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Subject: sci.fractals FAQ

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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 (&gt100 pages double-spaced) essay on chaos, fractals, and non-linear dynamics. It requires a moderate math background, though is not aimed at the mathematician. http://www.columbia.edu/~gae4/chaos/ Mathsnet this site has several pages devoted to fractals and complex numbers. http://www.anglia.co.uk/education/mathsnet/ Fractals in Your Future by Ronald Lewis <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 Jürgen 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 []. The first part is the protocol, FTP, rather than say "gopher", the second part "name.of.machine" is the machine you must ftp to. If your machine cannot determine the host from the name, you can try the numeric Internet address: "ftp". The part after the name: "/pub/path" is the file or directory to access once you are connected to the remote machine. _Q24b_: What if I can't use ftp to access files? _A24b_: If you don't have access to ftp because you are on a UUCP, Fidonet, BITNET network there is an e-mail gateway at 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 Krüger <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 Université 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. Jürgens, H. O Peitgen, & D. Saupe. 1990 August, The Language of Fractals. _Scientific American_, pp. 60-67. H. Jürgens, H. O. Peitgen, H.O., & D. Saupe, 1992, _Chaos and Fractals: New Frontiers of Science_. New York: Springer-Verlag. S. Levy, _Artificial life : the quest for a new creation_, Pantheon Books, New York, 1992. This book takes off where Gleick left off. It looks at many of the same people and what they are doing post-Gleick. B. Mandelbrot, _The Fractal Geometry of Nature_, W. H. FreeMan, New York. ISBN 0-7167-1186-9. In this book Mandelbrot attempts to show that reality is fractal-like. He also has pictures of many different 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, Jürgen Dollinger, Robert Drake, Detlev Droege, Gerald Edgar, Glenn Elert, Gordon Erlebacher, Yuval Fisher, Duncan Foster, David Fowler, Murray Frank, Jean-loup Gailly, Noel Giffin, Frode Gill, Terry W. Gintz, Earl Glynn, Lamont Granquist, John Holder, Jon Horner, Luis Hernandez-Urëa, Jay Hill, Arto Hoikkala, Carl Hommel, Robert Hood, Larry Husch, Oleg Ivanov, Henrik Wann Jensen, Simon Juden, J. Kai-Mikael, Leon Katz, Matt Kennel, Robert Klep, Dave Kliman, Pavel Kotulsky, Tal Kubo, Per Olav Lande, Paul N. Lee, Jon Leech, Otmar Lendl, Ronald Lewis, Jean-Pierre Louvet, Garr Lystad, Jose Oscar Marques, Douglas Martin, Brian Meloon, Tom Menten, Guy Metcalfe, Eugene Miya, Lori Moore, Robert Munafo, Miriam Nadel, Ron Nelson, Tom Parker, Dale Parson, Matt Perry, Cliff Pickover, Francois Pitt, Olaf G. Podlaha, Francesco Potortì, Kevin Ring, Michael Rolenz, Tom Scavo, Jeffrey Shallit, Ken Shirriff, Rollo Silver, Lee H Skinner, David Sharp, J. C. Sprott, Gerolf Starke, Bruce Stewart, Dwight Stolte, Michael C. Taylor, Rich Thomson, Tommy Vaske, Tim Wegner, Andrea Whitlock, David Winsemius, Erick Wong, Wayne Young, Giuseppe Zito, and 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