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10. Web Sites

The following are some sites involving fractal geometry and iterated function systems as garnered from the World Wide Web. Some also involve L-systems (see The Algorithmic Beauty of Plants by P. Prusinkiewicz and A. Lindenmayer). They involve mostly visual imagery, although there is some fractal-generated music here and there, and some other digressions farther afield. Read the first few for a general overview and an introduction to the mathematical basis. The rest are mostly pure poetry and other applications!

My comments are in black. Plagiarized text is in maroon.

The following references to sites will be sorted into a better arrangement by category later.


Web 1. Fractals and Chaos in Nature - Mark Cushman. Fractals and Chaos in Nature. A general introductory essay.

Web 2. See the glorious works of Ken Musgrave via the Ken Musgrave page.

Web 3. Fractal Geometry and Multifractal Formalism. Classical mathematical concepts and methods are concerned with smooth objects, ignoring irregular sets . . . Four transformations in the plane are iterated as a random iterated function system (IFS) to produce a  fractal attractor of the IFS -- in this case, a fern. Ignore for now the fine point about which set of four probabilities produces the highest-fidelity image. Note that you can see a blow-up of a leaf of the fern, and it would be possible to generate a leaf of that leaf, and ., ad infinitum -- all from 28 numbers!

Here is how to grow a fern!

Start with a single rectangle. Transform it into four smaller rectangles, suitably placed, and all lying within (or mostly within) the original rectangle. Denote these as transformations 1 to 4. Pick an arbitrary point on the plane. Choose at random a digit from one to four. Apply that numbered transformation to the point and plot the resulting point. To that resulting point, apply the next randomly chosen transformation. Repeat (iterate) this process several thousand times. What you may see emerging from a supposedly random fog of dots could be a pine tree or the famous Black Spleenwort Fern of Barnsley, depending on the placement and sizes of the four transformed rectangles. Ignore the first 500 or so points. All subsequent points will converge onto the resulting image, the so-called attractor of the ifs.

Choose any portion of that image and plot an additional million points to obtain as much additional detail of the attractor as you wish. Whether one started with four rectangles or four profiles of the Volkswagen Beetle*, one will end up with the same image. Why? Because the equations describing the transformations are invariant in relation to the particular starting image (rectangle) used as illustration. Iteration of the ifs converges to its unique attractor. As an important refinement, choose the transformations with probabilities that are relative to the areas of their associated rectangles. This process is referred to as the random iteration algorithm, or (by Barnsley) the chaos game.

        *I love the VW Beetle ad: "Reverse engineered from UFOs"! (Demand now is so high that VW can't paste them together fast enough!)

Have a look at what you could have wrought!

Web 303. The Fractory: An Interactive Tool for Creating and Exploring Fractals. This page will help you learn about fractals: what they are and how to design them, but it will also let you discover more on your own. Fractals are just now emerging as a science. They show an order in seemingly random things, and give us tools with which we can predict the weather, render natural looking objects, and help understand the order in our chaotic lives. You will also play a part in our interactive fractal creation center, designing and displaying fractals you have invented. You can even post your coolest fractals on our fractal message board. How to Understand Fractals: Fractals can be understood on many different levels. We have separated the information on this page into the five levels listed in increasing order of difficulty.

Web 310. Chaos in the Classroom. Robert L. Devaney, Department of Mathematics, Boston University. Technology lets teachers bring some topics of contemporary interest in research mathematics into both middle school and high school classrooms.

Web 315.  Fractals page 1 Fractals. Math Index, Index, A, B, C, . , Z. Ad Hoc Fractals. After landing on the site, start by clicking Mathematics | Fractals. Has good tutorials. A must-see site!

Web 320. Here is one deep zoom. A big image from the land of Mandelbrot. It takes a while to download, but it's worth it! It is probably a Julia set, generated from z = z**2 + c (with z = x + i*y, iterated in the complex plane.

Web325. Fractal Landscape Gallery Uses KPT Bryce software. See "Jesse's Cool Seascape" for a rendition of a problem that I have been considering: how to create waves and reflectionson them.

Web 330. Oliver's images. Computer generated plants generated by our plant modelling software xfrog ( ) (see also Web 335 below) demonstrate the state of the art in plant generation and imaging. These plants were generated by Marco Bubke and rendered with Maya: Computer generated ecosystems created in a project for SIGGRAPH 98 where we showed that whole ecosystems can be generated and rendered with today's workstations. The models were created using xfrog, the populations were specified by additional tools. The paper is: O. Deussen, P. Hanrahan, B. Lintermann, R. Mech, M. Pharr, P. Prusinkiewicz: Realistic Modeling and Rendering of Plant Ecosystems SIGGRAPH '98, Orlando, Florida, July 19-24, 1998, Be sure to see the final image of trees in a field of sunflowers -- among the finest of natural IFS images ever generated! Uses Xfrog. See also Web 335 below.

Web 335. Greenworks Organic Software, Home of Xfrog. The organic modeler: Lintermann und Deussen Gbr and the website were launched in 1996 in Karlsruhe Germany by Bernd Lintermann and Professor Oliver Deussen, with the aim to make their development efforts in the area of organic research available to the general public, i.e., publish their technical papers, and their software, XFrog. XFrog is based on mathematical building blocks of Nature, using a linked hierarchical graph of procedural nodes to build and animate(SGI only) organic models. many technical papers have been published by lintermann and deussen regarding this work and a select group of their research is available for download. At SIGGRAPHf 2000 we were excited to release our XFrog for PC product. Xfrog 3.0 is available for almost all flavors of Windows. You can download a 30 day fully functional trial version. Be sure to visit the following:

. . . Its gallery! See also Web 330 above.

. . . Its documents section: , including:

. . . . . . Interactive Modelling and Animation of Branching Botanical Structures: where you can see movies of plants growing up!

Web 340. Sierpinski Gasket: The Sierpinski gasket was originally described in two dimensions but represents a family of objects in other dimensions. This family of objects will be discussed in dimensions 1, 2, 3, and an attempt will be made to visualise it in the 4th dimension. Along with a few cans of Coke! See also:

. . . 3D Sierpinski Gasket (3dsg.jpg):

. . . Sierpinski Gasket:

. . . Sierpinski Gasket Generator Applet: Allows plotting lines, changing no. of vertices.

. . . The Sierpinski Triangle Fractal: Interactive Applet. Go ahead and play with it! Best demo of these!

. . . Sierpinski Pyramid. Variants.

. . . The Magic Sierpinski Triangle: Order dependent on randomness. Interactive Applet.

Web 360. Fractals and the Fractal Dimension. The Hausdorff Dimension: The "Richardson Effect". Mandelbrot Set depicted with different colors or altitudes. See also:

. . . Felix Hausdorff: In 1919 he introduced the notion of Hausdorff dimension, which was a real number lying between the topological dimension of an object and 3. It is used to study objects such as Koch's curve. He also introduced the Hausdorff measure and the term 'metric space' is due to him.

. . . FRACTAL HISTORY AND BACKGROUND. The most intriguing of the nonlinear fractals thus far has been the mathematical set named after Mandelbrot by the American mathematicians John Hubbard and Adrien Douady.

. . . Making Julia Set Fractals: Julia sets, named for the early twentieth century French mathematician Gaston Julia, who pioneered the method by which they are produced.

. . . Abram Samoilovitch Besicovitch: His work on sets of non-integer dimension was an early contribution to fractal geometry. Hausdorff, in 1918, had extended Carathéodory's theory of measure to sets having finite measure of non-integral order. Besicovitch, around 1930, extended his density properties of sets to those of finite Hausdorff measure. Domb writes: . . . Hausdorff and Besicovitch [were] the two mathematical pioneers on whose work Mandelbrot's development of fractals is based.

Web 370. Fractal Geometry: By John Hoggard.

Web 4. The World of Chaos. Introduction. Laplace's Demon. What is Chaos? The Butterfly Effect. Strange Attractors. Logistic Difference Equation. Fractals. Iterated . . . Includes more on how to generate the fern, etc.

Web 4000. The World of Chaos and Fractals. This paper was written in conjunction with a talk given for Intermediate Physics Seminar of the Department of Physics and Astronomy at the Johns Hopkins University. It provides an excellent overview, being neither too vague nor too detailed. It includes the following pages:

. . . Introduction:, including:

. . . . . . Laplace's Demon: Determinism.

. . . . . . What is Chaos? Chaos is indeterminism at its best.

. . . The Butterfly Effect: Weather prediction is difficult!

. . . Strange Attractors: Edward Lorenz started it!

. . . Logistic Difference Equation: Variability in animal populations: a simple quadratic equation, the logistic difference equation, leads to fantastically complex and chaotic behavior. Bifurcation and Lorenz and Henon attractors.

. . . Fractals: Sierpenski triangle. Fractional dimension. Snowflake curve of Von Koch.

. . . Iterated Function Systems (IFS): Black spleenwort fern. This image is infinitely complex -- a self-similar fractal on all scales.

. . . References: Books and more links, including:

. . . . . . The Computational Beauty of Nature: a slide show:

. . . . . . The Computational Beauty of Nature Computer Explorations of Fractals, Chaos, Complex Systems, and Adaptation, by Gary William Flake :

. . . . . . The Chaos Hypertextbook: Very detailed!

. . . . . . Chaos and Order:

. . . . . . Chaos & Fractals: Part of the ThinkQuest project. Has Applets.

. . . . . . The Chaos Experience: Includes real-life examples, e.g., washing machine.

. . . . . . The Chaos Metalink: Optimized for WebTV!

. . . . . . fractalus home: Over three hundred and ninety fractal images + . . .

. . . . . . Mu-Ency - The Encyclopedia of the Mandelbrot Set:

Web 5. J.P. Louvet : Fractals - A History. The discovery of fractals. 8-Feb-98. Excellent pics.

Web 6. No Title A Path to Fractals. Nature is filled with a beauty easily perceived at many levels. Trees and forests, rocks and mountains, streams and mighty rivers, 8-Mar-95. No pics.

Web 7. Fat Days: Fractal Links. Spanky Fractal Database. Huge archive of fractal-related programs, . . .

Web 400. Open Directory - Science Math Chaos and Fractals. Excellent links to Chaos and Fractals (184 links), Chaos (44), Fractal Art (77), and Software (30). Here are some:

. . . The Computational Beauty of Nature: Computer Explorations of Fractals, Chaos,
Complex Systems, and Adaptation. This is the home page for The Computational Beauty of Nature, affectionately known as ``The Fish and Chips Book.'' Here, you will find information about the book, source code for simulations involving fractals, chaos, complex systems, and adaptation, and a whole slew of goodies for people interested in multidisciplinary topics involving computers, philosophy, and science. Some recent news items concerning the book and author are highlighted. Excellent!

. . . Fractals: An excellent site with in-depth descriptions of many different types of fractals. It also describes the formulas behind the creation of the fractals.

. . . Our Fractal Universe: See Fractals in Nature | Fractals in Biology for interesting relationship between animal size, metabolic rate, and fractals. Good overview of IFS basics and of advanced IFS image compression via range and domain blocks.

. . . Fantastic Fractals. A comprehensive fractals site that features tutorials for all ages, fractal-generated music, realistic landscapes. Our Fantastic Fractals 98 software that can generate millions of fractals is available for download only at this site. Interact with others through the fractal workshop, message boards, and discussion forum.

. . . Fractal Adventures. Helps the beginner to create fractal images. Overview of programs. Program-related fractal Gallery.

. . . Fractals: Nature's Numbers?

. . . Fractal Seeks Wavelet. A research project for De MontfortUniversity, exploring links between Fractals and Wavelets for1D-Signals.

.. . . Fractals Unleashed.. The most comprehensive website about fractals on the net. Contains enourmous amounts of information, including fractal applications, Java applets, CGI programs, a well-organized gallery, and a fractal art contest. Download software and learn how to program fractals too!

. . . An Introduction to Fractals. Gives excellent description, math, image of fern, and C-code to generate it. Slow to download (from Russia with love), but worth it!

Web 410. Chaos Web Site -- Fractals and Resources: Includes many links, including:

. . . Interactive Fractal Servers, including:

. . . . . . Mandelbrot Viewer: with interactive Java Applet.

Web 14. IFS A general, way to generate fractals is through Iterated Function Systems (IFSs). 5-Sep-97 See some of the math. and try your own ifs!

Web 15. Fractal Image Compression Here is your chance to meet Lena! This site has excellent details of IFSs and images of Lena, including an example of a compressed image having more detail of her than the original -- not to be missed! No other compression technique (e.g., JPEG, MPEG, GIF) can do this because, although non-ifs compression techniques, and even the direct rendition of the original image, are pixel-dependent, the ifs technique is not! That is why, here, "an eye for an eye" takes on a whole new meaning, and why Barnsley calls this "fractal forgery"! It is also why HDTV could become possible, via ifss and extremely fast home computers (contemplate 100,000 MIPS!).

Web 16. Fractal Image Compression: More math. -- and another look at Lena!

Web 17. Groupe Fractales - INRIA. Fractal approaches for the analysis and modeling of complex signals. Fractal approach for studying complex systems have proven useful in mathematics . . .

Web 19. Iterated Function Systems, Technical Description, based on the extended mathematical development given in Barnsley's 1988 . . . More math!

Web 20. Fractal Image Compression. Michael Barnsley and Lyman Hurd ISBN 1-56881-000-8. AK Peters - 1993 Hardcover 254pp $54.95. How to encode 7 hours of sound, 100 animations, 800 color maps and more than 7000 photographs in less than 600 megabytes of data on a single computer compact disk. Encarta.

Web 22. Fractals link between the sculpture Temple of Fire of John Robinson and the world of fractals.

Web 1220. Center for Polymer Studies (CPS), a scientific visualization research center in the Physics Department and Science and Mathematics Education Center at Boston University, devoted to interdisciplinary research in polymer, random, and fractal systems. Natural images, molecular dynamics. Excellent!

Web 1224. Recursion: A Paradigm For Future Music? Potential uses of recursion, iteration and complex mathematics as an extension of traditional music-compositional practice. Includes an elementary introduction to fractal-related tools proposed to be useful for extrapolating new sequences of musical events.

Web 1226. Chaos and Music Ames, C. "Stylistic Automata in Gradient". Computer Music Journal Vol. 7, No. 4, 1988 Bibliography and extensive links.

Web 1260. Fractal Music Project. Fractals provide extremely interesting musical results, and the field is becoming one of the the most exciting fields of new music research. Fractal music enthusiasts are encouraged to publish their works here. Has Links ( ) to physics, mathematics, algorithms and artificial intelligence.

Web 1270. Fractal music. CLASSIFYING music using fractals could make it much easier for Internet search engines to pick out the kind of music people are looking for, say researchers in France. Scientists have long looked for self-similar patterns, or fractals, in music but without much success. Alain Iost and Maxime Bigerelle at the University of Science and Technology in Lille, France, made their breakthrough by looking at the amplitude of notes--their loudness--rather than their pitch. They found that the fractal patterns in the way loudness varies could be used to distinguish between different types of music, such as heavy metal and jazz. From New Scientist magazine, 05 August 2000.

Web 27. Complex Systems: The last few years have seen extraordinary growth of interest in complex systems. From ecology to economics . . . Excellent images, especially by Junpei Sekino after clicking on "various pretty fractal pictures". Has great math intro.!

Web 28. Brighten Godfrey's Web Site! Fractals, Funnies, and more: . Don't miss the "Humor" section!

Web 29. NeuroFractal Home Internet Centre of Excellence: srl is an all azimuth WWW consulting company based in Milan, Italy. Imputation of IFSs via neural nets -- an interesting concept! How a neural network, learning from a set of points plotted by the Chaos algorithm, is able to generate fractal shapes (a fern). In the second study, a neural network tries to reconstruct the IFS functions starting from the lighted pixels in a raster image. This is a first, rough tentative to reach the fractal image compression through neural networks. Excellent ideas!

Web 30. Fractal Image Compression Bibliography:

Web 506. The Fractal Art of Uwe Krueger and Heinz Wolf Two pioneers of image rendering. All of their rooms (1 to 4, at least) are must-see! . A must-see site! More images from Mandelbrot country, all from z = z**2 + c! Linkages obsolete. Looking for new ones.

Web 34. Fractal Art Umbrellas Order Form. Shaw Creations. How to get noticed on a rainy day! Not warranted for use atop bowler hats in London!

Web 35. Fractal Art Jigsaw Puzzles: Be puzzled no longer!

Web 38. Fractal - Art Gallery - David Neal Dubois: See "The story behind the image" for a simple explanation of how to generate Mandelbrot sets.

Web 41. Spanky Welcome THE SPANKY FRACTAL DATABASE. The Elastic Fractal. A major database!

Web 42. THE FRACTAL DATABASE (a large portion is mirrored from spanky). The capillary Fractal. Fractals and scale. A tutorial on . . . A major database!!!

Web 46. Fractals & the Set. Explanation of Fractals and the Mandelbrot Set, using material from Chan 4 program with Arthur C. Clarke. This page is intended to be a basic introduction to Fractal Geometry, and more specifically the Mandelbrot Set, aided by material from a television program presented by Arthur C. Clarke, Colours of Infinity. Well worth a visit!

Web 47. Gumbycat's Cyberhome - For Scott:

Web 48. A big image from the land of Mandelbrot: . It takes a while to download, but it's worth it! Talk about sunflowers!

Web 49.Fractint Fractal Gallery: Well worth a visit!

Web 50. Weed's Fractal Gallery: See the flaming brot (Weed). Well worth a visit!

Web 51. Fractint WWW pages. Fractint is a freeware fractal generator created for IBM PCs and compatible computers. It is the most versatile and extensive fractal program available for any price.

Web 52. Iterated Systems: The corporate home of Michael Barnsley, inventor of iterated function systems, co-founder of Iterated Systems, and author of Fractals Everywhere. A must-see site!

Web 55. Jay Hill's Home Page:

Web 57. Fractal Web Sites: Extensive, excellent links.

Web 58. Brain Games Web Guide: See the works of Ken (not Forest) Musgrave, a pioneer of fractal landscape art and fractal planets! Extensive, excellent links.

68. Graphics Archive -- Clouds 3 by Danek Duvall is one of four created using spectral . . . How to draw a cloud!

Web 77. Member of Faculty Dr. Richard Voss. Professor of Complex Systems and Brain Sciences. Professor of Physics and Mathematics:

Web 620. Cut The Knot! An interactive column using Java applets by Alex Bogomolny. The Collage Theorem. Has detailed math. Alex Bogomolny has started and still maintains a popular Web site Interactive Mathematics Miscellany and Puzzles to which he brought more than 10 years of college instruction and, at least as much, programming experience. He holds M.S. degree in Mathematics from the Moscow State University and Ph.D. in Applied Mathematics from the Hebrew University of Jerusalem. Broad educational resources. Lets you produce and plot your own ifss! Excellent site!

Web 658. Sites - Implicate Beauty: A Computational Art site: visitors are invited to draw curves . . . Includes conversion of Mandelbrot set into music!

Web 1070. Benoit Mandelbrot CONFERENCE ON A NEW SPACE FOR CULTURE AND SOCIETY. (NEW IDEAS IN SCIENCE AND ART). About Mandelbrot and lecture by him. Must see!

Web 81. Huge Harry, Institute of Artificial Art, Amsterdam, advocates complete automatization of art generation. He works with Arthur Elsenaar on computer-controlled facial expression and Faces and music.

96. algo Algorithmic composing. 10-Jan-97. Some links to unusual and/or useful tools: •Algorithmic Art •Fractal Music Project •Koan Pro •Seq303 •Aleatoric composer •Anklung

Web 500. IFD: Internet Fractal Database.The database contains 112 galleries and a list of resources to find more galleries. Excellent site!

Web 117. Evolutionary Computation and its application to art and design: behavior control programs for artificial agents.

Web 61. Carlson's Fractal Gallery:

Web 66. FRACTAL LANDSCAPE 2 Emergency Exit / Sortie de Secours. Page maintained by FLEETOX, © DMHK. Created: Tue 14 May 96 Ud: 11-06-96. Getting closer to the type of landscape that I would like to build -- in this case, mountains!!! Excellent! See more landscapes at: , , etc. or use navigator at site.

Web 69. Fractal Landscape Gallery Uses KPT Bryce software! See Jesse's Cool Seascape for a rendition of a problem that I have been considering: how to create waves and reflections on them. Excellent!

Web 72. Chaos and Fractals : New Frontiers of Science, by Heinz-Otto Peitgen, Hartmut Jurgens, Dietmar Saupe, H. O. Peitgen, E. Maletsky, L. Yunker: The big book. A treasure!!!

Web 73. Fractal Imaging: ISBN: 0124580106 Academic Press Publ Mar 97. Pages: 441 Price: £ 34.95 Author: Ning Lu.. 

KEY FEATURES * Focuses on fractal imaging using affine transformations * Applies various applications to real-world photographic imaging * Presents a new image model, 'image as a distribution', for the first time * Examines information on all current compression technologies * Includes a CD-ROM containing fractal images and a freeware version of Iterated Systems' fractal imaging utility * Describes why the fractal method gives a good model with respect to the human visual system * Foreword by Michael Barnsley * Projects key developments in the future of fractal optical image capturing devices

CONTENTS: Introduction. Mathematical Foundation. Fractal Imagery Model. Image Spatial Mask. Spatial Parameters. Brightness and Contrast. Clustering Searching. Decompression Techniques. Image in Motion. Color Image. Entropy Copy. Imagery Applications. Further Enhancement. Fractal Image File Format. Future Fractal Imagery.

Web 573. Laboratory of Eidomatics. Visitate il Museo Nazionale della Scienza e della Tecnica di Milano!
Reaearch topics:
Photorealism. Accurate photometric modelling, optimisation and parallelisation of ray-tracing and radiosity algorithms; Volume Rendering. EVA-VisualStudio: an interactive system for Volume Visualisation, Analysis and Classification; Visualization of Digital Terrain Models; Colour Visual perception Computer simulation of Edwin Land's Retinex Model; Fractal Image Compression. Yet another implementation of IFS image compression; Object Oriented geometric modelling and construction. GEObject: an object oriented system for interactive geometric modelling and geometric construction, that mimics plane constructive euclidean geometry.

End of part 1 of 3 of section 10. Web Sites. _______ Continued at part 2 of section 10. Web Sites, at Page A4.

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