41. Cool Math - Fractals - Fractal Art, Lessons And Generators To view the fractal in all its glory, just click on it! To learn about what a fractal is, check out my lesson The Math of fractals http://www.coolmath.com/fractals/gallery.htm

Your browser does not support the IFRAME tag. Your browser does not support the IFRAME tag. Coolmath's Fractal Gallery Very cool fractals made from math! Check out the new stuff at the bottom of the page! Sorry, your browser doesn't support Java. To view the fractal in all its glory, just click on it! Please do not link to any individual image - the URLs will change. To save an image, right click on the image - PC's only... Mac's, you're on your own! Students are the only ones who can use these images and it must be for a school project, but not if it's a website. fractals 1 fractals 2 fractals 3 fractals 4 fractals 5 fractals 6 fractals 7 fractals 8 fractals 9 fractals 10 fractals 11 fractals 12 fractals 13 fractals 14 fractals 15 fractals 16 fractals 17 fractals 18 fractals 19 fractals 20 fractals 21 fractals 22 fractals 23 fractals 24 fractals 25 fractals 26 fractals 27 fractals 28 fractals 29 fractals 30 To learn about what a fractal is, check out my lesson: The Math of Fractals Create your own fractals: Coolmath's Fractal Generators: Fractal Generator 1 Fractal Generator 2 Fractal Generator 3 Fractal Generator 4 ... Fractal Generator 5 Your browser does not support the IFRAME tag.

42. Fractals For Everyone -- Into The Mystic -- "Where The Endless Dimension Begins" Go beyond your imagination. Mystic Fractal features powerful yet easy to use 3D fractal generators with all the options you need to create intriguing http://www.mysticfractal.com/

Fractals For Everyone Into the Mystic " Where the Endless Dimension Begins " Site Updated: March 26, 2008 Go beyond your imagination Welcome to the dynamic worlds of Mystic Fractal! Inside you will find the latest fractal generators (for Windows and OS X), and galleries filled with timeless fractal landscapes. If you are looking for something outside the common and mundane, step into my domain... What do QuaSZ and other Mystic Fracta l programs offer you? The simplicity of random image creation You can create unique fractals without a long learning curve Hundreds of built-in formulas : you never need to know anything about the complex math to use it 3-D images with specular (Phong) highlights and Perlin noise textures for added realism Advanced 3-D export capabilities [obj, pov, dfx] to photo-realistic programs like Bryce, POV-Ray and Vue d'Esprit AVI (Win) / QuickTime (Mac) options to animate every image: when you finish an image the fun is just beginning! Email support , free bug fixes and minor updates for the life of the programs: we are always happy to see what you do with our programs, or point you in the right direction if you get lost Proceed at your own pace : when you are ready for the nuts and bolts there are plenty to tinker with Random batching create up to 1000 random images unattended Each of our fractal programs is a different world to discover, but all are driven by the same simplified interface. In the short time it takes to learn one program, you are ready to start on another. Then the programs begin to act like a suite of programs, since many options such as palettes, formulas and textures are exportable from one program to another.

43. Arcane Fractals Home Page Take a journey into the fascinating and beautiful world of fractals. http://www.arcanefractals.com/

Home Galleries The Final Frontier Fantasy Mythology The Spirit Apophysis Tutorial Creating The Forms Colouring the Flames Saving and Rendering Some FAQ's Apophysis Resources Tutorials Scripts Plugins Gradients ... Links Arcane Fractals is currently undergoing an upgrade and as a result, there may be some broken links or glitches. Please bear with me and things should be back to normal within the next couple of days. Fran, 16th January 2008 LATEST IMAGES OF MY IMAGINATION Fractals are graphic representations of mathematical equations. Without the aid of a computer to generate the graphics from the equations, these mathematical objects would remain invisible. Traditionally flame fractals have always been particularly beautiful because of their ethereal quality. But in recent times, with the advent of many new marvellous scripts, new previously unimagined forms are now available. Many of my images are actually collages of individual flame fractals. Most of these flames are made in Apophysis, though a few of the earliest ones were made in the Gimp using it's flame plugin. Therefore, all the elements that you see, for example, the trees, plants, stars, etc. are flame fractals. These are opened in my graphics programme and then arranged to make the final picture. On a rare occassion I may sharpen a flame if it is warranted, but I don't add any extra effects as I like their purity.

44. Fractals A fractal is a rough or fragmented geometric shape that can be subdivided in parts, each of which is (at least approximately) a reducedsize copy of the http://oldhome.adelphi.edu/~StemkosL/mathematrix/fractal.html

Fractals 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. This property is call self-similarity. Basically, if you zoom in on one part, its going to look like the bigger picture. The leaf of a fern is a good example. Fractals also describe many other real-world objects, such as clouds, mountains, turbulence, and coastlines, that do not correspond to simple geometric shapes. Here is a fractal called the Koch snowflake. To make it, you start with an equilateral triangle. Replace each of the lines: with something that looks like this: Next, replace each of the lines: with something that looks like this: And so on. Here is what happens to each line: Here are what the the first few stages of the Koch Snowflake look like: Voila. You have a fractal. Lots of other fractals can be built up in this manner, like the Sierpinski Triangle: And here's an interesting, somewhat mathematical side note. Imagine you drew a circle the initial triangle used in the Koch snowflake. At no point will the Koch snowflake extend beyond the boundaries of the circle. Thus the snowflake has a finite area (since its area is less than that of the circle, which is finite). However, at every stage in building the snowflake, the perimeter is multiplied by 4/3 - it is always increasing. So the ideal snowflake (ideal meaning you go through an infinite number of stages constructing the figure) has an infinite perimeter (initial perimeter * 4/3 * 4/3 * 4/3 * ...) yet a finite area. Its not a paradox; its simply a fractal.

45. Fractals & The Fractal Dimension It could be a fraction, as it is in fractal geometry. This generalized treatment of dimension is named after the German mathematician, Felix Hausdorff. http://www.vanderbilt.edu/AnS/psychology/cogsci/chaos/workshop/Fractals.html

Fractals and the Fractal Dimension Mandelbrot and Nature "Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line."(Mandelbrot, 1983). The Concept of Dimension So far we have used "dimension" in two senses: The three dimensions of Euclidean space (D=1,2,3) The number of variables in a dynamic system Fractals, which are irregular geometric objects, require a third meaning: The Hausdorff Dimension If we take an object residing in Euclidean dimension D and reduce its linear size by 1/r in each spatial direction, its measure (length, area, or volume) would increase to N=r D times the original. This is pictured in the next figure. We consider N=r D , take the log of both sides, and get log(N) = D log(r). If we solve for D. D = log(N)/log(r) The point: examined this way, D need not be an integer, as it is in Euclidean geometry. It could be a fraction, as it is in fractal geometry. This generalized treatment of dimension is named after the German mathematician, Felix Hausdorff. It has proved useful for describing natural objects and for evaluating trajectories of dynamic systems. The length of a coastline Mandelbrot began his treatise on fractal geometry by considering the question: "How long is the coast of Britain?" The coastline is irregular, so a measure with a straight ruler, as in the next figure, provides an estimate. The estimated length, L, equals the length of the ruler, s, multiplied by the N, the number of such rulers needed to cover the measured object. In the next figure we measure a part of the coastline twice, the ruler on the

46. Dynamical Systems And Fractals Lecture Notes Lecture Notes by David J. Wright in HTML. http://www.math.okstate.edu/mathdept/dynamics/lecnotes/lecnotes.html

Next: Contents Up: Dynamical Systems and Fractals Dynamical Systems and Fractals Lecture Notes David J. Wright Contents Introduction to Dynamical Systems Symbolic dynamical systems Discrete-time systems ... About this document ... David J. Wright Mon Aug 19 17:21:15 CDT 1996

47. Math.com Wonders Of Math Learn about fractals and create your own beautiful fractal images by Make your own fractals, take the fractal challenge and see fractal movies. http://www.math.com/students/wonders/fractals.html

Home Teacher Parents Glossary ... Email this page to a friend More Wonders Fractals Spirograph Conway's Game of Life Roman Numeral Calculator ... Lissajous Lab Resources Cool Tools References Test Preparation Study Tips ... Wonders of Math Search Enter your search terms Web Math.com Submit search form Play Sudoku Fall Into Fractals The word FRACTAL was invented by Benoit Mandelbrot Fractals are interesting because as you zoom in closer, the pattern is just as beautiful and complex as when you start. Learn about fractals and create your own beautiful fractal images by following the links below. Interactive Fractal Sites Mandelbrot Set Zoom into a fractal in your browser window. Mandelbrot Explorer Make and post your own images. The Fractory A site built by students for the Thinkquest contest. Build your own fractals and learn about the math behind the images. Mandelbrot and Julia Set Explorer Zoom into fractals. Fractal Galleries Fractalus The fractal from an artist's point of view.

48. MathsNet Fractals In order to understand fractals you need to understand a little bit of mathematics concerning complex numbers and the process of iteration. http://www.mathsnet.net/fractals.html

home geometry curriculum puzzles ... MathsNet.com "Mathematics possesses not only truth, but supreme beauty - a beauty cold and austere, like that of sculpture." Betrand Russell Fractals What are fractals? The properties of fractals What else are fractals? Logo fractals ... 4D Magic Cube A brief summary of the mathematics of fractals. What are fractals? In order to understand fractals you need to understand a little bit of mathematics concerning complex numbers and the process of iteration. First you need to know about the special number known as i . You can do arithmetic with i . A complex numbe r is a number made up of two parts added together. These parts are called the "real" part and the "imaginary" part . Examples of complex numbers are: 4 - 6i, 2.1 + 4.7i, 1-i, and so on... In fact any number of the form a + bi, where a and b are real numbers, and i is as above, is a complex number. Addition and multiplication of these numbers is achieved using the conventional rules of algebra. Complex numbers have a size. It is usually called their

49. LizardTech - Press Room - Press Release www.lizardtech.com/solutions/gf/ Fractal Panorama - Fractal Home PageLes St Clair s fractal images, Paint Shop Pro tubes and fractal Java applets. http://www.lizardtech.com/solutions/gf/

Home Products Industries Purchase Download Community Support GeoExpress Express Server Spatial Express Express Suite ... Retired Products Press Releases Company News Product News Partner News Archive ... Press Gallery Case Studies GeoExpress Document Express Express Server Press Room Press Release Extensis and LizardTech Sell Plug-ins to Third Party Software Developer onOne Software purchases popular line of productivity enhancement products for Photoshop and QuarkXPress. PORTLAND, Ore. - July 28, 2005 - Celartem, Inc. and its company brands - Extensis and LizardTech - today announced the sale of its popular Photoshop Plug-ins and QuarkXPress XTensions to independent software developer, onOne Software. Products included in this purchase are pxl SmartScale, PhotoFrame, Intellihance Pro, Mask Pro, PhotoTools, and QX-Tools Pro from Extensis and the Genuine Fractals product from LizardTech. Extensis and LizardTech will continue to support these products until July 31, 2005 when onOne Software will take control of all aforementioned titles. Celartem, Inc. is selling these products to refine its existing product lines focusing on digital content management and distribution - GeoExpress, Document Express, Portfolio, Font Reserve and Suitcase.

50. Blatte's Fractals Welcome to Blatte s fractals. I generally work in three different programs; KPT Fraxplorer, Ultra Fractal and Apophysis, and I previously did some work in http://exoteric.roach.org/frax/index.html

Welcome to Blatte's Fractals. I generally work in three different programs; KPT Fraxplorer, Ultra Fractal and Apophysis, and I previously did some work in KPT Fraxflame. All images are high-quality jpgs at approximately 640x480 pixels. If the background of this webpage appears black to you, you are likely seeing these images darker than I intend. If you would like to use these images on your personal webpage, you must read my Dimensionality Precious Things Benediction The Outsiders Apophysis Gallery 24 August 24, 2007 16 Images Ultra Fractal Gallery 15 July 15, 2007 16 Images Apophysis Gallery 23 July 2, 2007 16 Images Apophysis Gallery 22 June 11, 2007 16 Images Apophysis Gallery 21 May 21, 2007 16 Images Apophysis Gallery 20 January 27, 2007 16 Images Apophysis Gallery 19 January 8, 2007 16 Images Ultra Fractal Gallery 14 December 12, 2006 16 Images Fraxplorer Gallery 14 October 29, 2006 16 Images Fraxplorer Gallery 13 October 20, 2006 16 Images Ultra Fractal Gallery 13 August 18, 2006 16 Images Ultra Fractal Gallery 12 June 5, 2006 16 Images Apophysis Gallery 18 April 28, 2006

51. Linda S Fractals Galleries of Ultra Fractal, Frax Flames, and collage. http://www.lindabucklin.com/fractals/

NEW! Gallery One Gallery Four Frax Flame Gallery Click on a picture to enter a gallery Three NEW Galleries added 6/24/04 NEW! GalleryTwo Gallery Five Frax Flame 2 NEW! Gallery Three Apophysis One Fractal Collage Visit ‘Linda’s Universe’ - My Main Page, with links to all of my sites, including ‘Cats! Cats! Cats! and ‘Linda’s World’. Visit Linda’s World to see my paintings, drawings and digital images. There are also two galleries that exhibit mental illness - related art. My photos, paintings, drawings and digital images of cats. Visit my Renderosity Gallery to see a wide selection of my artwork. Lots of links to some gorgeous fractal galleries! Contact Linda Bucklin Linda's Fractals Gallery One Frax Flame1 ... Frax Flame 2

52. Ivars Peterson's MathTrek Pascal s fractals. Fascinating patterns can arise out of arrays of numbers defined by . It s a neat way to explore the fractal side of Pascal s triangle. http://www.maa.org/mathland/mathland_2_10.html

Ivars Peterson's MathTrek February 10, 1997 Pascal's Fractals Fascinating patterns can arise out of arrays of numbers defined by simple rules. For example, start with the number 1, and make it the apex of what will become a triangle of numbers. In the second row, write two 1s. For each subsequent line, add together adjacent numbers of the previous row and write the sums in the new row, then place 1s at both ends of the line. Here's what one gets for the first eight rows: This set of numbers is now widely known as Pascal's triangle, named for French philosopher and mathematician Blaise Pascal (1623-1662), who studied it intensively. Pascal, however, was not the first to identify the pattern. The mathematician, astronomer, and poet Omar Khayyam (1048-1122) described this number triangle in his writings. It was also well-known to the Chinese. A nine-row version was featured prominently in the introduction to the book "Precious Mirror of the Four Elements," which appeared in 1303. The book's author refers to the triangle as a "diagram of the old method for finding eighth and lower powers." Indeed, the triangle represents a simple way to determine, for example, that (

53. Index Of Www.fractals.net/ The fractals directory ought to be self explanatory. See office for a virtual tour of our company Empire Industries. And finally, the public directory http://www.fractals.net/

Welcome to www.fractals.net! Be sure to check out the fractals. The home directory has a page I like to use as a web browser's home page as it contains many useful and cute links. The fractals directory ought to be self explanatory. See office for a virtual tour of our company Empire Industries. And finally, the public directory contains some commonly used files. It is updated frequently, but you should get current versions from the original sites when possible. This is simply a convenience intended for use by our technical support personnel. Name Last modified Size Description ... home/ 01-Apr-2006 00:00 - fractals/ 01-Apr-2006 00:00 - 01-Apr-2006 00:00 - public/ 01-Apr-2006 00:00 - pics/ 05-Jun-2006 01:11 - reunion/ 05-Jun-2006 01:11 - ware/ 05-Jun-2006 01:11 -

54. Hidden Dimension Galleries Ron Barnett s fractal images, music, raytracing, Ultra Fractal formulas and links. http://www.hiddendimension.com/

Welcome to the wonderful world of algorithmic art. In your tour through this site you will be able to see algorithmic art created by a variety of programs and hear music created from mathematical algorithms. Come explore and learn more about fractals! Visit the art galleries and decide for yourself whether fractals can be art. Mouse over an image to see where it leads. Click on the image to visit the site. Email me Mathematics of Fractals: Visit this page for a description of the mathematics of various types of fractals - including Divergent, Convergent, Strange Attractors, IFS, Juliabrots, Quaternions, L-Systems, Kleinian Groups, Circle and Sphere Inversions and Hyperbolic Tessellations. Art Galleries: The main page for entrance to the various art galleries and animations. The memorial gallery for Red Williams is also here. Tutorials: The main page for UltraFractal tutorials for selected formulas. These are step-by-step how to instructions and a different from the help files available in the Downloads section below. Downloads: Some of my UltraFractal formulas are too large for inclusion the UltraFractal database and are available here. Windows style formula help files which can be used within UltraFractal are also available, along with a gradient editor designed for Fractint but which also works with UltraFractal.

55. Dauger Research, Inc. - Power Fractal Fractal calculations utilizing highperformance and cluster computing and vector hardware. http://daugerresearch.com/fractals/

56. Fantastic Fractals But it turns out that fractals are an excellent tool for math teachers. At least that s the opinion of fractal fan Cynthia Lanius, a math teacher at Milby http://whyfiles.org/046sci_project/fractal.html

Fractals aren't just cool they're comely. Fun with fractals If you think mathematics ricochets between the ancient and the mystical, never halting at the useful or comprehensible, if you think half of math was perfected by Euclid and Newton, and the other half can't be comprehended without a Ph.D., meet fractals. These geometric shapes are formed by repetition: Take a shape, do something to it, then do it again for as long as you want. Fractals generally start simple enough, but they can get hideously complicated. But it turns out that fractals are an excellent tool for math teachers. At least that's the opinion of fractal fan Cynthia Lanius , a math teacher at Milby High School, Houston, Texas. For one thing, she says, fractals make the shapes seen in nature. "You don't see a lot of triangles, squares or rectangles in the woods," but fractals show up in everything from cumulus clouds to fern fronds. For another, they are a field of modern math research that students can grasp. "Kids think that all mathematics was learned 500 years ago, that math is a dead study like Latin is a 'dead' language. I want them to understand that there are people living, breathing, and making a living as math researchers." Ten years ago, Lanius became captivated by fractals. When she consulted the web, she found the sites were either way too technical for her high-school students, or they were just pretty pictures with no explanations. So in June, 1996, Lanius created her own set of

57. Fractals fractals are one of the most interesting puzzles of mathematics. They are made by a simple formula, yet they have such beautiful and complex designs. http://www.bugman123.com/Fractals/Fractals.html

Contents Home Physics Simulations Engineering ... Links Fractals are one of the most interesting puzzles of mathematics. They are made by a simple formula, yet they have such beautiful and complex designs. This page provides simple Mathematica code for many of the most interesting types of fractals. Some of this code may not be very efficient, but it should be enough to give you a basic understanding of the mathematics involved. Mandelbrot Set Zoom - original version: Java , 5/24/01; animated version: C++, 2/16/05 z n+1 = z n +z c This animation zooms in on the Mandelbrot set by a factor of 10 code for double-doubles . See also my Java program and C program . Here is some Mathematica code for this fractal: (* runtime: 1 minute *) < 2, z = z^2 + zc; i++]; i]; Here is a faster version: (* runtime: 7 seconds *) POV-Ray has a built-in function for this fractal: // runtime: 0.5 second Links Deep-zooming Fractint can zoom in by magnification factors of up to 10 1/50 of a Light Year - zooming animation by Damien Jones Kleinian Double Cusp Group - calculated in Mathematica 4.2

58. Makin' Magic Fractals Freeware Fractal 3d Program Programs, fractals, utilities, freeware, shareware, source code, algorithms, MMFrac from Makin Magic. (Dave Makin s homesite) http://skyscraper.fortunecity.com/terabyte/966/

web hosting short URLs photo sharing Makin' Magic Programs, Fractals, Freeware, and Graphics This is the Makin' Magic main index page or, if you prefer, Dave Makin's index page. This site is best viewed at width 1024 or higher, the galleries should now view ok at width 640. Main site (or downloads) last updated 8th September 2002. This page last updated 14th January 2008. If you are interested in purchasing David Makin's fractal art prints or simply would like to view David's most recent artwork please visit our main website at fractalgallery.co.uk David produces fractal art in many styles which include impressionist, science fiction, abstract art and mineral art. Every print we supply is printed on good quality matte paper, we also produce larger prints such as Giclée on canvas or Durst Lambda. As fractals are generated using complex mathematical formulas we are also able to supply custom art prints using numerical data which you supply, this could be an anniversary for instance. David Makin's fractal art prints are available in size A4 through to A0 and licences to publish are available for a fee on request. For more information please visit fractalgallery.co.uk

59. Buddhabrot Fractal Method An alternative method of displaying the Mandelbrot set yields a lifelike image of a seated buddha. http://www.superliminal.com/fractals/bbrot/bbrot.htm

The Buddhabrot Technique by Melinda Green New ! Alex Boswell found an almost magical way to vastly speed the rendering of highly zoomed regions. Click here for an example. New Albert Lobo has produced a simply gorgeous music video exploration of the Buddhabrot in 4D. View it on YouTube or download the 22 MB high-res version The images on this page were all generated using a technique I developed to render the mandelbrot set. It's important to realize that it is not a different fractal from the mandelbrot set, but simply a different way of displaying it. Clicking on some of the images will take you to a normal rendering of the exact same area, but using the traditional mandelbrot technique. Note that even though the images resemble Hindu art, they were actually generated completely automatically, without any sort of human artistic intervention. When I first tried using the new technique, I had no idea what the images might look like and was completely surprised by the results. I was later pleased to learn that a computer artist named Lori Gardi, who I had described this technique to several years ago, has since devoted a great deal of her creative effort to generating various high-resolution images using the technique. She named it Buddhabrot which is a name I instantly loved and have adopted. Lori's web site contains some reduced examples of her work along with her writings into the mystical connections she's made between the Mandelbrot set and Buddhism.

60. African Fractals African fractals book (Rutger s University Press) Interactive software simulate African Reviews of African fractals in newspapers and journals http://www.rpi.edu/~eglash/eglash.dir/afractal/afractal.htm

African Fractals by Ron Eglash African Fractals book (Rutger's University Press) Interactive software: simulate African designs, create your own patterns, and learn about fractal mathematics African fractals: the TED video ... Reviews of African Fractals in newspapers and journals Related web sites International Study Group on Ethnomathematics Back to Ron Eglash home page

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