41. Internet Public Library: Mathematics related topic, applications from almost all areas where fractals are used by advertisers,the media, reporters, politicians, activists, and in general many non http://www.ipl.org/div/subject/browse/sci40.00.00/

dqmcodebase = "/javascript/" Subject Collections Business Computers Education ... Mathematics This collection All of the IPL Advanced The study of the measurement, properties, and relationships of quantities, using numbers and symbols. Resources in this category: You can also view Magazines Associations on the Net under this heading. About.com: Mathematics http://math.about.com/ Ask Dr. Math! http://mathforum.org/dr.math/dr-math.html This is primarily an archive of K-12 student mathematical questions and answers answered by a group of students, instructors and mathematicians. The archive is searchable both by a Table of Contents and by keyword. The site also includes problems suitable for students of varying grade levels. Biographies of Women Mathematicians http://www.agnesscott.edu/lriddle/women/women.htm "These pages are part of an on-going project by students in mathematics classes at Agnes Scott College, in Atlanta, Georgia, to illustrate the numerous achievements of women in the field of mathematics. There are brief comments on many of the women mathematicians and some photos (which look best at more than 256 colors). Our goal is for this list to continue to expand, and for more biographies to be completed." Not all entries are complete, but the ones that are include references. There are also links to excellent resources on the topic of women and mathematics. A Brief History of Algebra and Computing: An Eclectic Oxonian View http://vmoc.museophile.com/algebra/

42. Newton Fractals It is also possible to construct Newton fractals for general polynomialsof degree between three and ten by entering the coefficients. http://members.optusnet.com.au/~peterstone/nwtn.html

NEWTON FRACTALS Fractal Explorer Home Page Escape Fractals Newton Parameter Space Fractals Root-finding and the Zeta Function ... Download Fractal Explorer Fractal Explorer uses a special colouring method for fractals associated with the use of Newton's iterative formula for the calculation of complex roots of polynomials and other functions of a complex variable. The general idea is to associate a particular colour with each root chosen from the ten basic colours: red, green, blue, cyan, yellow, magenta, purple, orange, pink and lime, and then use up to 25 shades of the basic colour to indicate the number of iterations required to arrive close to the root. The icon for Fractal Explorer is based on the Newton fractal corresponding to the 4 roots of the polynomial x^4 - 1, while the following fractal is obtained by using Newton's formula to calculate the 5 roots of x^5 - 1 with different starting points. Fractal Explorer has numerous built-in fractals of the Newton type. It is also possible to construct Newton fractals for general polynomials of degree between three and ten by entering the coefficients. One can start with a built-in fractal, and then make alterations to the polynomial coefficients. Fractal Explorer calculates and displays the roots of the polynomial before starting to draw the fractal. Zoom into Newton fractal for z^10 + 0.2 i * z^5 - 1.

43. ThinkQuest : Library : Fractals Unleashed Tips. general Tips. While diagrams. Below are some of our own suggestionsfor programming fractals, which we came up with while doing it. http://library.thinkquest.org/26242/full/progs/t2.html

Index Math Fractals Fractals Unleashed For centuries mathematicians rejected complex figures, leaving them under a single description: formless. For centuries geometry was unable to describe trees, landscapes, clouds, and coastlines. However, in the late 1970s a revolution of our perception of the world was brought by the work of Benoit Mandelbrot. He introduced and developed the theory of fractals figures that were truly able to describe these shapes. The theory was continued to be used in a variety of applications. Out website will show fractals importance in areas ranging from special TV effects to economy and biology. Besides applications it will contain a course on fractals, making the user familiar with topics of various difficulties. The website will have an enhanced section on generating fractals, and the user will have a chance to create fractals using a Java applet. Visit Site 1999 ThinkQuest Internet Challenge Languages English Students Mikhail Stuyvesant High School, Brooklyn, NY, United States Ming Jack Stuyvesant High School, Fresh Meadows, NY, United States

44. Fractal Web Sites Excellent source of information on fractal art and fractals in general. If you area neophyte in fractal matters, this would be a good place to start learning. http://www.fractaldomains.com/html/sites.html

Fractal Web Sites Home News Download Register ... Fractal Domains Gallery The fractal sites listed on this page will lead you to dazzling images and mind boggling information about chaos and fractals. Even if you didn't find my site too entertaining, do yourself a favor and check out some of the sites listed below! Galleries Fractal Information Other Sites Of Interest Galleries Infinite Fractal Loop Index I already have a link to the Infinite Fractal Loop on my gallery page, but I want to put an extra plug in for the graphical index that was recently added for the Infinite Fractal Loop. The fractal gallery sites that are members of the loop are all of high quality, and this page has a representative thumbnail for practically every site in the loop. Takes a while for this page to load, but it's worth it! This is the single best starting point I know of for exploring fractal galleries on the Web an unparalleled portal to the world of fractal images. Fractalus (Damien M. Jones) Some of these fractals are really spectacular. Various formulas and coloring schemes are used. This is one of the first sites I've seen (besides my own) where anti-aliasing is was regularly used even on the large images. (Anti-aliasing is common at most of the best online galleries now.) Gumbycat's Cyberhome Gumbycat is the nom de plume of Linda Allison. Her talent continues to grow and amaze. She started out using Fractint, getting incredible shading effects from Fractint's limited color palette. Later she switched to Ultrafractal (a big favorite among PC fractal enthusiasts.)

45. CALResCo Major Complex Systems Software Links Osinga Evolutionary Art Andrew Rowbottom Evolutionary Computation - PDP Lab fractals- Paul N well known authors or open source code systems for general use) http://www.calresco.org/sos/software.htm

Major Complex Systems Software Collections Software Projects Collections of Programs (these contain the authors' links and often comments on the available software in their fields of interest) Artificial Intelligence - Generation 5 Artificial Life - PDP Lab Artificial Life - Santa Fe Artificial Life - Zooland Artificial Worlds - Open Directory Cellular Automata - Universidad Autónoma de Puebla - for Mac by Glenn Elert Complex Adaptive Systems for Macintosh Complex Systems - for Macintosh by Ishihama Yoshiaki Complex Systems - PDP Lab Complexity - CTCS Dynamical Systems Theory - Hinke Osinga Evolutionary Art - Andrew Rowbottom Evolutionary Computation - PDP Lab Fractals - Paul N. Lee Fuzzy Systems - PDP Lab Genetic Algorithms - Nova Genetica L-Systems Software - Unix, Mac, MS-Dos Multi-Agent Systems - tools for building MASs Neural Networks - Chris P. Hess Neural Networks - NEuroNet Neural Networks - PDP Lab Neural Network Software for Classification - KDnuggets Simulation - IDSEA Simulation - Michael Altmann Software Projects (these are either multi-application frameworks, well known authors or open source code systems for general use)

46. Historical Notes: History Of Fractals somewhat mixed success, leading to the introduction of multifractals with more parameters,but Mandelbrot’s general idea of the importance of fractals is now http://www.wolframscience.com/reference/notes/934a

SOME HISTORICAL NOTES From: Stephen Wolfram, A New Kind of Science Notes for Chapter 5: Two Dimensions and Beyond Section: Substitution Systems and Fractals Page History of fractals. The idea of using nested 2D shapes in art probably goes back to antiquity; some examples were shown on page 43. In mathematics, nested shapes began to be used at the end of the 1800s, mainly as counterexamples to ideas about continuity that had grown out of work on calculus. The first examples were graphs of functions: the curve on page 920 was discussed by Bernhard Riemann in 1861 and by Karl Weierstrass in 1872. Later came geometrical figures: example (c) on page 191 was introduced by Helge von Koch in 1906, the example on page 187 by Waclaw Sierpinski in 1916, examples (a) and (c) on page 188 by Karl Menger in 1926 and the example on page 190 by Paul Lévy in 1937. Similar figures were also produced independently in the 1960s in the course of early experiments with computer graphics, primarily at MIT. From the point of view of mathematics, however, nested shapes tended to be viewed as rare and pathological examples, of no general significance. But the crucial idea that was developed by Benoit Mandelbrot in the late 1960s and early 1970s was that in fact nested shapes can be identified in a great many natural systems and in several branches of mathematics. Using early raster-based computer display technology, Mandelbrot was able to produce striking pictures of what he called fractals. And following the publication of Mandelbrot’s 1975 book, interest in fractals increased rapidly. Quantitative comparisons of pure power laws implied by the simplest fractals with observations of natural systems have had somewhat mixed success, leading to the introduction of multifractals with more parameters, but Mandelbrot’s general idea of the importance of fractals is now well established in both science and mathematics.

47. Stephen Wolfram: A New Kind Of Science | Online somewhat mixed success, leading to the introduction of multifractals with more parameters,but Mandelbrot s general idea of the importance of fractals is now http://www.wolframscience.com/nksonline/page-934a-text

Cookies Required A New Kind of Science See http://www.wolframscience.com/nksonlineFAQs.html for more information or send email to support@wolframscience.com Search site Get the NKSwire newsletter Sign the Guestbook

48. 4D Fractals the different type of fractals, hcmplx.c implementing hypercomplex fractals, andquatern.c they take a 3dimensionnal point IPoint, a general fractal structure http://skal.planet-d.net/quat/f_gal.ang.html

4D fractals main Context One simple mean of creating fractals, whom these pages are dedicated to, is the use some iteration map. Here's how it work: First, decide in which space your fractal is built. More than its dimension (>=3), it's rather the multiplicative and additive laws between its elements that matters. See below for details. Then, choose a map (say Fc(P) ) acting in this space. c is an element that will be used as a parameter. For instance: Fc(q) = q*q + c Now, 'each' point of that space will be tested to see whether they belong to the set or not. The usual test is: Starting with P(0) , the point to be tested, built the sequence defined by the recurrence map: P(n+1) = Fc( P(n) ) If remains finite as n goes toward infinity, than P(0) was in the set. ( ) is a norm in our space. This simple definition faces several problems when the fractals it breeds are to be drawn with computers: First, the maximal dimension that can easily (and comprehensively) be represented on a screen is commonly 3. So will our fractal set need to be sliced with 3-d hyperplanes (or stranger things, if you prefer) to be handled correctly by such limited devices as our brains. Making n go to 'infinity' is not something a computer likes to do. One has to stop somewhere. Say at the

49. Fractalus Home Fractal Info, Information on fractals in general, and FractInt and Ultra fractalspecifically; how these images are created; tips, techniques, and tutorials. http://www.fractalus.com/home/

fractalus home Over three hundred and ninety fractal images, organized into small galleries, with thumbnails for every one. Information on fractals in general, and FractInt and Ultra Fractal specifically; how these images are created; tips, techniques, and tutorials. Information on purchasing high-quality prints of gallery images or a CD filled with high-resolution pictures. Fractal generators, formula and parameter files, screen savers, Windows themes, and mailing list archives. Mindless drivel about the artist (me). Mainly intended for family and friends. No really, you're not interested. I insist.

50. Sprott's Fractal Gallery The following fractals are generalized Julia sets of 2D quadratic maps with techniqueis described in a paper Automatic Generation of general Quadratic Map http://sprott.physics.wisc.edu/fractals.htm

Sprott's Fractal Gallery Awards Received MIDI Fractal background music courtesy of Forrest Fang Fractal of the Day Every day at a few minutes past midnight (local Wisconsin time), a new fractal is automatically generated by a variation of the program included with the book Strange Attractors: Creating Patterns in Chaos by Julien C. Sprott . The figure above is today's fractal displayed in low (320 x 200) resolution. Click on it or on any of the cases below to see them at higher (640 x 480) resolution with a code that identifies them according to a scheme described in the book. Older Fractals of the Day are saved in an archive , which you can access. If your browser supports Java, you might enjoy the applet that creates a new fractal image every five seconds or so. If you would like to place the Fractal of the Day on your Web page, you may do so provided you mention that it is from Sprott's Fractal Gallery and you provide a link back to this page. fracday0.gif Today's fractal of the day fracday1.gif

51. Cliff Pickover, Factals And The Pursuit Of Beauty Pickover s love of fractals and the general field of computerized visualizationhas made him one of the more prolific scientists at IBM. http://sprott.physics.wisc.edu/pickover/cliffpick.html

Contact News! Books Home Cliff Pickover Fractals and the pursuit of beauty Interview previously at IBM Research web site... For IBM Researcher Cliff Pickover, the passion that drives his work is the search for that delicate balance between chaos and order. At the center of his passion lie fractals, computer-generated patterns that can represent mathematical concepts. Fractals are often objects of beauty, not to mention useful tools for areas as diverse as medicine, computer science and education. When he's not busy developing code for IBM's IntelliStation (IBM's new high-end personal computer), or writing and explaining mathematical concepts to the public, Pickover is exploring ways in which fractals and computerized visualization can be stretched for aesthetic as well as practical reasons. Entering the fractal dimension "One of the most important things that fractals contribute is the concept of fractal dimension," Pickover explains. "By using a single number, scientists can characterize the behavior of systems, and the irregularity of shapes. For example, fractals can be used to characterize coastlines or the structure of blood vessels and cells. They give us insights into chaos theory, which looks at the effect of irregularity on very large systems, such as weather. Computers graphics can be used to produce visual representations with a myriad of perspectives, many of which are beautiful to the eye." The work of Pickover and his colleagues at IBM is branching into many areas to explore new ways for using computerized visualization to represent data, illuminate patterns and simulate natural forms. Pickover has worked on topics ranging from the graphical representation of genetic sequences and sounds to the simulation and rendering of huge lifelike caverns (which he calls "virtual caverns" because they can be explored using computer tools).

52. IFS Fractals Sorry, your browser doesn t support Java. http://www.ncku.edu.tw/~general/chinese/

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53. Fractals In general, fractals arising in a chaotic dynamical system have a far more complexscaling relation, usually involving a range of scales that can depend on http://www.drchaos.net/drchaos/Book/node9.html

Next: References and Notes Up: Some Terminology: MapsFlows, Previous: Binary Arithmetic Fractals Nature abounds with intricate fragmented shapes and structures, including coastlines, clouds, lightning bolts, and snowflakes. In 1975 Benoit Mandelbrot coined the term fractal to describe such irregular shapes. The essential feature of a fractal is the existence of a similar structure at all length scales. That is, a fractal object has the property that a small part resembles a larger part, which in turn resembles the whole object. Technically, this property is called self-similarity and is theoretically described in terms of a scaling relation. Chaotic dynamical systems almost inevitably give rise to fractals. And fractal analysis is often useful in describing the geometric structure of a chaotic dynamical system. In particular, fractal objects can be assigned one or more fractal dimensions, which are often fractional ; that is, they are not integer dimensions To see how this works, consider a Cantor set , which is defined recursively as follows (Fig.

54. Math 90Q General Information . This isa sophomore seminar. The topics we will cover include classical fractals, self......Math 90Q general Information Fall 2003. Back to home. Course http://math.stanford.edu/~dlevy/90/info.html

Math 90Q General Information Fall 2003 Back to home Course Description This is a sophomore seminar. The topics we will cover include: classical fractals, self similarity, fractal dimensions, discrete maps, the Mandelbrot set, the Julia set, chaotic dynamics and fractals, fractal interpolation. Homepage http://math.stanford.edu/~dlevy/90 You are expected to visit our homepage on a regular basis and check for updates, announcements, etc. Instructor Name Office E-mail Phone Office Hours Doron Levy dlevy@math.stanford.edu MW 2:15-3:30 and by appointment Course Time and Location Day Time Room Wednesday 5:15-7:15 pm Building 320 - Room 221 You can find a detailed schedule here Text The required textbooks for the course are available at the bookstore: Chaos, Fractals, and Dynamics , by Robert Devaney, Addison-Wesley. Fractals Everywhere , by Michael Barnsley, second edition, Morgan Kaufmann. Additional bibliography: Introduction to Fractals and Chaos , by Richard Crownover, Jones and Bartlett. Chaos and Fractals , by Peitgen, Jurgens, and Saupe, Springer-Verlag. Nonlinear Dynamics and Chaos , by Steven Strogatz, Perseus.

55. Fractals For fractals we have in general. where is the fractal dimension of theobject. From B.1 we obtain the definition of the fractal dimension http://www.fys.uio.no/~eaker/thesis/node67.html

Next: Scaling behavior and Power Up: No Title Previous: The Conjugate Gradient Method Fractals Figure B.1: Three generations of the Cantor set. The generations are denoted by n . The Cantor set is constructed by taking a straight line of unit length (0th generation). Remove from its middle a section of length . In the figure . What remains are two pieces each of length (1th generation) to which the same procedure is repeated by removing a relative section of length . The process is to be continued indefinitely. In this thesis we have frequently used the concept of fractals when the structures between the invading and the defending fluids have been analyzed. To clarify, I will give a comprehensive introduction to some aspects in fractal geometry used in this thesis. Excellent books such as Fractal written by Feder (1988) [ ] and Fractal Concepts in Surface Growth ] give a complete introduction to these topics and should be read for further details. A fractal is called self-similar if the structure is invariant under isotropic scale transformation which means that the structure is enlarged uniformly in every spatial direction. This is a special case of the more general self-affine fractal which is invariant under anisotropic transformation. That is, when the structure is rescaled with different scaling factors in different spatial directions. One of the simplest self-similar objects is the Cantor set shown in figure B.1

56. Workshop Announcement in allometric scaling laws) can be predicted to stem from general biological and questioncan be phrased in several ways for example, can fractals be used as http://discuss.santafe.edu/biofractals/

Fractals In Biology A SFI workshop site Santa Fe Institute Navigation Home Participants Agenda Abstracts Site Help Members Join Now Login Santa Fe Institute Last update: Monday, April 2, 2001 at 2:14:14 PM by Webmaster Workshop Announcement Fractals In Biology: Developing The Underlying Mechanistic Principles For Self-Similarity Brian J. Enquist, National Center for Ecological Analysis and Synthesis and The Santa Fe Institute David R. Morse, The Open University, UK Recent News Articles on understanding the linkage between fractal-like morphology and organismal structure and function. New York Times: New Scientist: Ruling Passions Workshop Goals We intend to bring together a diverse group of people who see the fractal nature of life as a window by which to synthesize general principles in biology. At this point we envisage the workshop to consist of a mixture of 20min talks, discussion, and breakout sessions. These will be used to provide a survey of fractal geometry in biology and an overview of mechanistic models for the origin of fractal-like structures. Depending upon the interests of the participants, some of these may run as parallel sessions.

57. Efg's Mathematics Page general, Dr. GA Edgar www.math.ohiostate.edu/~edgar. 3D Fractal html. AnimatedGIFs, http//sprott.physics.wisc.edu/fractals/animated. Bifurcation, http://www.efg2.com/Lab/Library/mathematics.htm

Mathematics in look for Mathematics Fractals and Chaos Contents A. Reference D. Functions G. Wavelets E. Fourier Analysis, FFTs ... F. Fractals and Chaos also see efg's Delphi Math Info and Links Delphi Math Functions Math Projects A. Reference Calculators The Calculator Home Page. Calculator.org has a number of calculator related resources, including an online scientific calculator, units conversion and constants database, and other information. www.calculator.org Calculators Online Center. Part II. Mathematics http://www-sci.lib.uci.edu/HSG/RefCalculators2.html Geometry Geometry Algorithms http://geometryalgorithms.com Geometry http://mathworld.wolfram.com/topics/Geometry.html Géométrie http://perso.wanadoo.fr/jpq/geometrie/index.htm Famous Curves http://www-groups.dcs.st-and.ac.uk/~history/Curves/Curves.html 2D Curves www.2dcurves.com Geometry, Solid The 5 Platonic Solids and the 13 Archimedean Solids www.scienceu.com/geometry/facts/solids Math Reference Tables math2.org (formerly Dave's Math Reference Tables) General, Algebra, Geometry, Trig, Discrete/Linear, Other, Stat, Calc, Advanced

58. Fractals, Chaos, Symmetry, Physical Laws And Computer Programs The There is a general way for computer programs to simulate natural phenomena?What Whyfractals seem to represent natural phenomena and shapes better than simple http://www.ba.infn.it/~zito/plaw.html

The essays How nature is described by mathematical laws Why are natural laws recursive? Chaotic attractors ,stability and life processes Julia and Mandelbrot set ... Recipes for chaos Previous essays Frattali e caos (in italian) La fabbrica di Newton (in italian) Dynamical Systems db Dynamical Systems tutorial ... Lettere a corrispondenti Links Fractal FAQ Chaos in Discrete Dynamical Systems Videofeedback Feigenbaum fractal ... Dynamical systems lecture notes Fractals, chaos, symmetry, physical laws and computer programs The Questions How does science describe natural phenomena?What is a mathematical law? Why natural laws are mathematical Why are they recursive What is the relationship between mathematical laws and computer programs? Do we have a general way of representing a natural phenomenon(or at least its more important features)? What is a dynamical system It is possible to describe a continuous physical phenomenon with a discrete dynamical system? What is the Poincare' map? What is an iteration? What is chaos? How many kinds of chaos we have? How a map (and perhaps the physical phenomenon described) can become chaotic? What is an orbit What is a strange attractor What is a basin of attraction?

59. General Pascal's Triangle As A Modulo Sum Rule Don t let it fool you that I use an experimental mathematics approach the generalapplication of this mathematics does not depend on who or what by what http://astronomy.swin.edu.au/~pbourke/fractals/pascalmodulo/

A Second Level Sierpinski from A Pascal's Triangle Sum Modulo Program By Roger Bagula Compiled and graphics by Paul Bourke 20 Aug 1998 Basic source code C source code I have been working on the connection between the number theory Pascal's triangle and the Sierpinski set generalizations I was worked out for along time. Last night in the middle of a very hot and uncomfortable night I had an idea. I programed it up this morning. It works! I don't think anyone else has this. My generalized Pascal's sum triangles are based on the seed triangle: 1 a 1 1 1+a 1+a 1 Where a is a real number and the recurrence formula is: 1) n(i,j) = Mod(n(i,j - 1) + n(i - 1,j). int(a)) for array n(i,j). This procedure works better than the combination forms based on factorials and binomials because it is always a number that can be calculated with out computer floating point (scientific notation) chopping. I made up a pattern of eight of these to fill the screen symmetrically as a pretty test pattern. The program and one results is part of this article. I had the idea last night of starting with the seed: 1 a 1 And making the recurrence go one level deeper: 2) n(i,j) = Mod(n(i,j - 1) + n(i - 1,j) + n(i - 1,j - 1), int(a) + 1)

60. General Information 1. general INFORMATION. Therefore it proves to be an ideal tool to study FRACTALSgenerated by iteration of COMPLEX VARIABLES FUNCTIONS, like MANDELBROT and http://www.ciram.unibo.it/~strumia/Fractals/FractalMatlab/GenInfo.html

1. GENERAL INFORMATION MATLAB is a very powerful package which allows to manipulate simultaneously: VECTORS and MATRICES COMPLEX FUNCTIONS of COMPLEX VARIABLES DENSITY PLOTS and 3D PLOTS thanks to pcolor and mesh instructions. Moreover it allows recursion in calculating functions. Therefore it proves to be an ideal tool to study FRACTALS generated by iteration of COMPLEX VARIABLES FUNCTIONS, like MANDELBROT and JULIA sets. THE HARDWARE ENVIRONMENT. The images we present have been generated by MATLAB starting from SCRIPTS of few rows on a Sun Spark 10 workstation. THE SPEED OF GENERATION Each image takes only FEW MINUTES of machine time to be generated, since MATLAB works simultaneously with the all the elements of the matrix involving the coordinates of the pixels which is created by the function meshgrid. PLOTS AND SCRIPTS Each plot is presented in the form of a clickable image and it is accompanied by the text of the MATLAB SCRIPT by which it has been generated. INDEX Back to "FRACTALS WITH MATLAB" Decorations with MATLAB Mandelbrot sets with MATLAB Julia sets with MATLAB ... Newton's method sets with MATLAB sets with MATLAB 3D fractals with MATLAB FRACTAL GALLERY CIRAM HOME PAGE

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