From: aa459@ccn.cs.dal.ca (Michael Charles Taylor) Followup-To: poster Newsgroups: sci.fractals,sci.answers,news.answers Subject: sci.fractals FAQ Approved: news-answers-request@MIT.EDU Summary: Frequently Asked Questions about Fractals Keywords: fractals Mandelbrot Julia chaos IFS Archive-name: sci/fractals-faq Posting-Frequency: monthly Last-modified: July 8 1996 Version: v3n7 URL: http://www.ccn.cs.dal.ca/~aa459/sci/fractals-faq-html/ sci.fractals FAQ (Frequently Asked Questions) _________________________________________________________________ Volume 3 Number 7 Date July 8, 1996 _________________________________________________________________ Copyright 1995-1996 by Michael C. Taylor. All Rights Reserved. _________________________________________________________________ Introduction This FAQ is posted monthly to sci.fractals, a Usenet newsgroup about fractals; mathematics and software. This document is aimed at being a reference about fractals, including answers to commonly asked questions, archive listings of fractal software, images, and papers that can be accessed via the Internet using FTP, gopher, or World-Wide-Web (WWW), and a bibliography for further readings. The FAQ does not give a textbook approach to learning about fractals, but a summary of information from which you can learn more about and explore fractals. This FAQ is posted monthly to the Usenet newsgroups: sci.fractals ("Objects of non-integral dimension and other chaos") , sci.answers, and news.answers. Like most FAQs it can be obtained freely with a WWW browser (such as Mosaic or Netscape), or by anonymous FTP from ftp://rtfm.mit.edu/pub/usenet/news.answers/sci/fractals-faq (USA) (which is also ftp://18.181.0.24/pub/usenet/news.answers/sci/fractals-faq if you have Domain Name lookup problems). It is also available from ftp://ftp.Germany.EU.net/pub/newsarchive/news.answers (Europe), http://spanky.triumf.ca/pub/fractals/docs/SCI_FRACTALS.FAQ (Canada), and http://www.chebucto.ns.ca/~aa459/sci/fractals-faq (Canada). The hypertext version is available from http://www.chebucto.ns.ca/~aa459/sci/fractals-faq-html/. Those without FTP or WWW access can obtain the FAQ via email, by sending a message to mail-server@rtfm.mit.edu with the message: send usenet/news.answers/sci/fractals-faq _________________________________________________________________ Suggestions, Comments, Mistakes Please send suggestions and corrections about the sci.fractals FAQ to aa459@chebucto.ns.ca. Without your contributions, the FAQ for sci.fractals will not grow in its wealth. "For the readers, by the readers." Rather than calling me a fool behind my back, if you find a mistake, whether spelling or factual, please send me a note. That way readers of future versions of the FAQ will not be misled. Also if you have problems with the appearance of the hypertext version. There should not be any Netscape only markup tags contained in the hypertext verion, but I have not followed strict HTML 2.0 specifications. If the appearance is "incorrect" let me know what problems you experience. Why the different name? The old FAQ about fractals has not not been updated for over a year and has not been posted by Dr. Ermel Stepp, in as long. So this is a new FAQ based on the previous FAQ's information. Hence it is now the sci.fractals FAQ. If you are viewing this file with a newsreader such as "rn" or "trn", you can search for a particular question by using "g^Qn" (that's lower-case g, up-arrow, Q, and n, the number of the question you wish). Or you may browse forward using to search for a Subject: line. The questions which are answered are: Q0: I am new to the 'Net what should I know about being online? NEW Q1: I want to learn about fractals. What should I read first? Q2: What is a fractal? What are some examples of fractals? Q3: What is chaos? Q4a: What is fractal dimension? How is it calculated? Q4b: What is topological dimension? Q5: What is a strange attractor? Q6a: What is the Mandelbrot set? Q6b: How is the Mandelbrot set actually computed? Q6c: Why do you start with z = 0? NEW Q6d: What are the bounds of the Mandelbrot set? When does it diverge? Q6e: How can I speed up Mandelbrot set generation? Q6f: What is the area of the Mandelbrot set? Q6g: What can you say about the structure of the Mandelbrot set? Q6h: Is the Mandelbrot set connected? Q6i: What is the Mandelbrot Encyclopedia? Q6j: What is the dimension of the Mandelbrot Set? Q7a: What is the difference between the Mandelbrot set and a Julia set? Q7b: What is the connection between the Mandelbrot set and Julia sets? Q7c: How is a Julia set actually computed? Q7d: What are some Julia set facts? Q8a: How does complex arithmetic work? Q8b: How does quaternion arithmetic work? Q9: What is the logistic equation? Q10: What is Feigenbaum's constant? NEW Q11a: What is an iterated function system (IFS)? Q11b: What is the state of fractal compression? Q12a: How can you make a chaotic oscillator? Q12b: What are laboratory demonstrations of chaos? Q13: What are L-systems? Q14: What is some information on fractal music? Q15: How are fractal mountains generated? Q16: What are plasma clouds? Q17a: Where are the popular periodically-forced Lyapunov fractals described? Q17b: What are Lyapunov exponents? Q17c: How can Lyapunov exponents be calculated? Q18: Where can I get fractal T-shirts and posters? Q19: How can I take photos of fractals? Q20: How can 3-D fractals be generated? Q21a: What is Fractint? NEW Q21b: How does Fractint achieve its speed? Q22: Where can I obtain software packages to generate fractals? NEW Q23a: How does anonymous ftp work? Q23b: What if I can't use ftp to access files? Q24a: Where are fractal pictures archived? NEW Q24b: How do I view fractal pictures from alt.binaries.pictures.fractals? Q25: Where can I obtain fractal papers? Q26: How can I join the FRAC-L fractal discussion? Q27: What is complexity? Q28a: What are some general references on fractals and chaos? NEW Q28b: What are some relevant journals? Q28c: What are some other Internet references? Q29: What is a multifractal? Q30: Are there any special notices? NEW Q31: Who has contributed to the Fractal FAQ? NEW Q32: Copyright? _________________________________________________________________ Subject: USENET and Netiquette Q0: I am new to the 'Net what should I know about being online? NEW A0: Read the guidelines and Frequently Asked Questions (FAQ) in news.announce.newusers. They are available from: Welcome to news.newusers.questions ftp://rtfm.mit.edu/pub/usenet/news.answers/news-newusers-intro ftp://garbo.uwasa.fi/pc/doc-net/usenews.zip A Primer on How to Work With the Usenet Community ftp://rtfm.mit.edu/pub/usenet/news.answers/usenet/primer/part1 Frequently Asked Questions about Usenet ftp://rtfm.mit.edu/pub/usenet/news.answers/usenet/faq/part1 Rules for posting to Usenet ftp://rtfm.mit.edu/pub/usenet/news.answers/usenet/posting-rules/part1 Emily Postnews Answers Your Questions on Netiquette ftp://rtfm.mit.edu/pub/usenet/news.answers/usenet/emily-postnews/part1 Hints on writing style for Usenet ftp://rtfm.mit.edu/pub/usenet/news.answers/usenet/writing-style/part1 What is Usenet? ftp://rtfm.mit.edu/pub/usenet/news.answers/usenet/what-is/part1 There are a couple of common mistakes people make, posting ads, posting large binaries (images or programs), and posting off-topic. Do Not Post Images to sci.fractals. If you follow this rule people will be your friend. There is a special place for you to post your images, alt.binaries.pictures.fractals. The other group is considered obsolete and may not be carried to as many people as a.b.p.f. In fact there is CancelBot which will delete any posts it finds in sci.fractals (and most other non-binaries newsgroup) so nearly no one will see it. Post only about fractals, this includes fractal mathematics, software to generate fractals, where to get fractal posters and T-shirts, and fractals as art. Do not bother posting about news events not directly related to fractals, or about which OS / hardware / language is better. These lead to flame wars. Do not post advertisements. I should not have to mention this, but I will. If you have some fractal software available as shareware or shrink-wrap do not post your brief announcement more than once. After than, you should limit yourself to notices of upgrades and responding via e-mail to people looking for fractal software. ______________________________________________________________________ Subject: Learning about fractals Q1: I want to learn about fractals. What should I read/view first? A1: Chaos is a good book to get a general overview and history that does not require an extensive math background. Fractals Everywhere is a textbook on fractals that describes what fractals are and how to generate them, but it requires knowing intermediate analysis. Chaos, Fractals, and Dynamics is also a good start. There is a longer book list at the end of this file (see "What are some general references?"). Also, there are networked resources available, such as: Exploring Fractals and Chaos http://www.lib.rmit.edu.au/fractals/exploring.html Fractal Microscope http://www.ncsa.uiuc.edu/Edu/Fractal/Fractal_Home.html Dynamical Systems and Technology Project: a introduction for high-school students http://math.bu.edu/DYSYS/dysys.html An Introduction to Fractals (Written by Paul Bourke) http://www.auckland.ac.nz/arch/pdbourke/fractals/fracintro.html Fractals and Scale (by David G. Green) http://life.csu.edu.au/complex/tutorials/tutorial3.html What are fractals? (by Neal Kettler) http://www.vis.colostate.edu/~user1209/fractals/fracinfo.html Fract-ED a fractal tutorial for beginners, targeted for high school/tech school students. http://www.ealnet.com/ealsoft/fracted.htm Mandelbrot Questions & Answers (without any scary details) by Paul Derbyshire http://chat.carleton.ca/~pderbysh/mandlfaq.html _________________________________________________________________ Subject: What is a fractal? Q2: What is a fractal? What are some examples of fractals? A2: A fractal is a rough or fragmented geometric shape that can be subdivided in parts, each of which is (at least approximately) a reduced-size copy of the whole. Fractals are generally self-similar and independent of scale. There are many mathematical structures that are fractals; e.g. Sierpinski triangle, Koch snowflake, Peano curve, Mandelbrot set, and Lorenz attractor. Fractals also describe many real-world objects, such as clouds, mountains, turbulence, and coastlines, that do not correspond to simple geometric shapes. Benoit B. Mandelbrot gives a mathematical definition of a fractal as a set of which the Hausdorff Besicovich dimension strictly exceeds the topological dimension. However, he is not satisfied with this definition as it excludes sets one would consider fractals. According to Mandelbrot, who invented the word: "I coined fractal from the Latin adjective fractus. The corresponding Latin verb frangere means "to break:" to create irregular fragments. It is therefore sensible - and how appropriate for our needs! - that, in addition to "fragmented" (as in fraction or refraction), fractus should also mean "irregular," both meanings being preserved in fragment." (The Fractal Geometry of Nature, page 4.) _________________________________________________________________ Subject: Chaos Q3: What is chaos? A3: Chaos is apparently unpredictable behavior arising in a deterministic system because of great sensitivity to initial conditions. Chaos arises in a dynamical system if two arbitrarily close starting points diverge exponentially, so that their future behavior is eventually unpredictable. Weather is considered chaotic since arbitrarily small variations in initial conditions can result in radically different weather later. This may limit the possibilities of long-term weather forecasting. (The canonical example is the possibility of a butterfly's sneeze affecting the weather enough to cause a hurricane weeks later.) Devaney defines a function as chaotic if it has sensitive dependence on initial conditions, it is topologically transitive, and periodic points are dense. In other words, it is unpredictable, indecomposable, and yet contains regularity. Allgood and Yorke define chaos as a trajectory that is exponentially unstable and neither periodic or asymptotically periodic. That is, it oscillates irregularly without settling down. The following resources may be helpful to understand chaos: sci.nonlinear FAQ (UK) http://www.fen.bris.ac.uk/engmaths/research/nonlinear/faq.html sci.nonlinear FAQ (US) http://amath.colorado.edu/appm/faculty/jdm/faq.html Exploring Chaos and Fractals http://www.lib.rmit.edu.au/fractals/exploring.html Chaos and Complexity Homepage (M. Bourdour) http://www.cc.duth.gr/~mboudour/nonlin.html The Institute for Nonlinear Science http://inls.ucsd.edu/ _________________________________________________________________ Subject: Fractal dimension Q4a : What is fractal dimension? How is it calculated? A4a: A common type of fractal dimension is the Hausdorff-Besicovich Dimension, but there are several different ways of computing fractal dimension. Roughly, fractal dimension can be calculated by taking the limit of the quotient of the log change in object size and the log change in measurement scale, as the measurement scale approaches zero. The differences come in what is exactly meant by "object size" and what is meant by "measurement scale" and how to get an average number out of many different parts of a geometrical object. Fractal dimensions quantify the static geometry of an object. For example, consider a straight line. Now blow up the line by a factor of two. The line is now twice as long as before. Log 2 / Log 2 = 1, corresponding to dimension 1. Consider a square. Now blow up the square by a factor of two. The square is now 4 times as large as before (i.e. 4 original squares can be placed on the original square). Log 4 / log 2 = 2, corresponding to dimension 2 for the square. Consider a snowflake curve formed by repeatedly replacing ___ with _/\_, where each of the 4 new lines is 1/3 the length of the old line. Blowing up the snowflake curve by a factor of 3 results in a snowflake curve 4 times as large (one of the old snowflake curves can be placed on each of the 4 segments _/\_). Log 4 / log 3 = 1.261... Since the dimension 1.261 is larger than the dimension 1 of the lines making up the curve, the snowflake curve is a fractal. For more information on fractal dimension and scale, via the WWW Fractals and Scale (by David G. Green) http://life.csu.edu.au/complex/tutorials/tutorial3.html Fractal dimension references: 1. J. P. Eckmann and D. Ruelle, Reviews of Modern Physics 57, 3 (1985), pp. 617-656. 2. K. J. Falconer, The Geometry of Fractal Sets, Cambridge Univ. Press, 1985. 3. T. S. Parker and L. O. Chua, Practical Numerical Algorithms for Chaotic Systems, Springer Verlag, 1989. 4. H. Peitgen and D. Saupe, eds., The Science of Fractal Images, Springer-Verlag Inc., New York, 1988. ISBN 0-387-96608-0. This book contains many color and black and white photographs, high level math, and several pseudocoded algorithms. 5. G. Procaccia, Physica D 9 (1983), pp. 189-208. 6. J. Theiler, Physical Review A 41 (1990), pp. 3038-3051. References on how to estimate fractal dimension: 1. S. Jaggi, D. A. Quattrochi and N. S. Lam, Implementation and operation of three fractal measurement algorithms for analysis of remote-sensing data., Computers & Geosciences 19, 6 (July 1993), pp. 745-767. 2. E. Peters, Chaos and Order in the Capital Markets , New York, 1991. ISBN 0-471-53372-6 Discusses methods of computing fractal dimension. Includes several short programs for nonlinear analysis. 3. J. Theiler, Estimating Fractal Dimension, Journal of the Optical Society of America A-Optics and Image Science 7, 6 (June 1990), pp. 1055-1073. There are some programs available to compute fractal dimension. They are listed in a section below (see Q22 "Fractal software"). Reference on the Hausdorff-Besicovitch dimension A clear and concise (2 page) write-up of the definition of the Hausdorff-Besicovitch dimension in MS-Word 6.0 format is available in zip format. hausdorff.zip (~26KB) http://www.newciv.org/jhs/hausdorff.zip Q4b : What is topological dimension? A4b: Topological dimension is the "normal" idea of dimension; a point has topological dimension 0, a line has topological dimension 1, a surface has topological dimension 2, etc. For a rigorous definition: A set has topological dimension 0 if every point has arbitrarily small neighborhoods whose boundaries do not intersect the set. A set S has topological dimension k if each point in S has arbitrarily small neighborhoods whose boundaries meet S in a set of dimension k-1, and k is the least nonnegative integer for which this holds. _________________________________________________________________ Subject: Strange attractors Q5: What is a strange attractor? A5: A strange attractor is the limit set of a chaotic trajectory. A strange attractor is an attractor that is topologically distinct from a periodic orbit or a limit cycle. A strange attractor can be considered a fractal attractor. An example of a strange attractor is the Henon attractor. Consider a volume in phase space defined by all the initial conditions a system may have. For a dissipative system, this volume will shrink as the system evolves in time (Liouville's Theorem). If the system is sensitive to initial conditions, the trajectories of the points defining initial conditions will move apart in some directions, closer in others, but there will be a net shrinkage in volume. Ultimately, all points will lie along a fine line of zero volume. This is the strange attractor. All initial points in phase space which ultimately land on the attractor form a Basin of Attraction. A strange attractor results if a system is sensitive to initial conditions and is not conservative. Note: While all chaotic attractors are strange, not all strange attractors are chaotic. Reference: 1. Grebogi, et al., Strange Attractors that are not Chaotic, Physica D 13 (1984), pp. 261-268. _________________________________________________________________ Subject: The Mandelbrot set Q6a : What is the Mandelbrot set? A6a: The Mandelbrot set is the set of all complex c such that iterating z -> z² + c does not go to infinity (starting with z = 0). Other images and resources are: Frank Rousell's hyperindex of clickable/retrievable Mandelbrot images http://www.cnam.fr/fractals/mandel.html Neal Kettler's Interactive Mandelbrot http://www.vis.colostate.edu/~user1209/fractals/explorer/ Panagiotis J. Christias' Mandelbrot Explorer http://www.softlab.ntua.gr/mandel/mandel.html 2D & 3D Mandelbrot fractal explorer (set up by Robert Keller) http://reality.sgi.com/employees/rck/hydra/ Mandelbrot viewer written in Java (by Simon Arthur) http://www.mindspring.com/~chroma/mandelbrot.html Mandelbrot Questions & Answers (without any scary details) by Paul Derbyshire http://chat.carleton.ca/~pderbysh/mandlfaq.html Quick Guide to the Mandelbrot Set (includes a tourist map) by Paul Derbyshire http://chat.carleton.ca/~pderbysh/manguide.html Beginner's guide to the Mandelbrot Set by Eric Carr http://www.cs.odu.edu/~carr/mandelbr.html Java program to view the Mandelbrot Set by Ken Shirriff ftp://ftp.cs.berkeley.edu/ucb/sprite/www/java/mandel.html Q6b : How is the Mandelbrot set actually computed? A6b: The basic algorithm is: For each pixel c, start with z = 0. Repeat z = z² + c up to N times, exiting if the magnitude of z gets large. If you finish the loop, the point is probably inside the Mandelbrot set. If you exit, the point is outside and can be colored according to how many iterations were completed. You can exit if |z| > 2, since if z gets this big it will go to infinity. The maximum number of iterations, N, can be selected as desired, for instance 100. Larger N will give sharper detail but take longer. Frode Gill has some information about generating the Mandelbrot Set at http://www.krs.hia.no/~fgill/mandel.html. Q6c : Why do you start with z = 0? A6c: Zero is the critical point of z = z² + c, that is, a point where d/dz (z² + c) = 0. If you replace z² + c with a different function, the starting value will have to be modified. E.g. for z -> z² + z, the critical point is given by 2z + 1 = 0, so start with z = -½. In some cases, there may be multiple critical values, so they all should be tested. Critical points are important because by a result of Fatou: every attracting cycle for a polynomial or rational function attracts at least one critical point. Thus, testing the critical point shows if there is any stable attractive cycle. See also: 1. M. Frame and J. Robertson, A Generalized Mandelbrot Set and the Role of Critical Points, Computers and Graphics 16, 1 (1992), pp. 35-40. Note that you can precompute the first Mandelbrot iteration by starting with z = c instead of z = 0, since 0² + c = c. Q6d: What are the bounds of the Mandelbrot set? When does it diverge? A6d: The Mandelbrot set lies within |c| <= 2. If |z| exceeds 2, the z sequence diverges. Proof: if |z| > 2, then |z² + c| >= |z ²| - |c| > 2|z| - |c|. If |z| >= |c|, then 2|z| - |c| > |z|. So, if |z| > 2 and |z| >= c, then |z² + c| > |z|, so the sequence is increasing. (It takes a bit more work to prove it is unbounded and diverges.) Also, note that |z| = c, so if |c| > 2, the sequence diverges. Q6e : How can I speed up Mandelbrot set generation? A6e: See the information on speed below (see "Fractint"). Also see: 1. R. Rojas, A Tutorial on Efficient Computer Graphic Representations of the Mandelbrot Set, Computers and Graphics 15, 1 (1991), pp. 91-100. Q6f: What is the area of the Mandelbrot set? A6f: Ewing and Schober computed an area estimate using 240,000 terms of the Laurent series. The result is 1.7274... However, the Laurent series converges very slowly, so this is a poor estimate. A project to measure the area via counting pixels on a very dense grid shows an area around 1.5066. (Contact rpm%mrob.uucp@spdcc.com for more information.) Hill and Fisher used distance estimation techniques to rigorously bound the area and found the area is between 1.503 and 1.5701. References: 1. J. H. Ewing and G. Schober, The Area of the Mandelbrot Set, Numer. Math. 61 (1992), pp. 59-72. 2. Y. Fisher and J. Hill, Bounding the Area of the Mandelbrot Set, Numerische Mathematik,. (Submitted for publication). Available via World Wide Web (in Postscript format) http://inls.ucsd.edu/y/Complex/area.ps.Z. Q6g: What can you say about the structure of the Mandelbrot set? A6g: Most of what you could want to know is in Branner's article in Chaos and Fractals: The Mathematics Behind the Computer Graphics. Note that the Mandelbrot set in general is not strictly self-similar; the tiny copies of the Mandelbrot set are all slightly different, mainly because of the thin threads connecting them to the main body of the Mandelbrot set. However, the Mandelbrot set is quasi-self-similar. However, the Mandelbrot set is self-similar under magnification in neighborhoods of Misiurewicz points (e.g. -.1011 + .9563i). The Mandelbrot set is conjectured to be self-similar around generalized Feigenbaum points (e.g. -1.401155 or -.1528 + 1.0397i), in the sense of converging to a limit set. References: 1. T. Lei, Similarity between the Mandelbrot set and Julia Sets, Communications in Mathematical Physics 134 (1990), pp. 587-617. 2. J. Milnor, Self-Similarity and Hairiness in the Mandelbrot Set, in Computers in Geometry and Topology, M. Tangora (editor), Dekker, New York, pp. 211-257. The "external angles " of the Mandelbrot set (see Douady and Hubbard or brief sketch in "Beauty of Fractals") induce a Fibonacci partition onto it. The boundary of the Mandelbrot set and the Julia set of a generic c in M have Hausdorff dimension 2 and have topological dimension 1. The proof is based on the study of the bifurcation of parabolic periodic points. (Since the boundary has empty interior, the topological dimension is less than 2, and thus is 1.) Reference: 1. M. Shishikura, The Hausdorff Dimension of the Boundary of the Mandelbrot Set and Julia Sets, The paper is available from anonymous ftp: ftp://math.sunysb.edu/preprints/ims91-7.ps.Z [IP: 129.49.18.1] Q6h: Is the Mandelbrot set connected? A6h: The Mandelbrot set is simply connected. This follows from a theorem of Douady and Hubbard that there is a conformal isomorphism from the complement of the Mandelbrot set to the complement of the unit disk. (In other words, all equipotential curves are simple closed curves.) It is conjectured that the Mandelbrot set is locally connected, and thus pathwise connected, but this is currently unproved. Connectedness definitions: Connected: X is connected if there are no proper closed subsets A and B of X such that A union B = X, but A intersect B is empty. I.e. X is connected if it is a single piece. Simply connected: X is simply connected if it is connected and every closed curve in X can be deformed in X to some constant closed curve. I.e. X is simply connected if it has no holes. Locally connected: X is locally connected if for every point p in X, for every open set U containing p, there is an open set V containing p and contained in the connected component of p in U. I.e. X is locally connected if every connected component of every open subset is open in X. Arcwise (or path) connected: X is arcwise connected if every two points in X are joined by an arc in X. (The definitions are from Encyclopedic Dictionary of Mathematics.) Reference: Douady, A. and Hubbard, J., "Comptes Rendus" (Paris) 294, pp.123-126, 1982. Q6i: What is the Mandelbrot Encyclopedia? A6i: The Mandelbrot Encyclopedia is a mail server which contains information about the Mandelbrot Set. It was setup by Robert Munafo but is not currently available. Further information will be available once it is available again. Q6j: What is the dimension of the Mandelbrot Set? A6j: The Mandelbrot Set has a dimension of 2. The Mandelbrot Set contains and is contained in a disk. A disk has a dimension of 2, thus so does the Mandelbrot Set. The Koch snowflake (dimension 1.2619...) does not satify this condition because it is a thin boundary curve, thus containing no disk. If you add the region inside the curve then it does have dimension of 2. _________________________________________________________________ Subject: Julia sets Q7a: What is the difference between the Mandelbrot set and a Julia set? A7a: The Mandelbrot set iterates z² + c with z starting at 0 and varying c. The Julia set iterates z² + c for fixed c and varying starting z values. That is, the Mandelbrot set is in parameter space (c-plane) while the Julia set is in dynamical or variable space (z-plane). Q7b: What is the connection between the Mandelbrot set and Julia sets? A7b: Each point c in the Mandelbrot set specifies the geometric structure of the corresponding Julia set. If c is in the Mandelbrot set, the Julia set will be connected. If c is not in the Mandelbrot set, the Julia set will be a Cantor dust. Q7c: How is a Julia set actually computed? A7c: The Julia set can be computed by iteration similar to the Mandelbrot computation. The only difference is that the c value is fixed and the initial z value varies. Alternatively, points on the boundary of the Julia set can be computed quickly by using inverse iterations. This technique is particularly useful when the Julia set is a Cantor Set. In inverse iteration, the equation z1 = z0² + c is reversed to give an equation for z0: z0 = ± sqrt(z1 - c). By applying this equation repeatedly, the resulting points quickly converge to the Julia set boundary. (At each step, either the positive or negative root is randomly selected.) This is a nonlinear iterated function system. In pseudocode: z = 1 (or any value) loop if (random number < .5) then z = sqrt(z - c) else z = -sqrt(z - c) endif plot z end loop Q7d: What are some Julia set facts? A7d: The Julia set of any rational map of degree greater than one is perfect (hence in particular uncountable and nonempty), completely invariant, equal to the Julia set of any iterate of the function, and also is the boundary of the basin of attraction of every attractor for the map. Julia set references: 1. A. F. Beardon, Iteration of Rational Functions : Complex Analytic Dynamical Systems, Springer-Verlag, New York, 1991. 2. P. Blanchard, Complex Analytic Dynamics on the Riemann Sphere, Bull. of the Amer. Math. Soc 11, 1 (July 1984), pp. 85-141. This article is a detailed discussion of the mathematics of iterated complex functions. It covers most things about Julia sets of rational polynomial functions. _________________________________________________________________ Subject: Complex arithmetic and quaternion arithmetic Q8a: How does complex arithmetic work? A8a: It works mostly like regular algebra with a couple additional formulas: (note: a,b are reals, x ,y are complex, i is the square root of -1) Powers of i: i² = -1 Addition: (a+i·b)+(c+i ·d) = (a+c)+i·(b+d) Multiplication: (a+i ·b)·(c+i·d) = a ·c-b·d + i·(a·d+b·c) Division: (a+i·b) ÷ (c+i·d) = (a+i·b)·(c-i·d) ÷ (c²+d²) Exponentiation: exp(a+i·b) = exp(a)(cos(b)+i ·sin(b)) Sine: sin(x) = (exp(i·x) - exp(-i·x))÷(2·i) Cosine: cos(x) = (exp(i·x) + exp(-i·x)) ÷ 2 Magnitude: |a+i·b| = sqrt(a²+b²) Log: log(a+i·b) = log(|a+i·b|)+i·arctan(b ÷ a) (Note: log is multivalued.) Log (polar coordinates): log(r·e^(i ·ø)) = log(r)+i·ø Complex powers: x^y = exp(y·log(x)) de Moivre's theorem: x^n = r^n · [cos(n ·ø) + i · sin(n·ø)] (where n is an integer) More details can be found in any complex analysis book. Q8b: How does quaternion arithmetic work? A8b: quaternions have 4 components (a + ib + jc + kd) compared to the two of complex numbers. Operations such as addition and multiplication can be performed on quaternions, but multiplication is not commutative. Quaternions satisfy the rules * i² = j² = k² = -1 * ij = -ji = k * jk = -kj = i, * ki = -ik = j See: Frode Gill's quaternions page http://www.krs.hia.no/~fgill/quatern.html _________________________________________________________________ Subject: Logistic equation Q9: What is the logistic equation? A9: It models animal populations. The equation is x -> c·x·(1 - x), where x is the population (between 0 and 1) and c is a growth constant. Iteration of this equation yields the period doubling route to chaos. For c between 1 and 3, the population will settle to a fixed value. At 3, the period doubles to 2; one year the population is very high, causing a low population the next year, causing a high population the following year. At 3.45, the period doubles again to 4, meaning the population has a four year cycle. The period keeps doubling, faster and faster, at 3.54, 3.564, 3.569, and so forth. At 3.57, chaos occurs; the population never settles to a fixed period. For most c values between 3.57 and 4, the population is chaotic, but there are also periodic regions. For any fixed period, there is some c value that will yield that period. See "An Introduction to Chaotic Dynamical Systems" for more information. _________________________________________________________________ Subject: Feigenbaum's constant Q10: What is Feigenbaum's constant? NEW A10: In a period doubling cascade, such as the logistic equation, consider the parameter values where period-doubling events occur (e.g. r[1]=3, r[2]=3.45, r[3]=3.54, r[4]=3.564...). Look at the ratio of distances between consecutive doubling parameter values; let delta[n] = (r[n+1]-r[n])/(r[n+2]-r[n+1]). Then the limit as n goes to infinity is Feigenbaum's (delta) constant. Based on computations by F. Christiansen, P. Cvitanovic and H.H. Rugh, it has the value 4.6692016091029906718532038... Note: several books have published incorrect values starting 4.66920166...; the last repeated 6 is a typographical error. The interpretation of the delta constant is as you approach chaos, each periodic region is smaller than the previous by a factor approaching 4.669... Feigenbaum's constant is important because it is the same for any function or system that follows the period-doubling route to chaos and has a one-hump quadratic maximum. For cubic, quartic, etc. there are different Feigenbaum constants. Feigenbaum's alpha constant is not as well known; it has the value 2.50290787509589282228390287272909. This constant is the scaling factor between x values at bifurcations. Feigenbaum says, "Asymptotically, the separation of adjacent elements of period-doubled attractors is reduced by a constant value [alpha] from one doubling to the next". If d[a] is the algebraic distance between nearest elements of the attractor cycle of period 2ª, then d[a]/d[a+1] converges to -alpha. References: 1. K. Briggs, How to calculate the Feigenbaum constants on your PC, Aust. Math. Soc. Gazette 16 (1989), p. 89. 2. K. Briggs, A precise calculation of the Feigenbaum constants, Mathematics of Computation 57 (1991), pp. 435-439. 3. K. Briggs, G. R. W. Quispel and C. Thompson, Feigenvalues for Mandelsets, J. Phys. A 24 (1991), pp. 3363-3368. 4. F. Christiansen, P. Cvitanovic and H.H. Rugh, "The spectrum of the period-doubling operator in terms of cycles", J. Phys A 23, L713 (1990). 5. M. Feigenbaum, The Universal Metric Properties of Nonlinear Transformations, J. Stat. Phys 21 (1979), p. 69. 6. M. Feigenbaum, Universal Behaviour in Nonlinear Systems, Los Alamos Sci 1 (1980), pp. 1-4. Reprinted in Universality in Chaos, compiled by P. Cvitanovic. Feigenbaum Constants http://www.mathsoft.com/asolve/constant/fgnbaum/fgnbaum.h tml _________________________________________________________________ Subject: Iterated function systems and compression Q11a: What is an iterated function system (IFS)? A11a: If a fractal is self-similar, you can specify mappings that map the whole onto the parts. Iteration of these mappings will result in convergence to the fractal attractor. An IFS consists of a collection of these (usually affine) mappings. If a fractal can be described by a small number of mappings, the IFS is a very compact description of the fractal. An iterated function system is By taking a point and repeatedly applying these mappings you end up with a collection of points on the fractal. In other words, instead of a single mapping x -> F(x), there is a collection of (usually affine) mappings, and random selection chooses which mapping is used. For instance, the Sierpinski triangle can be decomposed into three self-similar subtriangles. The three contractive mappings from the full triangle onto the subtriangles forms an IFS. These mappings will be of the form "shrink by half and move to the top, left, or right". Iterated function systems can be used to make things such as fractal ferns and trees and are also used in fractal image compression. Fractals Everywhere by Barnsley is mostly about iterated function systems. The simplest algorithm to display an IFS is to pick a starting point, randomly select one of the mappings, apply it to generate a new point, plot the new point, and repeat with the new point. The displayed points will rapidly converge to the attractor of the IFS. Interactive IFS Playground (Otmar Lendl) http://www.cosy.sbg.ac.at/rec/ifs/ Frank Rousell's hyperindex of IFS images http://www.cnam.fr/fractals/ifs.html Q11b: What is the state of fractal compression? A11b: Fractal compression is quite controversial, with some people claiming it doesn't work well, and others claiming it works wonderfully. The basic idea behind fractal image compression is to express the image as an iterated function system (IFS). The image can then be displayed quickly and zooming will generate infinite levels of (synthetic) fractal detail. The problem is how to efficiently generate the IFS from the image. Barnsley, who invented fractal image compression, has a patent on fractal compression techniques (4,941,193). Barnsley's company, Iterated Systems Inc (http://www.iterated.com/), has a line of products including a Windows viewer, compressor, magnifier program, and hardware assist board. Fractal compression is covered in detail in the comp.compression FAQ file (See "compression-FAQ"). ftp://rtfm.mit.edu/pub/usenet/comp.compression [18.181.0.24]. One of the best online references for Fractal Compress is Yuval Fisher's Fractal Image Encoding page (http://inls.ucsd.edu/y/Fractals/) at the Institute for Nonlinear Science, University for California, San Diego. It includes references to papers, other WWW sites, software, and books about Fractal Compression. Three major research projects include Waterloo Montreal Verona Fractal Research Initiative http://links.uwaterloo.ca/ Groupe FRACTALES http://www-syntim.inria.fr/fractales/ Bath Scalable Video Demo Software http://dmsun4.bath.ac.uk/bsvdemo/ Several books describing fractal image compression are: 1. M. Barnsley, Fractals Everywhere, Academic Press Inc., 1988. ISBN 0-12-079062-9. This is an excellent text book on fractals. This is probably the best book for learning about the math underpinning fractals. It is also a good source for new fractal types. 2. M. Barnsley and L. Anson, The Fractal Transform, Jones and Bartlett, April, 1993. ISBN 0-86720-218-1. Without assuming a great deal of technical knowledge, the authors explain the workings of the Fractal Transform(TM). 3. M. Barnsley and L. Hurd, Fractal Image Compression, Jones and Bartlett. ISBN 0-86720-457-5. This book explores the science of the fractal transform in depth. The authors begin with a foundation in information theory and present the technical background for fractal image compression. In so doing, they explain the detailed workings of the fractal transform. Algorithms are illustrated using source code in C. 4. Y. Fisher (Ed), Fractal Image Compression: Theory and Application. Springer Verlag, 1995. 5. Y. Fisher (Ed), Fractal Image Encoding and Analysis: A NATO ASI Series Book, Springer Verlag, New York, 1996 contains the proceedings of the Fractal Image Encoding and Analysis Advanced Study Institute held in Trondheim, Norway July 8-17, 1995. The book is currently being produced. The October 1993 issue of Byte discussed fractal compression. You can ftp sample code: ftp://ftp.uu.net/published/byte/93oct/fractal.exe. Andreas Kassler wrote a Fractal Image Compression with WINDOWS package for a Fractal Compression thesis. It is available at http://www-vs.informatik.uni-ulm.de/Mitarbeiter/Kassler.html An introductory paper is: 1. A. E. Jacquin, Image Coding Based on a Fractal Theory of Iterated Contractive Image Transformation, IEEE Transactions on Image Processing, January 1992. Many fractal image compression papers are available from ftp://ftp.informatik.uni-freiburg.de/documents/papers/fractal [IP 132.230.150.1]. A review of the literature is in Guide.ps.gz. See the README file for an overview of the available documents. Other references: Fractal Compression Bibliography http://dip1.ee.uct.ac.za/fractal.bib.html Fractal Video Compression http://inls.ucsd.edu/y/Fractals/Video/fracvideo.html _________________________________________________________________ Subject: Chaotic demonstrations Q12a: How can you make a chaotic oscillator? A12a: Two references are: 1. T. S. Parker and L. O. Chua, Chaos: a tutorial for engineers, Proceedings IEEE 75 (1987), pp. 982-1008. 2. New Scientist, June 30, 1990, p. 37. Q12b: What are laboratory demonstrations of chaos? A12b: Robert Shaw at UC Santa Cruz experimented with chaos in dripping taps. This is described in: 1. J. P. Crutchfield, Chaos, Scientific American 255, 6 (Dec. 1986), pp. 38-49. 2. I. Stewart, Does God Play Dice?: the Mathematics of Chaos, B. Blackwell, New York, 1989. Two references to other laboratory demonstrations are: 1. K. Briggs, Simple Experiments in Chaotic Dynamics, American Journal of Physics 55, 12 (Dec 1987), pp. 1083-1089. 2. J. L. Snider, Simple Demonstration of Coupled Oscillations, American Journal of Physics 56, 3 (Mar 1988), p. 200. _________________________________________________________________ Subject: L-Systems Q13: What are L-systems? A13: A L-system or Lindenmayer system is a formal grammar for generating strings. (That is, it is a collection of rules such as replace X with XYX.) By recursively applying the rules of the L-system to an initial string, a string with fractal structure can be created. Interpreting this string as a set of graphical commands allows the fractal to be displayed. L-systems are very useful for generating realistic plant structures. Some references are: 1. P. Prusinkiewicz and J. Hanan, Lindenmayer Systems, Fractals, and Plants, Springer-Verlag, New York, 1989. 2. P. Prusinkiewicz and A. Lindenmayer, The Algorithmic Beauty of Plants, Springer-Verlag, NY, 1990. ISBN 0-387-97297-8. A very good book on L-systems, which can be used to model plants in a very realistic fashion. The book contains many pictures. _________________________________________________________________ More information can be obtained via the WWW at: L-Systems Tutorial by David Green http://life.csu.edu.au/complex/tutorials/tutorial2.html L-system leaf http://www.csu.edu.au/complex_systems/iconfern.gif 3 Dim. L-system Tree program (P.J.Drinkwater) http://hill.lut.ac.uk/TestStuff/trees/ L-system images from the Center for the Computation and Visualization of Geometric Structures http://www.geom.umn.edu/pix/archive/subjects/L-systems.html _________________________________________________________________ Subject: Fractal music Q14: What is some information on fractal music? A14: One fractal recording is "The Devil's Staircase: Composers and Chaos" on the Soundprint label. Some references, many from an unpublished article by Stephanie Mason, are: 1. R. Bidlack, Chaotic Systems as Simple (But Complex) Compositional Algorithms, Computer Music Journal, Fall 1992. 2. C. Dodge, A Musical Fractal, Computer Music Journal 12, 13 (Fall 1988), p. 10. 3. K. J. Hsu and A. Hsu, Fractal Geometry of Music, Proceedings of the National Academy of Science, USA 87 (1990), pp. 938-941. 4. K. J. Hsu and A. Hsu, Self-similatrity of the '1/f noise' called music., Proceedings of the National Academy of Science USA 88 (1991), pp. 3507-3509. 5. C. Pickover, Mazes for the Mind: Computers and the Unexpected, St. Martin's Press, New York, 1992. 6. P. Prusinkiewicz, Score Generation with L-Systems, International Computer Music Conference 86 Proceedings, 1986, pp. 455-457. 7. Byte 11, 6 (June 1986), pp. 185-196. Online resources include: Well Tempered Fractal v3.0 from Spanky via FTP by Robert Greenhouse ftp://spanky.triumf.ca/pub/fractals/programs/ibmpc/wtf30.zip A fractal music C++ package is available at http://hamp.hampshire.edu/~gpzF93/inSanity.html The Fractal Music Project (Claus-Dieter Schulz) http://www-ks.rus.uni-stuttgart.de/people/schulz/fmusic Chua's Oscillator: Applications of Chaos to Sound and Music http://www.ccsr.uiuc.edu/People/gmk/Projects/ChuaSoundMusic/Chu aSoundMusic.html There is now a Fractal Music mailing list. It's purposes are: 1. To inform people about news, updates, changes on the Fractal Music Projects WWW pages. 2. To encourage discussion between people working in that area. The Fractal Music Mailinglist: fmusic@kssun7.rus.uni-stuttgart.de To subscribe to the list please send mail to fmusic-request@kssun7.rus.uni-stuttgart.de _________________________________________________________________ Subject: Fractal mountains Q15: How are fractal mountains generated? A15: Usually by a method such as taking a triangle, dividing it into 3 subtriangles, and perturbing the center point. This process is then repeated on the subtriangles. This results in a 2-d table of heights, which can then be rendered as a 3-d image. Two references are: 1. M. Ausloos, Proc. R. Soc. Lond. A 400 (1985), pp. 331-350. 2. H.O. Peitgen, D. Saupe, The Science of Fractal Images, Springer-Velag, 1988 Available online is an implementation of fractal Brownian motion (fBm) such as described in The Science of Fractal Images. Gforge (John Beale) http://jump.stanford.edu:8080/beale/land/index.html Other fractal landscape EECS News: Fall 1994: Building Fractal Planets by Ken Musgrave http://www.seas.gwu.edu/faculty/musgrave/article.html _________________________________________________________________ Subject: Plasma clouds Q16: What are plasma clouds? A16: They are a Fractint fractal and are similar to fractal mountains. Instead of a 2-d table of heights, the result is a 2-d table of intensities. They are formed by repeatedly subdividing squares. Robert Cahalan has fractal information about Earth's Clouds including how they differ from plasma clouds. Fractal Clouds Reference by Robert F. Cahalan (cahalan@clouds.gsfc.nasa.gov) http://climate.gsfc.nasa.gov/~cahalan/FractalClouds/ Also some plasma-based fractals clouds by John Walker are available. Fractal generated clouds http://ivory.nosc.mil/html/trancv/html/cloud-fract.html Two articles about the fractal nature of Earth's clouds: 1. "Fractal statistics of cloud fields," R. F. Cahalan and J. H. Joseph, Mon. Wea.Rev. 117, 261-272, 1989 2. "The albedo of fractal stratocumulus clouds," R. F. Cahalan, W. Ridgway, W. J. Wiscombe, T. L. Bell and J. B. Snider, J. Atmos. Sci. 51, 2434-2455, 1994 _________________________________________________________________ Subject: Lyapunov fractals Q17a: Where are the popular periodically-forced Lyapunov fractals described? A17a: See: 1. A. K. Dewdney, Leaping into Lyapunov Space, Scientific American, Sept. 1991, pp. 178-180. 2. M. Markus and B. Hess, Lyapunov Exponents of the Logistic Map with Periodic Forcing, Computers and Graphics 13, 4 (1989), pp. 553-558. 3. M. Markus, Chaos in Maps with Continuous and Discontinuous Maxima, Computers in Physics, Sep/Oct 1990, pp. 481-493. Q17b: What are Lyapunov exponents? A17b: Lyapunov exponents quantify the amount of linear stability or instability of an attractor, or an asymptotically long orbit of a dynamical system. There are as many lyapunov exponents as there are dimensions in the state space of the system, but the largest is usually the most important. Given two initial conditions for a chaotic system, a and b, which are close together, the average values obtained in successive iterations for a and b will differ by an exponentially increasing amount. In other words, the two sets of numbers drift apart exponentially. If this is written e^(n*(lambda)) for n iterations, then e^(lambda) is the factor by which the distance between closely related points becomes stretched or contracted in one iteration. Lambda is the Lyapunov exponent. At least one Lyapunov exponent must be positive in a chaotic system. A simple derivation is available in: 1. H. G. Schuster, Deterministic Chaos: An Introduction, Physics Verlag, 1984. Q17c: How can Lyapunov exponents be calculated? A17c: For the common periodic forcing pictures, the lyapunov exponent is: lambda = limit as N -> infinity of 1/N times sum from n=1 to N of log2(abs(dx sub n+1 over dx sub n)) In other words, at each point in the sequence, the derivative of the iterated equation is evaluated. The Lyapunov exponent is the average value of the log of the derivative. If the value is negative, the iteration is stable. Note that summing the logs corresponds to multiplying the derivatives; if the product of the derivatives has magnitude < 1, points will get pulled closer together as they go through the iteration. MS-DOS and Unix programs for estimating Lyapunov exponents from short time series are available by ftp: ftp://inls.ucsd.edu/pub/ncsu/ Computing Lyapunov exponents in general is more difficult. Some references are: 1. H. D. I. Abarbanel, R. Brown and M. B. Kennel, Lyapunov Exponents in Chaotic Systems: Their importance and their evaluation using observed data, International Journal of Modern Physics B 56, 9 (1991), pp. 1347-1375. 2. A. K. Dewdney, Leaping into Lyapunov Space, Scientific American, Sept. 1991, pp. 178-180. 3. M. Frank and T. Stenges, Journal of Economic Surveys 2 (1988), pp. 103- 133. 4. T. S. Parker and L. O. Chua, Practical Numerical Algorithms for Chaotic Systems, Springer Verlag, 1989. _________________________________________________________________ Subject: Fractal items Q18: Where can I get fractal T-shirts and posters? A18: One source is Art Matrix, P.O. box 880, Ithaca, New York, 14851, 1-800-PAX-DUTY. Another source is Media Magic; they sell many fractal posters, calendars, videos, software, t-shirts, ties, and a huge variety of books on fractals, chaos, graphics, etc. Media Magic is at PO Box 598 Nicasio, CA 94946, 415-662-2426. A third source is Ultimate Image; they sell fractal t- shirts, posters, gift cards, and stickers. Ultimate Image is at PO Box 7464, Nashua, NH 03060-7464. Yet another source is Dave Kliman (516) 625-2504 dkliman@pb.net, whose products are distributed through Spencer Gifts, Posterservice, 1-800-666-7654, and Scandecor International., this spring, through JC Penny, featuring all-over fractal t-shirts, and has fractal umbrellas available from Shaw Creations (800) 328-6090. Cyber Fiber produces fractal silk scarves, t-shirts, and postcards. Contact Robin Lowenthal, Cyber Fiber, 4820 Gallatin Way, San Diego, CA 92117. Chaos MetaLink website (http://www.industrialstreet.com/chaos/metalink.htm) also has postcards, CDs, and videos. Free fractal posters are available if you send a self-addressed stamped envelope to the address given on http://www.xmission.com/~legalize/. For foreign requests (outside USA) include two IRCs (international reply coupons) to cover the weight. ReFractal Design (http://www.refractal.com/) sells jewelry based on fractals. _______________________________________________________________________________ Subject: How can I take photos of fractals? Q19: How can I take photos of fractals? A19: Noel Giffin gets good results with the following setup: Use 100 ISO (ASA) Kodak Gold for prints or 64 ISO (ASA) for slides. Use a long lens (100mm) to flatten out the field of view and minimize screen curvature. Use f/4 stop. Shutter speed must be longer than frame rate to get a complete image; 1/4 seconds works well. Use a tripod and cable release or timer to get a stable picture. The room should be completely blackened, with no light, to prevent glare and to prevent the monitor from showing up in the picture. You can also obtain high quality images by sending your targa or gif images to a commercial graphics imaging shop. They can provide much higher resolution images. Prices are about $10 for a 35mm slide or negative and about $50 for a high quality 4x5 negative. _________________________________________________________________ Subject: 3-D fractals Q20: How can 3-D fractals be generated? A20: A common source for 3-D fractals is to compute Julia sets with quaternions instead of complex numbers. The resulting Julia set is four dimensional. By taking a slice through the 4-D Julia set (e.g. by fixing one of the coordinates), a 3-D object is obtained. This object can then be displayed using computer graphics techniques such as ray tracing. Frank Rousell's hyperindex of 3D images http://www.cnam.fr/fractals/mandel3D.html 4D Quaternions by Tom Holroyd http://bambi.ccs.fau.edu/~tomh/fractals/fractals.html The papers to read on this are: 1. J. Hart, D. Sandin and L. Kauffman, Ray Tracing Deterministic 3-D Fractals, SIGGRAPH, 1989, pp. 289-296. 2. A. Norton, Generation and Display of Geometric Fractals in 3-D, SIGGRAPH, 1982, pp. 61-67. 3. A. Norton, Julia Sets in the Quaternions, Computers and Graphics, 13, 2 (1989), pp. 267-278. Two papers on cubic polynomials, which can be used to generate 4-D fractals: 1. B. Branner and J. Hubbard, The iteration of cubic polynomials, part I., Acta Math 66 (1988), pp. 143-206. 2. J. Milnor, Remarks on iterated cubic maps, This paper is available from ftp://math.sunysb.edu/preprints/ims90-6.ps.Z. Published in 1991 SIGGRAPH Course Notes #14: Fractal Modeling in 3D Computer Graphics and Imaging. Instead of quaternions, you can of course use hypercomplex number such as in "FractInt", or other functions. For instance, you could use a map with more than one parameter, which would generate a higher-dimensional fractal. Another way of generating 3-D fractals is to use 3-D iterated function systems (IFS). These are analogous to 2-D IFS, except they generate points in a 3-D space. A third way of generating 3-D fractals is to take a 2-D fractal such as the Mandelbrot set, and convert the pixel values to heights to generate a 3-D "Mandelbrot mountain". This 3-D object can then be rendered with normal computer graphics techniques. POV-Ray 3.0, a freely available ray tracing package, has added 4-D fractal support. It takes a 3-D slice of a 4-D Julia set based on an arbitrary 3-D "plane" done at any angle. For more information see the POV Ray web site at http://www.povray.org/. _________________________________________________________________ Subject: Fractint Q21a: What is Fractint? A21a: Fractint is a very popular freeware (not public domain) fractal generator. There are DOS, Windows, OS/2, Amiga, and Unix/X versions. The DOS version is the original version, and is the most up-to-date. Please note: sci.fractals is not a product support newsgroup for Fractint. Bugs in Fractint/Xfractint should usually go to the authors rather than being posted. Fractint is on many ftp sites. For example: DOS 19.3 source via WWW from USA http://www.coast.net/cgi-bin/coast/dwn?msdos/graphics/frasr192. zip 19.3 executable via WWW from USA http://www.coast.net/cgi-bin/coast/dwn?msdos/graphics/frain192. zip 19.3 source via FTP from Canada ftp://spanky.triumf.ca/fractals/programs/ibmpc/frasr193.zip 19.3 executable via FTP from Canada ftp://spanky.triumf.ca/fractals/programs/ibmpc/frain193.zip (The suffix 193 will change as new versions are released.) Fractint is available on Compuserve: GO GRAPHDEV and look for FRAINT.EXE and FRASRC.EXE in LIB 4. Windows MS-Window FractInt 18.21 via FTP from Canada ftp://spanky.triumf.ca/fractals/programs/ibmpc/windows/winf1821 .zip MS-Window FractInt 18.21 via WWW from USA http://www.coast.net/cgi-bin/coast/dwn?win3/graphics/winf1821.z ip MS-Windows FractInt 18.21 source via FTP from Canada ftp://spanky.triumf.ca/fractals/programs/ibmpc/windows/wins1821 .zip MS-Windows FractInt 18.21 source via WWW from USA http://www.coast.net/cgi-bin/coast/dwn?win3/graphics/wins1821.z ip OS/2 Available on Compuserve in its GRAPHDEV forum. The files are PM*.ZIP. These files are also available by ftp://ftp-os2.nmsu.edu/os2/graphics/pmfra2.zip Unix The Unix version of FractInt, called XFractInt requires X-Windows. 3.02 source ftp://ftp.cs.berkeley.edu/pub/sprite/xfract302.shar.Z XFractInt is also available in LIB 4 of Compuserve's GO GRAPHDEV forum in XFRACT.ZIP. Macintosh There is NO Macintosh version of Fractint, although there may be several people working on a port. It is possibleto run Fractint on the Macintosh if you use Insignia Software's SoftAT, which is a PC AT emulator. Amiga There is an Amiga version also available: FractInt 2.6 via FTP from an AmiNET archive in USA ftp://wuarchive.wustl.edu/pub/aminet/gfx/fract/fractint26.lha FracInt 2.6 via WWW from an AmiNET archive in USA http://wuarchive.wustl.edu/pub/aminet/gfx/fract/fractint26.lha The latest version (3.02) via WWW from Norway http://login.eunet.no/~terjepe/aboutfractint.html FracXtra There is a collection of map, parameter, etc. files for FractInt, called FracXtra. It is available FracXtra Home Page by Dan Goldwater http://fatmac.ee.cornell.edu/~goldwada/fracxtra.html FracXtra via WWW (preferred) http://www.coast.net/cgi-bin/coast/dwn?mdos/graphics/fracxtr6.z ip FracXtra via FTP ftp://spanky.triumf.ca/fractals/programs/ibmpc/fracxtr6.zip FractInt PAR Exchange by Landon Kuhn "for all the fans of Fractint and fractal creation." Its purpose is the trading of parameter files created by Fractint. FractInt PAR Exchange http://www.hevanet.com/lkuhn/px For European users, these files are available from ftp://ftp.uni-koeln.de/. If you can't use ftp, see the mail server information below. Q21b: How does Fractint achieve its speed? A21b: Fractint's speed (such as it is) is due to a combination of: 1. Using fixed point math rather than floating point where possible (huge improvement for non-coprocessor machine, small for 486's). 2. Exploiting symmetry of the fractal. 3. Detecting nearly repeating orbits, avoid useless iteration (e.g. repeatedly iterating 0²+0 etc. etc.). 4. Reducing computation by guessing solid areas (especially the "lake" area). 5. Using hand-coded assembler in many places. 6. Obtaining both sin and cos from one 387 math coprocessor instruction. 7. Using good direct memory graphics writing in 256-color modes. The first four are probably the most important. Some of these introduce errors, usually quite acceptable. _________________________________________________________________ Subject: Fractal software NEW Q22: Where can I obtain software packages to generate fractals? A22: For X windows: xmntns xlmntn: fractal mountains ftp://ftp.uu.net/usenet/comp.sources.x/volume8/xmntns xfroot: fractal root window X11 distribution xmartin: Martin hopalong root window X11 distribution xmandel: Mandelbrot/Julia sets X11 distribution lyap: Lyapunov exponent images ftp://ftp.uu.net/usenet/comp.sources.x/volume17/lyapunov-xlib spider: Uses Thurston's algorithm, Kobe algorithm, external angles http://inls.ucsd.edu/y/Complex/spider.tar.Z xfractal_explorer: fractal drawing program ftp://ftp.x.org/contrib/applications/xfractal_explorer-v1.0.tar .gz Xmountains: A fractal landscape generator ftp://ftp.epcc.ed.ac.uk/pub/personal/spb/xmountains xfract: Mandelbrot with a color-cycling feature ftp://charm.il.ft.hse.nl/pub/X11/src/xfract.tar.gz xmfract v1.4: Needs Motif 1.2+, based on FractInt ftp://ftp.x.org/contrib/graphics/xmfract_1.4.tar.gz Fast Julia Set and Mandelbrot for X-Windows by Zsolt Zsoldos http://www.chem.leeds.ac.uk/ICAMS/people/zsolt/mandel.html Distributed X systems: MandelSpawn: Mandelbrot/Julia on a network ftp://ftp.x.org/R5contrib/mandelspawn-0.07.tar.Z ftp://ftp.funet.fi/pub/X11/R5contrib/mandelspawn-0.07.tar.Z gnumandel: Mandelbrot on a network ftp://ftp.elte.hu/pub/software/unix/gnu/gnumandel.tar.Z For SunView: Mandtool: Mandelbrot ftp://spanky.triumf.ca/fractals/programs/mandtool/M_TAR.Z For Unix/C: lsys: L-systems as PostScript (in C++) ftp://ftp.cs.unc.edu/pub/users/leech/lsys.tar.gz lyapunov: PGM Lyapunov exponent images ftp://ftp.uu.net/usenet/comp.sources.misc/volume23/lyapunov/ SPD: fractal mountain, tree, recursive tetrahedron ftp://ftp.povray.org/pub/povray/spd/ Fractal Studio: Mandelbrot set; handles distributed computing ftp://archive.cs.umbc.edu/pub/peter/fractal-studio fanal: analysis of fractal dimension by Jürgen Dollinger ftp://ftp.uni-stuttgart.de/pub/systems/linux/local/math/fanal-0 1b.tar.gz For Mac: LSystem, 3D-L-System, IFS, FracHill, Mandella http://wuarchive.wustl.edu/edu/math/software/mac/fractals/ ftp://ftp.auckland.ac.nz/ fractal-wizard.hqx, julias-dream-107.hqx, mandella-87.hqx ftp://mirrors.aol.com/pub/info-mac/app/ ftp://plaza.aarnet.edu.au/micros/mac/info-mac/app/ mandel-tv: a very fast Mandelbrot generator. ftp://mirrors.aol.com/pub/info-mac/sci/ ftp://plaza.aarnet.edu.au/micros/mac/info-mac/sci/ mandelzot, powerexplorer ftp://mirrors.aol.com/pub/info-mac/ There are also commercial programs, such as IFS Explorer and Fractal Clip Art, which are published by Koyn Software (314) 878-9125. Kai's Fractal Explorer (part of the Kai's Power Tools package for Adobe Photoshop) Note: This listing is quite old. If you have a Mac (especially a PowerMac) please do me a large favor and send me updates to this information. Thanks. (note: M-Set is short hand for Mandelbrot Set) For MSDOS: DEEPZOOM: a high-precision M-Set program for displaying highly zoomed fractals http://spanky.triumf.ca/pub/fractals/programs/ibmpc/depzm13.zip Fractal WitchCraft: a very fast fractal design program ftp://garbo.uwasa.fi/pc/demo/fw1-08.zip ftp://ftp.cdrom.com/pub/garbo/garbo_pc/show/fw1-08.zip CAL: generates more than 15 types of fractals including Lyapunov, IFS, user-defined, logistic, and Quaternion Julia ftp://ftp.coast.net/SimTel/msdos/graphics/frcal040.zip Fractal Discovery Laboratory: designed for use in a science museum or school setting. The Lab has five sections: Art Gallery, Microscope, Movies, Tools, and Library Sampler available from Compuserve GRAPHDEV Lib 4 in DISCOV.ZIP, or send high-density disk and self-addressed, stamped envelope to: Earl F. Glynn, 10808 West 105th Street, Overland Park, Kansas 66214-3057. WL-Plot 2.59 : plots functions including bifurcations and recursive relations ftp://archives.math.utk.edu/software/msdos/graphing/wlplt/wlplt 259.zip From ftp://ftp.coast.net/SimTel/msdos/graphics/ forb01a.zip: Displays orbits of M-Set mapping. C/E/VGA fract30.zip: Mandelbrot/Julia set 2D/3D EGA/VGA Fractal Gen fractfly.zip: Create Fractal flythroughs with FRACTINT fdesi313.zip: Program to visually design IFS fractals frain192.zip: FRACTINT v19.2 EGA/VGA/XGA fractal generator frasr192.zip: C & ASM src for FRACTINT v19.2 frcal040.zip: Fractal drawing program: 15 formulae available frcaldmo.zip: 800x600x256 demo images for FRCAL040.ZIP vlotkatc uses VESA 640x480x16 Million colour mode to generate Volterra-Lotka images. http://spanky.triumf.ca/pub/fractals/programs/ibmpc/vlotkatc.zi p http://spanky.triumf.ca/pub/fractals/programs/ibmpc/vlotkatc.do c ftp://spanky.triumf.ca/pub/fractals/programs/ibmpc/vlotkatc.zip ftp://spanky.triumf.ca/pub/fractals/programs/ibmpc/vlotkatc.doc Fast FPU Fractal Fun 2.0 (FFFF2.0) is the first M-Set generator working in hicolor gfx modes thus using up to 32768 different colors on screen by Daniele Paccaloni requires 386DX+ and VESA support http://spanky.triumf.ca/pub/fractals/programs/IBMPC/FFFF20.ZIP ftp://spanky.triumf.ca/pub/fractals/programs/IBMPC/FFFF20.ZIP 3DFract generates 3-D fractals including Sierpinski cheese and 3-D snowflake http://www.cstp.umkc.edu/users/bhugh/home.html FracTrue 2.00 - Hi/TrueColor Generator including a formular parser. 286+ VGA by Bernd Hemmerling http://www.cs.tu-berlin.de/~hemmerli/ HOP based on the HOPALONG fractal type. Math coprocessor (386DX and above) and SuperVGA required. shareware ($30) Places to download HOPZIP.EXE from: Compuserve GRAPHDEV forum, lib 4 The Well under ibmpc/graphics http://ourworld.compuserve.com/homepages/mpeters/hop.htm ftp://ftp.uni-heidelberg.de/pub/msdos/graphics/ http://spanky.triumf.ca/pub/fractals/programs/ibmpc/ ftp://spanky.triumf.ca/pub/fractals/programs/ibmpc/ ZsManJul 1.0 (requires 386DX+) by Zsolt Zsoldos http://www.chem.leeds.ac.uk/ICAMS/people/zsolt/zsmanjul.html Fractal Movie a real-time 3D IFS fractal movie renderer (running on the 486DX+) http://home.pacific.net.sg/~yqchen/ FracZoom shareware by Niels Ulrik Reinwald 386DX+ http://www.geocities.com/siliconvalley/4602/index.html RMandel 1.2 80-bit floating point M-Set animation generator by Marvin R. Lipford ftp://ftp.cnam.fr/pub/Fractals/anim/FRACSOFT/rmandel.zip TruMand 1.0 by Mike Freeman 486DX+ True-colour M-Set generator http://spanky.triumf.ca/pub/fractals/programs/ibmpc/TRMAND10.ZI P FAE - Fractal Animation Engine shareware by Brian Towles http://spanky.triumf.ca/pub/fractals/programs/ibmpc/FAE210B.ZIP For Windows: dy-syst: Explores Newton's method, Mandelbrot and Julia sets ftp://cssun.mathcs.emory.edu/pub/riddle/ bmand 1.1 shareware by Christopher Bare M-Set program http://www.gi.net/MSDOS_A/PM-1995/95-01/95-01-24/0012.html For Amiga: (all entries marked "ff###" are directories where the inividual archives of the Fish Disk set available at ftp://ftp.funet.fi/pub/amiga/fish/ and other sites) General Mandelbrot generators with many features: Mandelbrot (ff030), Mandel (ff218), Mandelbrot (ff239), TurboMandel (ff302), MandelBltiz (ff387), SMan (ff447), MandelMountains (ff383, in 3-D), MandelPAUG (ff452, MandFXP movies), MandAnim (ff461, anims), ApfelKiste (ff566, very fast), MandelSquare (ff588, anims) Mandelbrot and Julia sets generators: MandelVroom (ff215), Fractals (ff371, also Newton-R and other sets) With different algorithmic approaches (shown): FastGro (ff188, DLA), IceFrac (ff303, DLA), DEM (ff303, DEM), CPM (ff303, CPM in 3-D), FractalLab (ff391, any equation) Iterated Function System generators (make ferns, etc): FracGen (ff188, uses "seeds"), FCS (ff465), IFSgen (ff554), IFSLab (ff696, "Collage Theorem"") Unique fractal types: Cloud (ff216, cloud surfaces), Fractal (ff052, terrain), IMandelVroom (strange attractor contours?), Landscape (ff554, scenery), Scenery (ff155, scenery), Plasma (ff573, plasma clouds) Fractal generators: PolyFractals (ff015), FFEX (ff549) Lyapunov fractals ftp://ftp.luth.se/pub/aminet/gfx/misc/Lyapunovia15.lha Commercial packages: Fractal Pro 5.0, Scenery Animator 2.0, Vista Professional, Fractuality (reviewed in April '93 Amiga User International). MathVISION 2.4. Generates Julia, Mandelbrot, and others. Includes software for image processing, complex arithmetic, data display, general equation evaluation. Available for $223 from Seven Seas Software, Box 1451, Port Townsend WA 98368. Software for computing fractal dimension: Fractal Dimension Calculator is a Macintosh program which uses the box-counting method to compute the fractal dimension of planar graphical objects. http://wuarchive.wustl.edu/edu/math/software/mac/fractals/FDC/ http://wuarchive.wustl.edu/packages/architec/Fractals/FDC2D.sea.hqx http://wuarchive.wustl.edu/packages/architec/Fractals/FDC3D.sea.hqx FD3: estimates capacity, information, and correlation dimension from a list of points. It computes log cell sizes, counts, log counts, log of Shannon statistics based on counts, log of correlations based on counts, two-point estimates of the dimensions at all scales examined, and over-all least-square estimates of the dimensions. ftp://inls.ucsd.edu/pub/cal-state-stan ftp://inls.ucsd.edu/pub/inls-ucsd for an enhanced Grassberger-Procaccia algorithm for correlation dimension. A MS-DOS version of FP3 is available by request to gentry@altair.csustan.edu. Java applets Chaos! http://www.vt.edu:10021/B/bwn/Chaos.html Take's Online http://www.geocities.com/Hollywood/3618/java.html Fractal Lab http://www.wmin.ac.uk/~storyh/fractal/frac.html Mandelbrot Set Escape Iterations http://www.voidstar.org/java/escape.html The Mandelbrot Set http://www.mindspring.com/~chroma/mandelbrot.html Paton J. Lewis: Graphics Projects http://www.cs.brown.edu/people/pjl/mandelbrot.html Mark's Java Julia Set Generator http://liberty.uc.wlu.edu/~mmcclure/java/Julia/ Fractals by Sun Microsystems http://java.sun.com/java.sun.com/applets/applets/Fractal/example1.html The Mandelbrot set http://www.franceway.com/java/fractale/mandel_b.htm Mandelbrot Java Applet http://www.mit.edu:8001/people/mkgray/java/Mandel.html Ken Shirriff Java language pages http://www.sunlabs.com/~shirriff/java/ _________________________________________________________________ Subject: FTP questions Q23a: How does anonymous ftp work? A23a: Anonymous ftp is a method of making files available to anyone on the Internet. In brief, if you are on a system with ftp (e.g. Unix), you type "ftp lyapunov.ucsd.edu", or whatever system you wish to access. You are prompted for your name and you reply "anonymous". You are prompted for your password and you reply with your email address. You then use "ls" to list the files, "cd" to change directories, "get" to get files, an "quit" to exit. For example, you could say "cd /pub", "ls", "get README", and "quit"; this would get you the file "README". See the man page ftp(1) or ask someone at your site for more information. In this FAQ, anonymous ftp addresses are given in the form ftp://name.of.machine:/pub/path [1.2.3.4]. The first part is the protocol, FTP, rather than say "gopher", the second part "name.of.machine" is the machine you must ftp to. If your machine cannot determine the host from the name, you can try the numeric Internet address: "ftp 1.2.3.4". The part after the name: "/pub/path" is the file or directory to access once you are connected to the remote machine. Q23b: What if I can't use ftp to access files? A23b: If you don't have access to ftp because you are on a UUCP, Fidonet, BITNET network there is an e-mail gateway at ftpmail@decwrl.dec.com that can retrieve the files for you. To get instructions on how to use the ftp gateway send a message to ftpmail@decwrl.dec.com with one line containing the word "help". Warning, these archives can be very large, sometimes several megabytes (MB) of data which will be sent to your e-mail address. If you have a disk quota for incoming mail, be careful not exceed it. _________________________________________________________________ Subject: Archived pictures Q24a: Where are fractal pictures archived? NEW A24a: Fractal images (GIFs, etc.) used to be posted to alt.fractals.pictures; this newsgroup has been replaced by alt.binaries.pictures.fractals. Pictures from 1990 and 1991 are available via anonymous ftp at ftp://csus.edu/pub/alt.fractals.pictures Many Mandelbrot set images are available via ftp://ftp.ira.uka.de/pub/graphic/fractals Fractal images including some recent alt.binaries.pictures.fractals images are archived at ftp://spanky.triumf.ca/fractals. This can also be accessed via WWW at http://spanky.triumf.ca/ From Paris, France one of the largest collections (>= 460MB) is Frank Roussel's at http://www.cnam.fr/fractals.html. These images are also available via FTP at ftp://ftp.cnam.fr/pub/Fractals. Fractal animations in MPG and FLI format are in ftp://ftp.cnam.fr/pub/Fractals/anim or http://www.cnam.fr/fractals/anim.html. In Bordeaux (France) there is a mirror of this site, http://www.bdx1.u-bordeaux.fr/MAPBX/roussel/fractals.html. Another collection of fractal images is archived at ftp.maths.tcd.ie/pub/images/Computer A collection of interesting smoke- and flame-like jpeg iterated function system images is available on the WWW at http://www.cs.cmu.edu/afs/cs.cmu.edu/user/spot/web/images.html. Some images are also available from: ftp://hopeless.mess.cs.cmu.edu/spot/film/ Other tutorials, resources, and galleries of images are: Cliff Pickover http://sprott.physics.wisc.edu/pickover/home.htm Fractal Gallery (J. C. Sprott) http://sprott.physics.wisc.edu/fractals.htm Fractal Microscope http://www.ncsa.uiuc.edu/Edu/Fractal/Fractal_Home.html "Contours of the Mind" http://online.anu.edu.au/ITA/ACAT/contours/contours.html Computer Graphics Gallery http://www.maths.tcd.ie/pub/images/images.html The San Francisco Fractal Factory. http://www.awa.com/sfff/sfff.html Spanky Fractal Datbase (Noel Giffin) http://spanky.triumf.ca/www/spanky.html Fractal Gallery (Frank Rousell) http://www.cnam.fr/fractals.html Fractal Animations Gallery (Frank Rousell) http://www.cnam.fr/fractals/anim.html Yahoo Index of Fractal Art http://www.yahoo.com/Art/Computer_Generated/Fractals/ Geometry Centre at University of Minnesota http://www.geom.umn.edu/pix/archive/subjects/fractals.html Fractal from Ojai (Art Baker) http://www.fishnet.net/~ayb/ Skal's 3D-fractal collection (Pascal Massimino) http://acacia.ens.fr:8080/home/massimin/quat/f_gal.ang.html 3d Fractals (Stewart Dickson) via Mathart.com http://www.wri.com/~mathart/portfolio/SPD_Frac_portfolio.html Softsource http://www.softsource.com/softsource/fractal.html Favourite Fractals (Ryan Grant) http://www.ncsa.uiuc.edu/SDG/People/rgrant/fav_pics.html A.F.P. Fractal FTP Archive ftp://csus.edu/pub/alt.fractals.pictures Eric Schol http://hydra.cs.utwente.nl/~schol/video.html Mandelbrot and Julia Sets (David E. Joyce) http://aleph0.clarku.edu/~djoyce/home.html Newton's method http://aleph0.clarku.edu/~djoyce/newton/newton.html Gratuitous Fractals (evans@ctrvax.vanderbilt.edu) http://www.vanderbilt.edu/VUCC/Misc/Art1/fractals.html Xmorphia http://www.ccsf.caltech.edu/ismap/image.html Fractal Prairie Page (George Krumins) http://www.prairienet.org/astro/fractal.html Fractal Gallery (Paul Derbyshire) http://chat.carleton.ca/~pderbysh/fractgal.html David Finton's homepage http://www.d.umn.edu/~dfinton/ Algorithmic Image Gallery (Giuseppe Zito) http://www.ba.infn.it/gallery Octonion Fractals built using hyper-hyper-complex numbers by Onar Em http://www.stud.his.no/~onar/Octonion.html B' Plasma Cloud (animated gif) http://www.az.com/~rsears/fractp1.html John Bailey's fractal images http://www.servtech.com/public/jmb184/images Q24b: How do I view fractal pictures from alt.binaries.pictures.fractals? A24b: A detailed explanation is given in the "alt.binaries.pictures FAQ" (see "pictures-FAQ"). This is posted to the pictures newsgroups and is available by ftp: ftp://rtfm.mit.edu:/pub/usenet/news.answers/pictures-faq/ [18.181.0.24]. In brief, there is a series of things you have to do before viewing these posted images. It will depend a little on the system your working with, but there is much in common. Some newsreaders have features to automatically extract and decode images ready to display ("e" in trn) but if you don't you can use the following manual method: 1. Save/append all posted parts sequentially to one file. 2. Edit this file and delete all text segments except what is between the BEGIN-CUT and END-CUT portions. This means that BEGIN-CUT and END-CUT lines will disappear as well. There will be a section to remove for each file segment as well as the final END-CUT line. What is left in the file after editing will be bizarre garbage starting with begin 660 imagename.GIF and then about 6000 lines all starting with the letter "M" followed by a final "end" line. This is called a uuencoded file. 3. You must uudecode the uuencoded file. There should be an appropriate utility at your site; "uudecode filename " should work under Unix. Ask a system person or knowledgeable programming type. It will decode the file and produce another file called imagename.GIF. This is the image file. 4. You must use another utility to view these GIF images. It must be capable of displaying color graphic images in GIF format. (If you get a JPG or JPEG format file, you may have to convert it to a GIF file with yet another utility.) In the XWindows environment, you may be able to use "xv", "xview", or "xloadimage" to view GIF files. If you aren't using X, then you'll either have to find a comparable utility for your system or transfer your file to some other system. You can use a file transfer utility such as Kermit to transfer the binary file to an IBM-PC. _________________________________________________________________ Subject: Where can I obtain fractal papers? Q25: Where can I obtain fractal papers? A25: There are several Internet sites with fractal papers: There is an ftp archive site for preprints and programs on nonlinear dynamics and related subjects at: ftp://inls.ucsd.edu/pub. There are also articles on dynamics, including the IMS preprint series, available from ftp://math.sunysb.edu/preprints. A collection of short papers on fractal formulas, drawing methods, and transforms is available by ftp://ftp.coe.montana.edu:/pub/fractals (this site hasn't been working lately). The WWW site http://inls.ucsd.edu/y/Complex/ has some fractal papers. The site life.csu.edu.au has a collection of fractal programs, papers, information related to complex systems, and gopher and World Wide Web connections. The ftp path is: ftp://life.csu.edu.au/pub/complex/. Look in fractals, tutorial, and anu92. via WWW: http://life.csu.edu.au/complex/. _________________________________________________________________ Subject: How can I join the FRAC-L fractal discussion? Q26: How can I join the FRAC-L fractal discussion? A26: FRAC-L is a mailing list "Forum on Fractals, Chaos, and Complexity". The purpose of frac-l is to be a globally networked forum for discourse and collaboration on fractals, chaos, and complexity in multiple disciplines, professions, and arts. To subscribe to frac-l an email message to listproc@archives.math.utk.edu containing the sole line of text: SUBSCRIBE FRAC-L Your_first_name Your_last_name (substituting your actual first and last names, of course). To unsubscribe from frac-l, send LISTPROC (NOT frac-l) the message: UNSUBSCRIBE FRAC-L Messages may be posted to frac-l by sending email to: frac-l@archives.math.utk.edu If there is any difficulty contact the listowner: Ermel Stepp (stepp@marshall.edu). _________________________________________________________________ Subject: Complexity Q27: What is complexity? A27: Emerging paradigms of thought encompassing fractals, chaos, nonlinear science, dynamic systems, self-organization, artificial life, neural networks, and similar systems comprise the science of complexity. Several helpful online resources on complexity are: Institute for Research on Complexity http://www.marshall.edu/~stepp/vri/irc/irc.html The site life.csu.edu.au has a collection of fractal programs, papers, information related to complex systems, and gopher and World Wide Web connections. LIFE via WWW http://life.csu.edu.au/complex/ Complex Systems (UPENN) http://www.seas.upenn.edu/~ale/cplxsys.html Center for Complex Systems Research (UIUC) http://www.ccsr.uiuc.edu/ Complexity International Journal http://www.csu.edu.au/ci/ci.html Nonlinear Science Preprints ftp://xyz.lanl.gov/nlin-sys Nonlinear Science Preprints via email: To subscribe to public bulletin board to receive announcements of the availability of preprints from Los Alamos National Laboratory, send email to nlin-sys@xyz.lanl.gov containing the sole line of text: subscribe your-real-name _________________________________________________________________ Subject: References Q28a: What are some general references on fractals, chaos, and complexity? NEW A28a: Some references are: M. Barnsley, Fractals Everywhere, Academic Press Inc., 1988, 1993. ISBN 0-12-079062-9. This is an excellent text book on fractals. This is probably the best book for learning about the math underpinning fractals. It is also a good source for new fractal types. M. Barnsley, The Desktop Fractal Design System Versions 1 and 2. 1992, 1988. Academic Press. Available from Iterated Systems. M. Barnsley and P H Lyman, Fractal Image Compression. 1993. AK Peters Limited. Available from Iterated Systems. M. Barnsley and L. Anson, The Fractal Transform, Jones and Bartlett, April, 1993. ISBN 0-86720-218-1. This book is a sequel to Fractals Everywhere. Without assuming a great deal of technical knowledge, the authors explain the workings of the Fractal Transform(tm). The Fractal Transform is the compression tool for storing high-quality images in a minimal amount of space on a computer. Barnsley uses examples and algorithms to explain how to transform a stored pixel image into its fractal representation. R. Devaney and L. Keen, eds., Chaos and Fractals: The Mathematics Behind the Computer Graphics, American Mathematical Society, Providence, RI, 1989. This book contains detailed mathematical descriptions of chaos, the Mandelbrot set, etc. R. L. Devaney, An Introduction to Chaotic Dynamical Systems, Addison- Wesley, 1989. ISBN 0-201-13046-7. This book introduces many of the basic concepts of modern dynamical systems theory and leads the reader to the point of current research in several areas. It goes into great detail on the exact structure of the logistic equation and other 1-D maps. The book is fairly mathematical using calculus and topology. R. L. Devaney, Chaos, Fractals, and Dynamics, Addison-Wesley, 1990. ISBN 0-201-23288-X. This is a very readable book. It introduces chaos fractals and dynamics using a combination of hands-on computer experimentation and precalculus math. Numerous full-color and black and white images convey the beauty of these mathematical ideas. R. Devaney, A First Course in Chaotic Dynamical Systems, Theory and Experiment, Addison Wesley, 1992. A nice undergraduate introduction to chaos and fractals. A. K. Dewdney, (1989, February). Mathematical Recreations. Scientific American, pp. 108-111. G. A. Edgar, Measure Topology and Fractal Geometry, Springer-Verlag Inc., 1990. ISBN 0-387-97272-2. This book provides the math necessary for the study of fractal geometry. It includes the background material on metric topology and measure theory and also covers topological and fractal dimension, including the Hausdorff dimension. K. Falconer, Fractal Geometry: Mathematical Foundations and Applications, Wiley, New York, 1990. J. Feder, Fractals, Plenum Press, New York, 1988. This book is recommended as an introduction. It introduces fractals from geometrical ideas, covers a wide variety of topics, and covers things such as time series and R/S analysis that aren't usually considered. Y. Fisher (Ed), Fractal Image Compression: Theory and Application. Springer Verlag, 1995. L. Gardini(Editor), Chaotic Dynamics in Two-Dimensional Noninvertive Maps. World Scientific 1996, ISBN: 9810216475 J. Gleick, Chaos: Making a New Science, Penguin, New York, 1987. B. Hao, ed., Chaos, World Scientific, Singapore, 1984. This is an excellent collection of papers on chaos containing some of the most significant reports on chaos such as "Deterministic Nonperiodic Flow" by E.N. Lorenz. I. Hargittai and C. Pickover. Spiral Symmetry 1992 World Scientific Publishing, River Edge, New Jersey 07661. ISBN 981-02-0615-1. Topics: Spirals in nature, art, and mathematics. Fractal spirals, plant spirals, artist's spirals, the spiral in myth and literature... Loads of images. H. Jurgens, H. O Peitgen, & D. Saupe. (1990, August). The Language of Fractals. Scientific American, pp. 60-67. H. Jurgens, H. O. Peitgen, H.O., & D. Saupe. (1992). Chaos and Fractals: New Frontiers of Science. New York: Springer-Verlag. S. Levy, Artificial life : the quest for a new creation, Pantheon Books, New York, 1992. This book takes off where Gleick left off. It looks at many of the same people and what they are doing post-Gleick. B. Mandelbrot, The Fractal Geometry of Nature, W. H. FreeMan, New York. ISBN 0-7167-1186-9. In this book Mandelbrot attempts to show that reality is fractal-like. He also has pictures of many different fractals. E. R. Mac Cormac(Ed), M. Stamenov(Ed), Fractals of Brain, Fractals of Mind : In Searchg of a Symmetry Bond (Advances in Consciousness Research, No 7), John Benjamins, ISBN: 1556191871, Subjects include: Neural networks (Neurobiology), Mathematical models, Fractals, and Consciousness G.V. Middleton, (ed.), 1991: Nonlinear Dynamics, Chaos and Fractals (w/ application to geological systems) Geol. Assoc. Canada, Short Course Notes Vol. 9, 235 p. This volume contains a disk with some examples (also as pascal source code) ($25 CDN) T.F. Nonnenmacher, G.A Losa, E.R Weibel (ed.) Fractals in Biology and Medicine Birkhaeuser Verlag L. Nottale, Fractal Space-Time and Microphysics, Towards a Theory of Scale Relativity, World Scientific (1993). D. Peak and M. Frame, Chaos Under Control: The Art and Science of Complexity, W.H. Freeman and Company, New York 1994, ISBN 0-7167-2429-4 "The book is written at the perfect level to help a beginner gain a solid understanding of both basic and subtler appects of chaos and dynamical systems." - a review on the back cover H. O. Peitgen and P. H. Richter, The Beauty of Fractals, Springer-Verlag, New York, 1986. ISBN 0-387-15851-0. This book has lots of nice pictures. There is also an appendix giving the coordinates and constants for the color plates and many of the other pictures. H. Peitgen and D. Saupe, eds., The Science of Fractal Images, Springer-Verlag, New York, 1988. ISBN 0-387-96608-0. This book contains many color and black and white photographs, high level math, and several pseudocoded algorithms. H. Peitgen, H. Juergens and D. Saupe, Fractals for the Classroom, Springer-Verlag, New York, 1992. These two volumes are aimed at advanced secondary school students (but are appropriate for others too), have lots of examples, explain the math well, and give BASIC programs. H. Peitgen, H. Juergens and D. Saupe, Chaos and Fractals: New Frontiers of Science, Springer-Verlag, New York, 1992. C. Pickover, Computers, Pattern, Chaos, and Beauty: Graphics from an Unseen World, St. Martin's Press, New York, 1990. This book contains a bunch of interesting explorations of different fractals. C. Pickover, Keys to Infinity, (1995) John Wiley: NY. ISBN 0-471-11857-5. C. Pickover, (1995) Chaos in Wonderland: Visual Adventures in a Fractal World. St. Martin's Press: New York. ISBN 0-312-10743-9. (Devoted to the Lyapunov exponent.) C. Pickover, Computers and the Imagination (Subtitled: Visual Adventures from Beyond the Edge) (1993) St. Martin's Press: New York. C. Pickover. The Pattern Book: Fractals, Art, and Nature (1995) World Scientific. ISBN 981-02-1426-X Some of the patterns are ultramodern, while others are centuries old. Many of the patterns are drawn from the universe of mathematics. C. Pickover, Visualizing Biological Information (1995) World Scientific: Singapore, New Jersey, London, Hong Kong. on the use of computer graphics, fractals, and musical techniques to find patterns in DNA and amino acid sequences. J. Pritchard, The Chaos Cookbook: A Practical Programming Guide, Butterworth-Heinemann, Oxford, 1992. ISBN 0-7506-0304-6. It contains type in and go listings in BASIC and Pascal. It also eases you into some of the mathematics of fractals and chaos in the context of graphical experimentation. So it's more than just a type-and-see-pictures book, but rather a lab tutorial, especially good for those with a weak or rusty (or even nonexistent) calculus background. P. Prusinkiewicz and A. Lindenmayer, The Algorithmic Beauty of Plants, Springer-Verlag, NY, 1990. ISBN 0-387-97297-8. A very good book on L-systems, which can be used to model plants in a very realistic fashion. The book contains many pictures. Edward R. Scheinerman, Invitation to Dynamical Systems, Prentice-Hall, 1996, ISBN 0-13-185000-8, xvii + 373 pages M. Schroeder, Fractals, Chaos, and Power Laws: Minutes from an Infinite Paradise, W. H. Freeman, New York, 1991. This book contains a clearly written explanation of fractal geometry with lots of puns and word play. J. Sprott, Strange Attractors: Creating Patterns in Chaos, M&T Books (subsidary of Henry Holt and Co.), New York. ISBN 1-55851-298-5. This book describes a new method for generating beautiful fractal patterns by iterating simple maps and ordinary differential equations. It contains over 350 examples of such patterns, each producing a corresponding piece of fractal music. It also describes methods for visualizing objects in three and higher dimensions and explains how to produce 3-D stereoscopic images using the included red/blue glasses. The accompanying 3.5" IBM-PC disk contain source code in BASIC, C, C++, Visual BASIC for Windows, and QuickBASIC for Macintosh as well as a ready-to-run IBM-PC executable version of the program. Available for $39.95 + $3.00 shipping from M&T Books (1-800-628-9658). D. Stein, ed., Proceedings of the Santa Fe Institute's Complex Systems Summer School, Addison-Wesley, Redwood City, CA, 1988. See especially the first article by David Campbell: "Introduction to nonlinear phenomena". R. Stevens, Fractal Programming in C, M&T Publishing, 1989 ISBN 1-55851-038-9. This is a good book for a beginner who wants to write a fractal program. Half the book is on fractal curves like the Hilbert curve and the von Koch snow flake. The other half covers the Mandelbrot, Julia, Newton, and IFS fractals. I. Stewart, Does God Play Dice?: the Mathematics of Chaos, B. Blackwell, New York, 1989. Y. Takahashi, Algorithms, Fractals, and Dynamics, Plenum Pub Corp, (May) 1996, ISBN: 0306451271 Subjects: Differentiable dynamical syste, Congresses, Fractals, Algorithms, Differentiable Dynamical Systems, Algorithms (Computer Programming) T. Wegner and B. Tyler, Fractal Creations, 2nd ed. The Waite Group, 1993. ISBN 1-878739-34-4 This is the book describing the Fractint program. Q28b: What are some relevant journals? A28b: Some relevant journals are: "Chaos and Graphics" section in the quarterly journal Computers and Graphics. This contains recent work in fractals from the graphics perspective, and usually contains several exciting new ideas. "Mathematical Recreations" section by I. Stewart in Scientific American. Fractal Report. Reeves Telecommunication Labs. West Towan House, Porthtowan, TRURO, Cornwall TR4 8AX, U.K. WWW: http://ourworld.compuserve.com/homepages/JohndeR/fractalr Email: John@longevb.demon.co.uk (John de Rivaz) FRAC'Cetera. This is a gazetteer of the world of fractals and related areas, supplied on IBM PC format HD disk. FRACT'Cetera is the home of FRUG - the Fractint User Group. For more information, contact: Jon Horner, Editor, FRAC'Cetera Le Mont Ardaine, Rue des Ardains, St. Peters Guernsey GY7 9EU Channel Islands, United Kingdom. Email: 100112.1700@compuserve.com Fractals, An interdisciplinary Journal On The Complex Geometry of Nature This is a new journal published by World Scientific. B.B Mandelbrot is the Honorary Editor and T. Vicsek, M.F. Shlesinger, M.M Matsushita are the Managing Editors). The aim of this first international journal on fractals is to bring together the most recent developments in the research of fractals so that a fruitful interaction of the various approaches and scientific views on the complex spatial and temporal behavior could take place. Q28c: What are some other Internet references? A28c: Some other Internet references: Web references to nonlinear dynamics Dynamical Systems (G. Zito) http://alephwww.cern.ch/~zito/chep94sl/sd.html Scanning huge number of events (G. Zito) http://alephwww.cern.ch/~zito/chep94sl/chep94sl.html The Who Is Who Handbook of Nonlinear Dynamics http://www.nonlin.tu-muenchen.de/chaos/Dokumente/WiW/wiw.html _________________________________________________________________ Multifractals Q29: What are multifractals? NEW A29: It is not easy to give a succinct definition of multifractals. Following Feder (1988) one may distinguish a measure (of probability, or some physical quantity) from its geometric support -- which might or might not have fractal geometry. Then if the measure has different fractal dimension on different parts of the support, the measure is a multifractal. Hastings and Sugihara (1993) distinguish multifractals from multiscaling fractals -- which have different fractal dimensions at different scales (e.g. show a break in slope in a dividers plot, or some other power law). I believe different authors use different names for this phenomenon, which is often confused with true multifractal behaviour. ______________________________________________________________________ Subject: Notices Q30: Are there any special notices? NEW From: dfinton@ub.d.umn.edu (David Finton) 1. Well, I've been doing some computation of fractals, and thought, "You know, it would be really cool if there was another fractal art contest." So I thought I'd coordinate the next contest (giving Tim Wegner a break :). Here are the contest rules I propose: 2. The files must be in PAR file format. If it turns out you can only post a GIF or JPEG, post it on a binaries newsgroup, and post the location of the image on here. Do not post binary images like GIFs or JPEGs on this newsgroup! 3. Mandelbrot images only (from the equation z <- z² + c). Apologies to all the talented formula fractal artists out there, but I wanted to narrow down the parameters of the contest to make it easier to judge. :) 4. Deep Zoom. Yes that's right. I'd like to see some deep zoom fractals out there, and if anybody has found something interesting or amazing hidden in the depths of the Mandelbrot Set, please post them! Remember, originality counts. It would be nice to see something I would have never found rather than to see a generic image that everyone and their grandmother could have found. I hope I get a lot of responses out of this. Thanks! - Dave NOTICE from J. C. (Clint) Sprott (SPROTT@juno.physics.wisc.edu): The program, Chaos Data Analyzer, which I authored is a research and teaching tool containing 14 tests for detecting hidden determinism in a seemingly random time series of up to 16,382 points provided by the user in an ASCII data file. Sample data files are included for model chaotic systems. When chaos is found, calculations such as the probability distribution, power spectrum, Lyapunov exponent, and various measures of the fractal dimension enable you to determine properties of the system Underlying the behavior. The program can be used to make nonlinear predictions based on a novel technique involving singular value decomposition. The program is menu-driven, very easy to use, and even contains an automatic mode in which all the tests are performed in succession and the results are provided on a one-page summary. Chaos Data Analyzer requires an IBM PC or compatible with at least 512K of memory. A math coprocessor is recommended (but not required) to speed some of the calculations. The program is available on 5.25 or 3.5" disk and includes a 62-page User's Manual. Chaos Data Analyzer is peer-reviewed software published by Physics Academic Software, a cooperative Project of the American Institute of Physics, the American Physical Society, And the American Association of Physics Teachers. Chaos Data Analyzer and other related programs are available from The Academic Software Library, North Carolina State University, Box 8202, Raleigh, NC 27695-8202, Tel: (800) 955-TASL or (919) 515-7447 or Fax: (919) 515-2682. The price is $99.95. Add $3.50 for shipping in U.S. or $12.50 for foreign airmail. All TASL programs come with a 30-day, money-back guarantee. _________________________________________________________________ Subject: Acknowledgements Q31: Who has contributed to the Fractal FAQ? A31: Participants in the Usenet group sci.fractals and the listserv forum frac-l have provided most of the content of sci.fractals FAQ. For their help with this FAQ, special thanks go to: Alex Antunes, Simon Arthur, John Beale, Steve Bondeson, Erik Boman, Jacques Carette, John Corbit, Predrag Cvitanovic, Paul Derbyshire, John de Rivaz, Abhijit Deshmukh, Tony Dixon, Jürgen Dollinger, Robert Drake, Detlev Droege, Gerald Edgar, Gordon Erlebacher, Yuval Fisher, Duncan Foster, David Fowler, Murray Frank, Jean-loup Gailly, Noel Giffin, Frode Gill, Earl Glynn, Lamont Granquist, John Holder, Jon Horner, Luis Hernandez- Urëa, Jay Hill, Arto Hoikkala, Carl Hommel, Robert Hood, Larry Husch, Oleg Ivanov, Simon Juden, J. Kai-Mikael, Leon Katz, Matt Kennel, Robert Klep, Dave Kliman, Tal Kubo, Jon Leech, Otmar Lendl, Douglas Martin, Brian Meloon, Tom Menten, Guy Metcalfe, Eugene Miya, Lori Moore, Robert Munafo, Miriam Nadel, Ron Nelson, Tom Parker, Dale Parson, Matt Perry, Cliff Pickover, Francois Pitt, Olaf G. Podlaha, Francesco Potortì, Kevin Ring, Michael Rolenz, Tom Scavo, Jeffrey Shallit, Rollo Silver, J. C. Sprott, Ken Shirriff, Gerolf Starke, Bruce Stewart, Dwight Stolte, Tommy Vaske, Tim Wegner, Andrea Whitlock, Erick Wong, Wayne Young, Giuseppe Zito, and others. Special thanks to Matthew J. Bernhardt (mjb@acsu.buffalo.edu) for collecting many of the chaos definitions. If I have missed you, I am very sorry, let me know and I will add you to the list. Without the help of these contributors, the sci.fractals FAQ would be not be possible. _________________________________________________________________ Subject: Copyright Q32: Copyright? A32: This document, "sci.fractals FAQ", is Copyright 1995-1996 by Michael C. Taylor. All Rights Reserved. Previous versions: Copyright 1995 Ermel Stepp (edition v2n1) Copyright 1993-1994 Ken Shirriff The Fractal FAQ was created by Ken Shirriff and edited by him through September 26, 1994. The second editor of the Fractal FAQ is Ermel Stepp (Feb 13, 1995). Since December 2, 1995 the "acting editor" has been Michael C. Taylor. Standing permission is granted for non-profit reproduction and distribution of this issue of the sci.fractals FAQ as a complete document. This does not mean automatic permission for usage in CD-ROM collections or commerical educational products. If you would like to include sci.fractals FAQ in a commerical product, in whole or in part, contact Michael Taylor. If you would like to send a review sample of a program, or books, feel free send them to the editor: * Michael Taylor * P.O. Box 36 * Centreville (Kings) * Nova Scotia, B0P 1J0 * CANADA email: * aa459@chebucto.ns.ca * mctaylor@mailserv.mta.ca (until August 1996)