41. General, Photo » Photo Of The Day And Natural History Commentary fractals. Filed under. general. Recently we were approached about producingFractal Images for a client. With the correct software, and a sufficiently http://www.oceanlight.com/log/category/general/

@import url( http://www.oceanlight.com/log/style.css ); Photo Of The Day and Natural History Commentary Road Trip 2005 Filed under: General Just returned from a three week road trip. It was amazing. We had lots of fun and shot thousands of photos. It will take several weeks before I get the images organized and available online, so for now some summary statistics of the trip will have to suffice: 21 days 4500 miles with lots of hikes and mountain biking. 5700 photos occupying 130 gigabytes 7 National Parks and Monuments: Zion, Grand Teton, Yellowstone, Olympic, Mt. Rainier, Mt. Saint Helens, Crater Lake. 7 States: Utah, Idaho, Wyoming, Montana, Washington, Oregon, California. 3 Aquariums (Seattle, Steinhart, Monterey Bay) 13 Geysers including Old Faithful, Riverside, Daisy, Grand, White Dome, Pink Cone, Castle, Great Fountain, Grotto, Echinus, Anemone and Lion geysers. 17 Waterfalls including Upper and Lower Yellowstone, Gibbon, Lewis, Moose, Kepler, Firehole, Virginia Cascade, Undine, Rustic, Sol Duc, Marymere, Myrtle, Christine and Narada Falls. Many wild animals including gray wolf, moose, grizzly bear, black bear and cinnamon-colored black bear, elk, bison, coyote, golden and bald eagle, osprey, blacktail and whitetail deer, bighorn sheep, pronghorn antelope, marmot, chipmunk, squirrels, banana slug, wild turkey, long-tailed weasel, rabbits and some rabbit-like thing known as a pika.

42. [ Wu :: Fractals ] an introduction to fractals. topics discussed include fractals in nature and about fractals. i hope you now have a general idea of what a fractal is, http://www.ocf.berkeley.edu/~wwu/fractals/fractals.html

FRACTALS an introduction to fractals. topics discussed include fractals in nature and industry, and the basic mathematics behind generating several classic structures, including the mandelbrot set and sierpinski triangle. also included is a gallery of choice fractal art, including a few pieces made by myself. Fractal Intro Mandelbrot Sierpinski Gallery ... Return to Homepage A Short And Entertaining Introduction to Fractals a fractal is a geometric shape that can subdivided into parts, each of which is a reduced-size copy of the whole. the term was coined in the 1960s by benoit mandelbrot, a mathematician at IBM who adapted it from the latin adjective fractus , meaning "fragmented." to get a feel for what a fractal is, imagine inspecting a long, craggly, leafless tree branch on a frosty winter day. as your eyes scan the branch from its base to its tips, you notice that many sub-branches are generated along the way, and each sub-branch has a structure symmetric to the original, but smaller in scale. these sub-branches in turn fork off self-symmetric branches of their own. in the purest sense of a fractal, we imagine this branching process as never ending. you could compare it to the effect produced when two mirrors are faced toward each other, producing a claustrophobic, tunnel-like view of infinite imitation at smaller and smaller scales. a nice example of this is seen in the mandelbrot fractal at left, designed by Paul deCelle. usually one's first response to fractals is simply this: they are beautiful! indeed, they are visually arresting, and there are many reasons why. perhaps one reason is that they exhibit extreme levels of

43. The Strange World Of Fractals In general, fractals are not described by closedform analytic expressions, butare instead generated by simple iterative sets of rules, possibly with the http://users.forthnet.gr/ath/kimon/Fractals/Fractal.html

The Strange World of Fractals by Giota Makri General Overview A revolutionary step in the description of many natural shapes and phenomena was taken by Mandelbrot, when he discovered the meaning of fractality and fractal objects. "Fractal" came from the latin word fractus, meaning broken. While a formal definition of a fractal set is possible, the more intuitive notion is usually offered, that in a fractal, the part is reminiscent of the whole . Fractals have two important properties: Self similarity, and Self affinity Characteristic examples of self similar objects: Koch curves and the Sierpinski gasket and carpet. Self affine objects of unstable growth: DLA aggregates or Julia sets. An example of Koch snowflake An example of a Julia set In general, fractals are not described by closed-form analytic expressions, but are instead generated by simple iterative sets of rules, possibly with the addition of randomness at each step. The characterization of a fractal set is accomplished by the complete knowledge of the dependence of various moments of a measure of a set. Various fractal dimensions may, thus, be obtained. In fact, an infinity of such dimensions in general exists. However, the fractal dimension most commonly used is that of the support D, which is the fractal dimension casually invoked in describing a fractal object. A simple method of obtaining D is box counting , qualitatively explained as follows: The fractal set is completed covered by non-overlapping spheres (in a general sense) of (Euclidean) size

44. Fractal Geometry transformations that produce more general fractals by Iterated Function Systems.E. An elegant application of plane transformations to growing fractals http://classes.yale.edu/fractals/IntroToFrac/welcome.html

1. Introduction to Fractals Here we introduce some basic geometry of fractals, with emphasis on the Iterated Function System (IFS) formalism for generating fractals. In addition, we explore the application of IFS to detect patterns, and also several examples of architectural fractals. First, though, we review familiar symmetries of nature, preparing us for the new kind of symmetry that fractals exhibit. A . The geometric characterization of the simplest fractals is self-similarity : the shape is made of smaller copies of itself. The copies are similar to the whole: same shape but different size. B More examples of self-similarity examples, and variations including nonlinear self-similarity, self-affinity, and statistical self-similarity. Also, some fractal forgeries of nature. C Initiators and Generators is the simplest method for producing fractals. It is also the oldest, dating back 5000 years to south India. D Geometry of plane transformations is the mechanics of transformations that produce more general fractals by Iterated Function Systems. E . An elegant application of plane transformations to growing fractals is Iterated function systems . This method has been used in image compression. F Inverse problems finding the transformations to produce a given fractal. This is a geometrical version of Johnny Carson's "Karnak the Magnificent" routine.

45. Fractal Geometry Boxcounting dimension can be computed for mathematical fractals. In general,more detail is revealed with smaller r values going to 0, so for mathematical http://classes.yale.edu/fractals/Labs/CoastlineLab/BoxCountingDef.html

Dimension by Box-Counting Background - Box-Counting Dimension Natural fractals do not contain similar copies of themselves. How can we measure their dimension if there is no apparent scaling factor? Motivated by our Euclidean observations, we cover the shape with a grid of boxes and count the number of boxes that touch any part of the shape. Repeat for grids of smaller and smaller boxes. Denoting by N(r) the number of grid boxes of side length r touching any part of the shape, we adopt the assumption , and it is an assumption, that N(r) = k((1/r) d where k is the constant of proportionality, 1 in our previous examples. Solve for d by taking the log of both sides, obtaining log(N(r)) = dlog(1/r) + log(k). Here d and k are constants, log(N(r)) and log(1/r) vary with r, so this has the form y = mx + b of the equation of a straight line with slope d. Because d is interpreted as a dimension in our original scaling relation , we interpret it as a dimension in this more general setting. This d is called the box-counting dimension Suppose we have counted boxes for grids with side lengths r , r , ..., r

46. ► Fractals: History/formula/definition/meaning/Mandelbrot Set Generator fractals, described by Benoit Mandelbrot, is any image that is infinitely but Mandelbrot’s general idea of the importance of fractals is now well http://webweevers.com/fractals.htm

INTER ACTIVE PLAYGROUND BACK ... DONATE Send us your favorite landscape photographs newspaper format news sports college sports weather ... religions market place groceries vehicles shopping I shopping II ... financing guidance and direction directories searches I searches III world searches web site management website design web hosting endorsements visitors ... tell a friend administrative pages site hints I.P. mission contact us front page ... aol user notice Another WebWeevers.com Production A division of HidalCorp ... Should you have any questions, comments or otherwise, please click on this text, my picture above. A "Stumble Upon" Community Favorite to top Fractals Interactive fractals The Fractal explorer below will zoom in on a given section chosen by YOU of the Mandelbrot Set. The explorer will recalculate the given output for every new input and show you how Fractals continue forever into infinity. What is a fractal? A fractal is usually a rough or broken geometric shape which can be subdivided into parts. A fractal is the result of the input and output of a formula in a special software program.

47. General general MATHEMATICS SITES CHAOS AND fractals at http//hypertextbook.com/chaos/92.shtml (Various freeware programs for Koch curve, Mandelbrot Julia sets, http://mzone.mweb.co.za/residents/profmd/general.html

GENERAL MATHEMATICS SITES CHAOS AND FRACTALS at http://hypertextbook.com/chaos/92.shtml http://www.ed.gov/free/s-math.html Organized by the U.S. Department of Education, this site provides a lengthy list of links to free resources that support mathematics education and are available to teachers and students. http://www.enc.org/redirect/dd/?dd_id=1121 Contains a host of facts and figures about mathematical topics such as Pascal's triangle, Fibonacci numbers, and the Golden Section, as well as the connections between the concepts. FRACTALS at http://spanky.triumf.ca/ Gives information about what is new in fractal geometry. There is a collection of interactive software, fractal programs, and images. It is a good site to explore what fractals can really do. FRIEDMAN'S MATH MAGIC http://www.stetson.edu/~efriedma/mathmagic/archive.html INTERMATH at http://www.intermath-uga.gatech.edu/ KEY ISSUES FOR THE MATH COMMUNITY at http://forum.swarthmore.edu/social/index.html Key Issues: Key questions, key problems and opportunities, equity and access, minorities and mathematics, women and mathematics, the job marker and new teachers, mathematics and the public policy, public understanding of mathematics and ethical guidelines of the American Mathematical Society. MATHPAPERS at http://www.mathpapers.net/

48. Fractals And Chaos (Mandelbrot)-Springer Mathematics (general) Book It has only been a couple of decades since Benoit Mandelbrot published his famouspicture of what is now called the Mandelbrot set. http://www.springeronline.com/sgw/cda/frontpage/0,11855,4-120-22-26682365-0,00.h

Please enable Javascript in your browser to browse this website. Select your subdiscipline Algebra Analysis Applications Mathematical Biology Mathematical Physics Probability Theory Quantitative Finance Home Mathematics Select a discipline Biomedical Sciences Chemistry Computer Science Economics Education Engineering Environmental Sciences Geography Geosciences Humanities Law Life Sciences Linguistics Materials Mathematics Medicine Philosophy Popular Science Psychology Public Health Social Sciences Statistics preloadImage('/sgw/cda/pageitems/designobject/cda_displaydesignobject/0,11978,4-0-17-900180-0,00.gif'); preloadImage('/sgw/cda/pageitems/designobject/cda_displaydesignobject/0,11978,4-0-17-900170-0,00.gif'); preloadImage('/sgw/cda/pageitems/designobject/cda_displaydesignobject/0,11978,4-0-17-900190-0,00.gif'); preloadImage('/sgw/cda/pageitems/designobject/cda_displaydesignobject/0,11978,4-0-17-900200-0,00.gif'); preloadImage('/sgw/cda/pageitems/designobject/cda_displaydesignobject/0,11978,4-0-17-900369-0,00.gif'); preloadImage('/sgw/cda/pageitems/designobject/cda_displaydesignobject/0,11978,4-0-17-900344-0,00.gif'); Please select Africa Asia Australia / Oceania Europe Germany North America South America Switzerland United Kingdom All Author/Editor Title ISBN/ISSN Series preloadImage('/sgw/cda/pageitems/designobject/cda_displaydesignobject/0,11978,4-0-17-900050-0,00.gif');

49. Documents For An Access Point 2, Harold M. Hastings, fractals A user s guide for the natural sciences, 010677,1993, Book, 0 010677, general, 536.758FRACT/HAS/010677, On Shelf http://libibm.iucaa.ernet.in/slim/wslxRSLT.php?A1=8888

50. Chaos Theory Resources - Fractals - Directories - Academic Info A Directory of Internet resources on chaos theory and fractals. with aninterest in chaos, fractals, nonlinear dynamics, or mathematics in general. http://www.academicinfo.net/mathchaos.html

Academic Info Mathematics Home Search Index Contact ... Mathematics Chaos Theory Huge Savings on Scholarly and General Books at our Bookstore Advertisers University of Phoenix Online - Earn your degree 100% online. Online coupon codes Questia - Search over 70,000 books and journals online. Student Loan Consolidation Student Loans Search Engine Marketing Online Bingo ... Natural products for the laundry quit smoking smoke away liquid vitamins . Also weight loss diet diet products cortisol ... cortislim info; Libido orgasm info. Stop smoking help. - 100% Pure Hoodia Gordonii Diet Pills hoodia and hoodia gordonii diet pills and carb blocker Teeth Whitening - Low prices on all brands of teeth whitening products. The exact same teeth whitening prescribed by your dentist, delivered to you for less. MyUSBusinessFinder - The Business Search Directory Business search engine and business directory focussing mainly on the US. - Diet Reviews Zone provides reviews on Diet Pills and Diet Patches - Natural diet pills and Liposafe weight loss pills at discount prices.

51. The Educational Encyclopedia, Mathematics general overview. Aplusmath this web site is developed to help students improve Apollonius problem, Pythagoras, arbelos, fractals, fractal dimension, http://users.pandora.be/educypedia/education/mathematics.htm

EDUCYPEDIA The educational encyclopedia Home Electronics General Information technology ... Science Science Automotive Biology Biology-anatomy Biology-animals ... Space Social science Atlas - maps Countries Dinosaurs Environment ... Sitemap Mathematics Algebra Complex numbers Formulas Fractals ... Fourier General overview Geometry Integrals and differentials Logarithms and exponentials Matrices and determinants ... Trigonometry General overview Aplusmath this web site is developed to help students improve their math skills interactively, algebra, addition, subtraction, multiplication, division, fractions, geometry for kids Ask Dr. Math Ask Dr. Math a question using the Dr. Math Web form, or browse the archive Calculus tutorial Karl's calculus tutorial, limits, continuity, derivatives, applications of derivatives, exponentials and logarithms, trig functions (sine, cosine, etc.), methods of integration Cut the knot! algebra, geometry, arithmetic, proofs, butterfly theorem, chaos, conic sections, Cantor function, Ceva's theorem, Fermat point, cycloids, Collage Theorem, Carnot's theorem, bounded distance, barycentric coordinates, Pythagorean theorem, Napoleon's theorem, Ford's touching circles, Euclid's Fifth postulate, Non-Euclidean Geometry, Projective Geometry, Moebius Strip, Ptolemy's theorem, Sierpinski gasket, space filling curves, iterated function systems, Heron's formula, Euler's formula, Hausdorff distance, isoperimetric theorem, isoperimetric inequality, Shoemaker's Knife, Van Obel theorem, Apollonius problem, Pythagoras, arbelos, fractals, fractal dimension, chaos, Morley, Napoleon, barycentric, nine point circle, 9-point, 8-point, Miquel's point, shapes of constant width, curves of constant width, Kiepert's, Barbier's

52. Chaos And Fractals : New Frontiers Of Science By Heinz-Otto Peitgen, Dietmar Sau Chaos and fractals New Frontiers of Science by HeinzOtto Peitgen, fractals remains an authoritative general reference on chaos theory and fractals. http://www.berkhan.com/easy/books/english/peitg01.htm

Search: All Products Books Popular Music Classical Music Video DVD Electronics Software Outdoor Living Wireless Phones Keywords: Chaos and Fractals: New Frontiers of Science by Heinz-Otto Peitgen, Dietmar Saupe, H. Jurgens (Contributor), L. Yunker (Contributor) Price: $64.95 Hardcover - 984 pages (August 1992) Springer Verlag; ISBN: 0387979034 ; Dimensions (in inches): 2.25 x 9.53 x 7.89 Sales Rank: 45,578 Category: Science Editorial Reviews (Amazon.com) Fascinating and authoritative , Chaos and Fractals: New Frontiers of Science is a truly remarkable book that documents recent discoveries in chaos theory with plenty of mathematical detail, but without alienating the general reader. In all, this text offers an extremely rich and engaging tour of this quite revolutionary branch of mathematical research. The most appealing aspect about Chaos and Fractals has to be its hundreds of images and graphics (with dozens in full-color) used to illustrate key concepts. Even the math-averse reader should be able to follow the basic presentation of chaos and fractals here. Since fractals often mimic natural shapes such as mountains, plants, and other biological forms, they lend themselves especially well to visual representation. Early chapters here document the mathematical oddities (or "monsters") such as the Sierpinski Gasket and the Koch Curve, which laid the groundwork for later discoveries in fractals. The book does a fine job of placing recent discoveries about chaos into a tradition of earlier mathematical research. Its description of the work of mathematicians like Pascal, Kepler, Poincaré, Sierpinski, Koch, and Mandelbrot makes for a fine read, a detective story that ends with the discovery of order in chaos. (For programmers, the authors provide short algorithms and BASIC code, which lets you try out plotting various fractals on your own.)

53. Chaos And Fractals In Financial Markets, Part 7, By J. Orlin Grabbe Chaos and fractals in Financial Markets. Part 7 Grow Brain and the Flooding ofthe Nile In general, the set of Y s. {Y(1), Y(2), . . . , Y(99)} http://www.orlingrabbe.com/chaos7.htm

Chaos and Fractals in Financial Markets Part 7: Grow Brain and the Flooding of the Nile by J. Orlin Grabbe Grow Brain Many dynamical systems create solution paths, or trajectories, that look strange and complex. These solution plots are called "strange attractors". Some strange attractors have a fractal structure. For example, we saw in Part 3 that it was easy to create a fractal called a Sierpinski Carpet by using a stochastic dynamical system (one in which the outcome at each step is partially determined by a random component that either selects among equations or forms part of at least one of the equations, or both). Here is a dynamical system that I ran across while doing computer art. I labeled it "Grow Brain" because of its structure. To see Grow Brain in action, make sure Java is enabled on your browser (you can turn it off afterward) and click here . (The truly paranoid can, of course, compile their own applet, since I provide the source code, as usual.) The trajectory of Grow Brain is amazingly complex. But is it a fractal? That is, at some larger or smaller scale, will similar structures repeat themselves? Unlike the case of the Sierpinski Carpet, the answer to this question is not obvious for Grow Brain. Some dynamical systems create fractal structures in time (as Brownian motion does, in

54. Per Noergaard - Per Nørgårds Uendelighedsrække - Og Fraktaler In relation to the general description of fractals below, it may be said that the fractals in general. The term fractal is connected with the ideas of http://www.pernoergaard.dk/eng/strukturer/uendelig/ufraktal.html

Per Nørgård's infinity series - and fractals By Jørgen Mortensen The extent to which the infinity series possesses characteristics in common with fractals is really quite striking; equally remarkable is the fact that it offered an outline of Chaos theory long before this was formulated by science. In relation to the general description of fractals below, it may be said that the infinity series and fractals share the following qualities: Scale invariance - the same picture reappears in various sizes Self-similarity - the series reappears within the series Construction by iteration (projection of intervals) Construction by recursion (by repeated reduplication, or 'folding') Simple formula, detailed picture Fractals in general The term 'fractal' is connected with the ideas of 'fracture surface' and 'break'. Whilst geometry universally works with straight lines or smooth curves, nature would seem to avoid these regularities. Take the uneven bark of a tree, jagged mountain tops, or an irregular coastline. In fact, completely regular or straight lines do not seem to appear in nature at all. The only exception is a beam of light. A coastline is always uneven. It represents both 'self-similarity' and 'scale invariance', because from a distance one sees the same uneven picture as close to.

55. Sierpiñski Fractals In fact Sierpiñski fractals are really just a special kind of CAT In one waythey produce images that are more truly fractal than general CAT fractals, http://www.alunw.freeuk.com/sierpinskiroom.html

Sierpiñski fractals Sierpiñski was the mathematician who discovered one of the first fractals, the Sierpiñski triangle. You can find out more about it on the Spirofractal tour . (Possibly the very first fractal was the Koch snowflake curve, but this is not a C.A.T. fractal, though it is closely related to one). The process used to construct the triangle can be generalised slightly by picking a number of fixed points and picking one of them at random as a starting point. Then we pick one of the other fixed points at random, and move a fixed proportion (less than one) of the distance between the two points, either towards the point or away from it. We repeat this indefinitely, coloring points according to how often they are visited. In fact Sierpiñski fractals are really just a special kind of C.A.T. fractal. However they are usually very recognisable. In one way they produce images that are more truly fractal than general C.A.T. fractals, because if you magnify a Sierpinski fractal, the structure of the whole is repeated in each part, without the distortions that occur in general C.A.T. fractals. This is a randomly generated Sierpiñski pentagon. Notice that whereas the Sierpiñski triangle can be decomposed into its component parts, which are all identical (except half size) copies of the whole image, the same cannot be done for the Pentagon. Different scaled down copies of the fractal overlap. Fractals that do not overlapping parts are called

56. UserLand Frontier Server Error fractals In Biology Developing The Underlying Mechanistic Principles For of life as a window by which to synthesize general principles in biology. http://discuss.santafe.edu/biofractals/

Sorry! There was an error: The error was detected by Frontier 9.1b4 in mainResponder.respond. Webmaster: webmaster@santafe.edu. Time: Wed, 21 Sep 2005 19:38:38 GMT.

57. Books About Fractals And Fractal Art Books about fractals and Fractal Art buy them online at BrainGames Bazaar. Part I is concerned with the general theory of fractals and their geometry, http://thinks.com/books/fractals.htm

Home Books Fractals and Fractal Art Just click on the title of any book that interests you and you'll be automatically linked to Amazon.com - where you'll find that many books are offered at discounts of up to 40%. If you decide to buy, your transaction will be processed safely using Secure Server Technology. Next thing you know, that new book's on your coffee table and providing hours of entertainment. See also Books on Mathematic Recreations and Books by Cliff Pickover . You might also be interested in other fractal pages at Thinks.com. Chaos and Fractals R L Devaney, Linda Kean : American Mathematical Society, 1989 : Hardcover Fractal Cosmos : The Art of Mathematical Design Jeff Berkowitz : Amber Lotus, 1998 : Paperback The dynamic interplay between order and chaos is explored in 350 color images in this unique coffee table book that explains the mechanics of mathematical art. Berkowitz is one of the world's most widely recognized fractal artists. Fractal Cosmos is the first art book to feature fractal imagery and the largest collection of fractal art published anywhere. Fractal Geometry : Mathematical Foundations and Applications Kenneth Falconer An accessible introduction to fractals, useful as a text or reference. Part I is concerned with the general theory of fractals and their geometry, covering dimensions and their methods of calculation, plus the local form of fractals and their projections and intersections. Part II contains examples of fractals drawn from a wide variety of areas in mathematics and physics, including self-similar and self-affine sets, graphs of functions, examples from number theory and pure mathematics, dynamical systems, Julia sets, random fractals, and some physical applications. Also contains many diagrams and illustrative examples, includes computer drawings of fractals, and shows how to produce further drawings.

58. Grand Canyon Natural Fractals Presentation And Slide Notes Natural fractals is hyperlinked to the Earth Monitoring System website s Rocks. I have linked to some general information about geology and fractals. http://www.public.asu.edu/~starlite/SlideNotes.html

HOME FREQUENTLY ASKED QUESTIONS NATURAL FRACTALS IN GRAND CANYON NATIONAL PARK FOLLOW-UP IMAGES (screensaver sized) THE SIERPINSKI TETRAHEDRON BUILD YOUR OWN SIERPINSKI TETRAHEDRA PHOTOGRAPHS OF SIERPINSKI TETRAHEDRA (with/at) (Lemna gibba) at the Phoenix Zoo THE PLATONIC SOLIDS THE FRACTALS OF THE PLATONIC SOLIDS MATH STRUCTURES COLORING PAGES: CONTACT INFORMATION Natural Fractals in Grand Canyon National Park by Gayla Chandler Introduction I presented this as a special program at Grand Canyon National Park on Friday 31 December 2004 and Saturday 1 January 2005. The slideshow and notes are now accessible from the web for viewing by individuals and/or use by teachers in classrooms. I ask that no changes be made to the materials, although suggestions for changes are welcome. The presentation should project well provided the projector is set for 800x600 image size. If for any reason, links are inaccessible while viewing the presentation, they are listed in a Word document here I have also put up a page of Follow-Up Images , specially selected for post-presentation viewing. Sometimes one, sometimes several aspects of similarity discussed in the presentation are particularly prominent in each image (no hints are given, however). I put these up full-sized (1600 x 1200) at medium quality, so they may be used as screensavers or as desktop backgrounds for any size monitor.

59. Nrich.maths.org::Mathematics Enrichment::NRICH about how much you already know about IFS s, and fractals in general? It gives quite a flexible and beautiful set of fractals (which I have seen at http://nrich.maths.org/askedNRICH/edited/2191.html

Skip over navigation About Contact Mailing Lists ... maths finder past issues Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Welcome to NRICH. Iterated Function Systems (Fractals) By Ashish Bhambhani (P4136) on Tuesday, March 27, 2001 - 09:06 pm Does anyone here know much about Iterated Function Systems and how to program them ? They are very versatile algorithms, with which it is possible to create almost any image, given a relatively few numbers to describe that image in terms of transformations. These transformations are carried out recursively to get the final image, according to the desired accuracy. Infact an image compression technique is based on this technology. These fractals were introduced by Michael Barnsley. If you have ever used FRACTINT, it supports these beautiful fractal definitions. By David Loeffler (P865) on Thursday, March 29, 2001 - 09:44 pm Could you give a little more information about how much you already know about IFS's, and fractals in general? When you say "how to program them" do you mean that you are trying to write a computer program from scratch that can plot a picture of an IFS, given a description of the functions that make it up? This will depend a great deal on what programming languages you are using. Have you tried using a programming language such as Visual Basic before? David By Ashish Bhambhani (P4136) on Sunday, April 1, 2001 - 10:47 pm

60. Programming Tips - General Tips general Tips. While creating this website, we had a great deal of programming todo for Below are some of our own suggestions for programming fractals, http://library.thinkquest.org/26242/full/progs/t2.html

Tips General Tips While creating this website, we had a great deal of programming to do for the applets and many of the diagrams. Below are some of our own suggestions for programming fractals, which we came up with while doing it. Although some of the things here will be obvious to anyone who ever programmed, they are especially relevant to fractals. AVOID REDRAWING Letting your program redraw parts of the fractal has three major negative effects. First of all, it takes time without producing any image. Second of all, if your program is intended to be used by other people, to them it will look as though the program hung, since they cannot know that calculation is taking place meanwhile. Lastly, because of rounding errors that computers make when doing many calculations, the redrawn image will often be misplaced from the original one, producing an unclear image. There are several thing you can do about this problem. One of them is making the program check if the current part of the fractal was drawn already. This is especially efficient in strange attractors . For Julia Sets , you can use the MIIM instead of the IIM to implement this method. Another thing, which can be done for

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