains many color and black and white photographs, high level math,
- and several pseudocoded algorithms.
-
- H. Peitgen, H. Juergens and D. Saupe, _Fractals for the Classroom_,
- Springer-Verlag, New York, 1992. These two volumes are aimed at
- advanced secondary school students (but are appropriate for others
- too), have lots of examples, explain the math well, and give BASIC
- programs.
-
- H. Peitgen, H. Juergens and D. Saupe, _Chaos and Fractals: New
- Frontiers of Science_, Springer-Verlag, New York, 1992.
-
- E. Peters, _Fractal Market Analysis - Applying Chaos Theory to
- Investment & Economics_, John Wiley & Sons, 1994, ISBN 0-471-58524-6.
-
- C. Pickover, _Computers, Pattern, Chaos, and Beauty: Graphics from an
- Unseen World_, St. Martin's Press, New York, 1990. This book contains
- a bunch of interesting explorations of different fractals.
-
- C. Pickover, _Keys to Infinity_, (1995) John Wiley: NY. ISBN
- 0-471-11857-5.
-
- C. Pickover, (1995) _Chaos in Wonderland: Visual Adventures in a
- Fractal World._ St. Martin's Press: New York. ISBN 0-312-10743-9.
- (Devoted to the Lyapunov exponent.)
-
- C. Pickover, _Computers and the Imagination_ (Subtitled: Visual
- Adventures from Beyond the Edge) (1993) St. Martin's Press: New York.
-
- C. Pickover. _The Pattern Book: Fractals, Art, and Nature_ (1995)
- World Scientific. ISBN 981-02-1426-X Some of the patterns are
- ultramodern, while others are centuries old. Many of the patterns are
- drawn from the universe of mathematics.
-
- C. Pickover, _Visualizing Biological Information_ (1995) World
- Scientific: Singapore, New Jersey, London, Hong Kong.
- on the use of computer graphics, fractals, and musical techniques to
- find patterns in DNA and amino acid sequences.
-
- C. Pickover, _Fractal Horizons: The Future Use of Fractals._ (1996)
- St. Martin's Press, New York.
- Speculates on advances in the 21st Century. Six broad sections:
- Fractals in Education, Fractals in Art, Fractal Models and Metaphors,
- Fractals in Music and Sound, Fractals in Medicine, and Fractals and
- Mathematics. Topics include: challenges of using fractals in the
- classroom, new ways of generating art and music, the use of fractals
- in clothing fashions of the future, fractal holograms, fractals in
- medicine, fractals in boardrooms of the future, fractals in chess.
-
- J. Pritchard, _The Chaos Cookbook: A Practical Programming Guide_,
- Butterworth-Heinemann, Oxford, 1992. ISBN 0-7506-0304-6. It contains
- type in and go listings in BASIC and Pascal. It also eases you into
- some of the mathematics of fractals and chaos in the context of
- graphical experimentation. So it's more than just a
- type-and-see-pictures book, but rather a lab tutorial, especially good
- for those with a weak or rusty (or even nonexistent) calculus
- background.
-
- P. Prusinkiewicz and A. Lindenmayer, _The Algorithmic Beauty of
- Plants_, Springer-Verlag, NY, 1990. ISBN 0-387-97297-8. A very good
- book on L-systems, which can be used to model plants in a very
- realistic fashion. The book contains many pictures.
-
- Edward R. Scheinerman, _Invitation to Dynamical Systems_,
- Prentice-Hall, 1996, ISBN 0-13-185000-8, xvii + 373 pages
-
- M. Schroeder, _Fractals, Chaos, and Power Laws: Minutes from an
- Infinite Paradise_, W. H. Freeman, New York, 1991. This book contains
- a clearly written explanation of fractal geometry with lots of puns
- and word play.
-
- J. Sprott, _Strange Attractors: Creating Patterns in Chaos_, M&T Books
- (subsidary of Henry Holt and Co.), New York. ISBN 1-55851-298-5. This
- book describes a new method for generating beautiful fractal patterns
- by iterating simple maps and ordinary differential equations. It
- contains over 350 examples of such patterns, each producing a
- corresponding piece of fractal music. It also describes methods for
- visualizing objects in three and higher dimensions and explains how to
- produce 3-D stereoscopic images using the included red/blue glasses.
- The accompanying 3.5" IBM-PC disk contain source code in BASIC, C,
- C++, Visual BASIC for Windows, and QuickBASIC for Macintosh as well as
- a ready-to-run IBM-PC executable version of the program. Available for
- $39.95 + $3.00 shipping from M&T Books (1-800-628-9658).
-
- D. Stein (ed), _Proceedings of the Santa Fe Institute's Complex
- Systems Summer School_, Addison-Wesley, Redwood City, CA, 1988. See
- especially the first article by David Campbell: "Introduction to
- nonlinear phenomena".
-
- R. Stevens, _Fractal Programming in C_, M&T Publishing, 1989 ISBN
- 1-55851-038-9. This is a good book for a beginner who wants to write a
- fractal program. Half the book is on fractal curves like the Hilbert
- curve and the von Koch snow flake. The other half covers the
- Mandelbrot, Julia, Newton, and IFS fractals.
-
- I. Stewart, _Does God Play Dice?: the Mathematics of Chaos_, B.
- Blackwell, New York, 1989.
-
- Y. Takahashi, _Algorithms, Fractals, and Dynamics_, Plenum Pub Corp,
- (May) 1996, ISBN: 0306451271 Subjects: Differentiable dynamical syste,
- Congresses, Fractals, Algorithms, Differentiable Dynamical Systems,
- Algorithms (Computer Programming)
-
- T. Wegner and B. Tyler, _Fractal Creations_, 2nd ed. The Waite Group,
- 1993. ISBN 1-878739-34-4 This is the book describing the Fractint
- program.
-
- _Q29b_: What are some relevant journals?
-
- _A29b_: Some relevant journals are:
-
- "Chaos and Graphics" section in the quarterly journal _Computers and
- Graphics_. This contains recent work in fractals from the graphics
- perspective, and usually contains several exciting new ideas.
-
- "Mathematical Recreations" section by I. Stewart in _Scientific
- American_.
-
- "Fractal Trans-Light News" published by Roger Bagula
- (<tftn@earthlink.com>). Roger Bagula 11759 Waterhill Road, Lakeside,
- CA 92040 USA. Fractal Trans-Light News is a newsletter of mathematics,
- computer programs, art and poetry. To subscribe, send USD $20 (USD $50
- for overseas delivery) to the address above.
-
- _Fractal Report_. Reeves Telecommunication Labs.
- West Towan House, Porthtowan, TRURO, Cornwall TR4 8AX, U.K.
- WWW: http://ourworld.compuserve.com/homepages/JohndeR/fractalr.htm
- 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
-
- _Q30_: What are multifractals?
-
- _A30_: It is not easy to give a succinct definition of multifractals.
- Following Feder (1988) one may distinguish a measure (of probability,
- or some physical quantity) from its geometric support - which might or
- might not have fractal geometry. Then if the measure has different
- fractal dimension on different parts of the support, the measure is a
- multifractal.
-
- Hastings and Sugihara (1993) distinguish multifractals from
- multiscaling fractals - which have different fractal dimensions at
- different scales (e.g. show a break in slope in a dividers plot, or
- some other power law). I believe different authors use different names
- for this phenomenon, which is often confused with true multifractal
- behaviour.
-
- Aliasing
-
- _Q31a_: What is aliasing?
-
- _A31a_: In computer graphics circles, "aliasing" refers to the
- phenomenon of a high frequency in a continuous signal masquerading as
- a lower frequency in the sampled output of the continuous signal. This
- is a consequence of the discrete sampling used by the computer.
-
- Put another way, it is the appearance of "chuckiness" in an still
- image. Because of the finite resolution of a computer screen, a single
- pixel has an associate width, whereas in mathematics each point is
- infintesimely small, with _no width_. So a single pixel on the screen
- actually visually represents an infinite number of mathematical
- points, each of which may have a different correct visual
- representation.
-
- _Q31b_: What does aliasing have to do with fractals?
-
- _A31b_: Fractals, are very strange objects indeed. Because they have
- an infinite amount of arbitrarily small detail embedded inside them,
- they have an infinite number of frequencies in the images. When we use
- a program to compute an image of a fractal, each pixel in the image is
- actually a sample of the fractal. Because the fractal itself has
- arbitrarily high frequencies inside it, we can never sample high
- enough to reveal the "true" nature of the fractal. _Every_ fractal
- ever computed has aliasing in it. (A special kind of aliasing is
- called "Moire' patterns" and are often visible in fractals as well.)
-
- _Q31c_: How Do I "Anti-Alias" Fractals?
-
- _A31c_: We can't eliminate aliasing entirely from a fractal but we can
- use some tricks to reduce the aliasing present in the fractal. This is
- what is called "anti-aliasing." The technique is really quite simple.
- We decide what size we want our final image to be, and we take our
- samples at a higher resolution than our final size. So if we want a
- 100x100 image, we use at least 3 times the number of pixels in our
- "supersampled" image - 300x300, or 400x400 for even better results.
-
- But wait, we want a 100x100 image, right? Right. So far, we haven't
- done anything special. The anti-aliasing part comes in when we take
- our supersampled image and use a filter to combine several adjacent
- pixels in our supersampled image into a single pixel in our final
- image. The choice of the filter is very important if you want the best
- results! Most image manipulation and paint programs have a resize with
- anti-aliasing option. You can try this and see if you like the
- results. Unfortunately, most programs don't tell you exactly what
- filter they are applying when they "anti-alias," so you have to
- subjectively compare different tools to see which one gives you the
- best results.
-
- The most obvious filter is a simple averaging of neighbouring pixels
- in the supersampled image. Being the most obvious choice, it is
- generally the one most widely implemented in programs. Unfortunately
- it gives poor results. However, many fractal programs are now
- beginning to incorporate anti-aliasing directly in the fractal
- generation process along with a high quality filter. Unless you are a
- programmer, your best bet is to take your supersampled image and try
- different programs and filters to see which one gives you the best
- results.
-
- An example of such filtering in a fractal program can be found on
- Dennis C. De Mars' web page on anti-aliasing in his FracPPC program:
- http://members.aol.com/dennisdema/anti-alias/anti-alias.html
-
- References
-
- The original submission from Rich Thomson is available from
- http://www.mta.ca/~mctaylor/fractals/aliasing.html
-
- To read more about Digital Signal Processing, a good but technical
- book is "Digital Signal Processing", by Alan V. Oppenheim and Ronald
- W. Schafer, ISBN 0-13-214635-5, Prentice-Hall, 1975.
-
- For more on anti-aliasing filters and their application to computer
- graphics, you can read "Reconstruction Filters in Computer Graphics",
- Don P. Mitchell, Arun N. Netravali, Computer Graphics, Volume 22,
- Number 4, August 1988. (SIGGRAPH 1988 Proceedings).
-
- If you're a programmer type and want to experiment with lots of
- different filters on images, or if you're looking for an efficient
- sample implementation of digital filtering, check out Paul Heckbert's
- zoom program at ftp://ftp.cs.utah.edu/pub/painter/zoom.tar.gz
-
- Science Fair Projects
-
- _Q32_: Ideas for science fair projects?
-
- _A32_: You should check with your science teacher about any special
- rules and restrictions. Fractals are really an area of mathematics and
- mathematics may be a difficult topic for science fairs with an
- experimental bias.
-
- 1. Modelling real-world phenomena with fractals, e.g. Lorenz's
- weathers models or fractal plants and landscapes
- 2. Calculate the fractal (box-counting) dimension of a leaf, stone,
- river bed
- 3. _How long is a coastline?_, see The Fractal Geometry of Nature
- 4. Check books and web sites aimed at high school students.
-
- Subject: Notices
-
- _Q33_: Are there any special notices?
-
- _A33_:
-
- From: Lee Skinner <LeeHSkinner@CompuServe.COM>
- Date: Sun, 26 Oct 1997 12:37:33 -0500
- Subject: Explora Science Exhibit
-
- Explora Science Exhibit
-
- The newly combined Explora Science Center and Children's Museum of
- Albuquerque had its Grand Opening on Saturday October 25 1997. One of
- the best exhibits is one illustrating fractals and fractal art.
- Posters made by Doug Czor illustrate how fractals are computed.
- Fractal-art images were exhibited by Lee Skinner, Jon Noring, Rollo
- Silver and Bob Hill. The exhibit will probably be on display for about
- 6 months. Channel 13 News had a brief story about the opening and
- broadcasted some of the fractal-art images. The museum's gift shop is
- selling Rollo's Fractal Universe calendars and 4 different mouse-pad
- designs of fractals by Lee Skinner. Two of the art pieces are
- 18432x13824/65536 Cibachrome prints using images recalculated by Jon
- Noring.
-
- Lee Skinner
-
- _________________________________________________________________
-
- From: Javier Barrallo
- Date: Sun, 14 Sep 1997 18:06:14 +0200
- Subject: Mathematics & Design - 98
-
- INVITATION AND CALL FOR PAPERS
- Second International Conference on Mathematics & Design 98
-
- Dear friend,
-
- This is to invite you to participate in the Second International
- Conference on Mathematics & Design 98 to be held at San Sebastian,
- Spain, 1-4 June 1998.
-
- The main objective of these Conferences is to bring together
- mathematicians, engineers, architects, designers and scientists
- interested on the interaction between Mathematics and Design, where
- the world design is understood in its more broad sense, including all
- types of design.
-
- Further information and a regularly updated program is available
- under:
-
- http://www.sc.ehu.es/md98
-
- We will be pleased if you kindly forward this message to colleagues of
- yours who might be interested in this announcement.
-
- Hoping to be able to have your valuable collaboration and assistance
- to the Conference,
-
- The Organising Committee
- E-mail: mapbacaj@sa.ehu.es
-
- _________________________________________________________________
-
- From: John de Rivaz <John@longevb.demon.co.uk>
-
- Mr Roger Bagula, publisher of The Fractal Translight Newsletter, is seeking
- new articles. Write to him for a sample copy - he is not on the Internet -
- and he appreciates something for materials and postage.
-
- Mr Roger Bagula,
- 11759 Waterhill Road
- Lakeside
- CA 90240-2905
- USA
-
- _________________________________________________________________
-
- 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.
-
- _________________________________________________________________
-
- From Clifford Pickover <cliff@watson.ibm.com>
-
- You are cordially invited to submit interesting, well-written articles
- for the "Chaos and Graphics Section" of the international journal
- Computers and Graphics. I edit this on-going section which appears in
- each issue of the journal. Topics include the mathematical,
- scientific, and artistic application of fractals, chaos, and related.
- Your papers can be quite short if desired, for example, often a page
- or two is sufficient to convey an idea and a pretty graphic. Longer,
- technical papers are also welcome. The journal is peer-reviewed. I
- publish color, where appropriate. Write to me for guidelines. Novelty
- of images is often helpful.
-
- Goals
-
- The goal of my section is to provide visual demonstrations of
- complicated and beautiful structures which can arise in systems based
- on simple rules. The section presents papers on the seemingly
- paradoxical combinations of randomness and structure in systems of
- mathematical, physical, biological, electrical, chemical, and artistic
- interest. Topics include: iteration, cellular automata, bifurcation
- maps, fractals, dynamical systems, patterns of nature created from
- simple rules, and aesthetic graphics drawn from the universe of
- mathematics and art.
-
- Subject: Acknowledgements
-
- _Q34_: Who has contributed to the sci.fractals FAQ?
-
- _A34_: Former editors, participants in the Usenet group sci.fractals
- and the listserv forum frac-l have provided most of the content of
- sci.fractals FAQ. For their help with this FAQ, "thank you" to:
-
- Alex Antunes, Donald Archer, Simon Arthur, Roger Bagula, John Beale,
- Matthew J. Bernhardt, Steve Bondeson, Erik Boman, Jacques Carette,
- John Corbit, Douglas Cootey, Charles F. Crocker, Michael Curl, Predrag
- Cvitanovic, Paul Derbyshire, John de Rivaz, Abhijit Deshmukh, Tony
- Dixon, Jⁿrgen Dollinger, Robert Drake, Detlev Droege, Gerald Edgar,
- Glenn Elert, Gordon Erlebacher, Yuval Fisher, Duncan Foster, David
- Fowler, Murray Frank, Jean-loup Gailly, Noel Giffin, Frode Gill, Terry
- W. Gintz, Earl Glynn, Lamont Granquist, John Holder, Jon Horner, Luis
- Hernandez-Urδa, Jay Hill, Arto Hoikkala, Carl Hommel, Robert Hood,
- Larry Husch, Oleg Ivanov, Henrik Wann Jensen, Simon Juden, J.
- Kai-Mikael, Leon Katz, Matt Kennel, Robert Klep, Dave Kliman, Pavel
- Kotulsky, Tal Kubo, Per Olav Lande, Paul N. Lee, Jon Leech, Otmar
- Lendl, Ronald Lewis, Jean-Pierre Louvet, Garr Lystad, Jose Oscar
- Marques, Douglas Martin, Brian Meloon, Tom Menten, Guy Metcalfe,
- Eugene Miya, Lori Moore, Robert Munafo, Miriam Nadel, Ron Nelson, Tom
- Parker, Dale Parson, Matt Perry, Cliff Pickover, Francois Pitt, Olaf
- G. Podlaha, Francesco Potort∞, Kevin Ring, Michael Rolenz, Tom Scavo,
- Jeffrey Shallit, Ken Shirriff, Rollo Silver, Lee H Skinner, David
- Sharp, J. C. Sprott, Gerolf Starke, Bruce Stewart, Dwight Stolte,
- Michael C. Taylor, Rich Thomson, Tommy Vaske, Tim Wegner, Andrea
- Whitlock, David Winsemius, Erick Wong, Wayne Young, Giuseppe Zito, and
- others.
-
- A special thanks to Jean-Pierre Louvet, who has taken on the task of
- maintaining the sections for fractal software and where fractal
- pictures are archived.
-
- If I have missed you, I am very sorry, let me know
- (fractal-faq@mta.ca) and I will add you to the list. Without the help
- of these contributors, the sci.fractals FAQ would be not be possible.
-
- Subject: Copyright
-
- _Q35_: Copyright?
-
- _A35_: This document, "sci.fractals FAQ", is _Copyright ⌐ 1997-1998 by
- Michael C. Taylor and Jean-Pierre Louvet._ All Rights Reserved. This
- document is published in New Brunswick, Canada.
-
- Previous versions:
- Copyright 1995-1997 Michael Taylor
- Copyright 1995 Ermel Stepp (edition v2n1)
- Copyright 1993-1994 Ken Shirriff
-
- The Fractal FAQ was created by Ken Shirriff and edited by him through
- September 26, 1994. The second editor of the Fractal FAQ is Ermel
- Stepp (Feb 13, 1995). Since December 2, 1995 the acting editor has
- been Michael C. Taylor.
-
- Permission is granted for _non-profit_ reproduction and distribution
- of this issue of the sci.fractals FAQ as a complete document. You may
- product complete copies, including this notice, of the sci.fractals
- FAQ for classroom use. This _does not_ mean automatic permission for
- usage in CD-ROM collections or commercial educational products. If you
- would like to include sci.fractals FAQ, in whole or in part, in a
- commercial product contact Michael C. Taylor.
-
- Warranty
-
- This document is provided as is without any express or implied
- warranty.
-
- Contacting the editors
-
- If you would like to contact the editors, you may do so in writing at
- the following addresses:
-
- Attn: Michael Taylor
- Computing Services
- Mount Allison University
- 49A York Street
- Sackville, New Brunswick E4L 1C7
- CANADA
-
- email: fractal-faq@mta.ca
-