1. Cynthia Lanius' Lessons: A Fractals Lesson - Introduction Mathematics lessons for elementary school, middle school fractals, fractals, fractals. http://math.rice.edu/~lanius/frac/

Cynthia Lanius Fractals Pictured: A Famous Fractal - The Mandelbrot Set A Fractals Unit for Elementary and Middle School Students That Adults are Free to Enjoy Table of Contents Introduction Why study fractals? What's so hot about fractals, anyway? Making fractals Sierpinski Triangle Using Java Math questions Sierpinski Meets Pascal ... Using Java Fractal Properties Self-similarity Fractional dimension Formation by iteration For Teachers Teachers' Notes Teacher-to-Teacher Send mail Fractals on the Web The Math Forum Other Math Lessons by Cynthia Lanius Awards This Site has received What are Fractals? They're everywhere, those bright, weird, beautiful shapes called fractals. But what are they, really? Fractals are geometric figures, just like rectangles, circles and squares, but fractals have special properties that those figures do not have. There's lots of information on the Web about fractals, but most of it is either just pretty pictures or very high-level mathematics. So this fractals site is for kids, to help them understand what the weird pictures are all about - that it's math - and that it's fun! Teachers: Every lesson has a print version for classroom use.

2. Sprott's Fractal Gallery Daily updated gallery of automatically created fractals. Fractal generating software. 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 . 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 provide a link back to this page. If you want to make your own fractals, I recommend the Chaoscope freeware. fracday0.gif

3. What Are Fractals? Ahh, the beauty of noninterger self-similar dimensions. An explanation of fractals and some cool picts. http://www.jracademy.com/~jtucek/math/fractals.html

What are fractals? Dimensions Uses Pictures Resources What are fractals? Fractals have come up as an important question two times before the invention of computers. The first time was when British map makers discovered the problem with measuring the length of Britain's coast. On a zoomed out map, the coastline was measured to be 5,000 something or other. Sorry, I've forgotten the units. But anyway, by measuring the coast on more zoomed in maps, it got to be longer, like 8,000. And by looking at really detailed maps, the coastline was over double the original. You see, the coastline of Britain that's on a map of the world doesn't have all the bay's and harbors. A map of just Britain has more of these, but not all the little coves and sounds. The closer they looked, the more detailed and longer the coastline got. Little did they know that this is a property of fractals. (A finite area, aka Britain, being bounded by an infinite line) The second instance of pre-computer fractals was noted by the French mathematician Gaston Julia. He wondered what a complex polynomial function would look like, such as the ones named after him (in the form of z^2 + c, where c is a complex constant with real and imaginary numbers). The idea behind the formula is that you take the x and y coordinates of a point, and plug them into z in the form of x + y*i, where i is the square root of negative one, square this number, and then add c, a constant. Then plug the resulting pair of real and imaginary numbers back into z, run the equation again, and keep doing that until the result is greater than some number. The number of times you have to run the equations to get out of its "orbit" can be assigned a color and then the pixel (x,y) gets turned that color, unless those coordinates can't get out of their orbit, in which case they're made black.

4. Coolmath4kids - Fractals ( Geometry ) I ve really been beefing up the fractal area over on the main Coolmath.com site, so I m just going to send you over there to check it out. http://www.coolmath4kids.com/fractals.html

Your browser does not support the IFRAME tag. Coolmath's Fractals Your browser does not support the IFRAME tag. What are fractals? The easiest way to explain them is that they are pictures made with math! Like this... Pretty cool, huh? They were discovered just 20 or so years ago by a man named Benoit Mandelbrot (that's French). The reason that they weren't discovered before this is that you need a computer to make them! I've really been beefing up the fractal area over on the main Coolmath.com site, so I'm just going to send you over there to check it out. At the top of every page, you can click on the "kids" link to get back to this site. We've got a fractal gallery and generators so you can make your own fractals! Just click the link to go: Coolmath Fractals HELP SUPPORT COOLMATH link to us advertise with us about us safe surfing ... continuing teacher education Thanks for visiting Coolmath4kids.com

5. 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. http://library.thinkquest.org/3288/

A ThinkQuest project by David Green Alex Kulesza , and Keith Bergstresser Welcome to The Fractory! 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 below in increasing order of difficulty. We suggest that you start at the first one, and work your way up. You may be surprised at how far you are able to get. You are encouraged to continue as long as you are interested, but each level requires more math and more time than the previous, so you can decide what is reasonable for you. 1.The first way in which anyone can appreciate fractals is simply for their beauty. To do this, we suggest that you go to our

6. Mathematics Archives - Topics In Mathematics - Fractals KEYWORDS Monte Carlo Estimation of Pi, OneDimensional Random Walk, Two-Dimensional Random Walk, Diffusion Limited Aggregation, Self-Avoiding Walk, Fractal http://archives.math.utk.edu/topics/fractals.html

Topics in Mathematics Fractals Chaos, Fractals, and Arcadia ADD. KEYWORDS: The Chaos Game, The Sierpinski Hexagon, Iterated Function Systems Chaos in the Classroom, Boston University ADD. KEYWORDS: The Chaos Game, Self-similarity, Fractal Dimension, Iterated Function Systems Clifford A. Pickover's Homepage ADD. KEYWORDS: Cellular Automata, Carotid-Kundalini Functions, Books, Puzzles, JAVA Geometry Explorer The Dance of Chance ADD. KEYWORDS: Random Processes, Branching Structures, Patterns in Nature, Student Activities, Laboratory Experiments, Visualization, Software, Fractal Dimension, Fractal Coastlines Diffusion-limited Aggregation Exploring Fractals ADD. KEYWORDS: Lesson Plan, Tutorial, Bibliography Fantastic Fractals ADD. KEYWORDS: Tutorials, Gallery, Fractal Music, Bibliography, Software, Newsgroups Fract-ED ADD. KEYWORDS: Iteration, Attractor, Software, Tutorial, Bifurcation, Self-similarity, Strange attractor Fractal Clouds The Fractal Geometry of the Mandelbrot Set, The Periods of the Bulbs

7. Fractals Get 4 large shiny Christmas balls, some coloured wrapping paper, some lights from the Christmas tree, and you have you own fractal laboratory to amuse http://local.wasp.uwa.edu.au/~pbourke/fractals/

Fractals, Chaos New FAQ: Spherical mirror projection for hemispherical domes Digital Fulldome Techniques and Technologies course (Presented at Graphite 2007) Representing and modelling geometry in SecondLife Announcements Round 5 of the POVRay short code competition Index Papers, Articles, Seminars Exhibitions Projection Data Formats Fractals, Chaos Surfaces, Curves Modelling, Rendering Texture, Colour Geometry Fun, puzzles Other Old stuff Representing geometry in SecondLife Some comments on the suitability of SecondLife as a way of presenting fractal geometry POVRay Fractal Raytracing Contest Rendering Wada-type basins of attraction Here's a party trick for over Christmas. Get 4 large shiny Christmas balls, some coloured wrapping paper, some lights from the Christmas tree, and you have you own fractal laboratory to amuse yourself with over the festive season. Fractal Dimension Calculator - FDC Box counting software for the Macintosh and Linux. Ruler or Compass Dimension Time exists so that not everything happens at once. Multifractal spectrum Lacunarity Recurrence plots By Gayla Chandler Natural Fractals in Grand Canyon National Park Fractals: A Symmetry Approach Gasket Photography Self similarity Examples of self similarity in fractals with examples from mathematics and photos of the physical world.

8. Exploring Fractals It is is based on a curriculum, entitled Exploring Fractal Dimension, developed by Mary Ann Connors and Anna Rose Haralampus at an NSF funded Institute http://www.math.umass.edu/~mconnors/fractal/fractal.html

Exploring Fractals by Mary Ann Connors publications and photos Department of Mathematics and Statistics University of Massachusetts Amherst This World Wide Web project commenced in July 1994. It is is based on a curriculum, entitled "Exploring Fractal Dimension," developed by Mary Ann Connors and Anna Rose Haralampus at an NSF funded Institute for High School Mathematics Teachers at Georgetown University July - August, 1991. Its revision entitled "Exploring Fractals: From Cantor Dust to The Fractal Skewed Web" has been edited by Mary Ann Connors 1994, 1995, 1996, 1997, 1998, 1999, 2000, 2001, 2002, 2003, and 2004. Exploring Fractal Dimensions of Strictly Self-Similar Fractals: Cantor Dust to the Fractal Skewed Web What is a fractal? A fractal is a geometric shape which is self-similar and has fractional (fractal) dimension. An introduction So, what are fractals? What does it mean for a shape to be self-similar Strictly Self-similar shapes What is dimension? How do we assign dimension to an object intuitively? What is the dimension of a geometric object? Mathematical Interpretation Dimension of a Fractal Non-integer and integer dimensions Investigations Questions and Answers Generating Fractals Strictly Self-similar Fractals Teacher Information Instructions Sources Sources for Exploring Fractals Some Interesting Links for Further Exploration If you wish to use any of the text or images in Exploring Fractals please contact its author Mary Ann Connors at the following address.

9. Spanky Welcome The Spanky Fractal Database is a collection of programs, images, and other material all relating to the subject of fractals. http://spanky.triumf.ca/

THE SPANKY FRACTAL DATABASE The Elastic Fractal Welcome to the Spanky Fractal Database I also point to a great deal of material that is not at this site. Consequently I have no control of when this information or service is removed or am I notified if the URL becomes invalid. I try to check my indexes periodically to see what is still valid, but it is not an automated process. If you find URL's that I point to have become outdated and constantly fail to respond on your attempts to load them, (over several days) then please inform me of this problem and I will remove them from my lists. Mirrors of Spanky Fractal Database The main site for this fractal database spanky.triumf.ca is now completely mirrored at spanky.fractint.org Many thanks to Damien M. Jones at Fractalus for hosting the mirror site and for his help in setting it up. New PNG image format used Enjoy and discover. A Table of Fractal Information. What's New in Fractals An index of Fractal material at Spanky. Some General information about Spanky.

10. Fractals - Fractal Recursions Jock Cooper s gallery of still images and animations. http://www.fractal-recursions.com/

News! Order the new The Fractal Recursions DVD through Amazon or Createspace Welcome to the ever-evolving exhibition of Jock Cooper's Fractal Art. A noteworthy recent addition to the site is the Zoomable Fractal Gallery; an interactive playground that allows one to zoom through six levels in each quadrant of a given fractal image. As always, the newer images are in the highest gallery numbers, and new images are added several times a week. Traditional Gallery 766 images Mechanical Gallery 274 images Miscellaneous Gallery 233 images Limited Palette Gallery 160 images Fractal Animations Gallery 29 animations Zoomables Gallery 6 Zoomable Fractal images Some (mostly older) images are available as high quality prints on photo paper or posters . If you are interested in an image not available let me know, I can generally add it within a day or two. Please send comments to me at Jock.Cooper At Gmail Dot Com For commercial use please contact Bruststrokes At Comcast Dot Net You can now view gallery images on your Hires or Lores mobile device. Most recent images are shown first.

11. Fractals: An Introductory Lesson The purpose of this site is to present material to learn very simple methods to construct fractals, to do and practice some math, to appreciate the beauty http://www.arcytech.org/java/fractals/

Fractals: An Introductory Lesson Fractals An introduction A Process to Generate Fractals A bit of Fractal Math The Koch Curve Fractal ... Check awards and feedback Last Updated: Friday, 11-Aug-2000 22:59:35 GMT Arcytech Java Home Page Provide ... Feedback

12. Fractal Geometry An educational resource on the mathematical framework and formalism from the Yale University, covering the concept of self similarity. http://classes.yale.edu/fractals/

13. IconBAZAAR - Fractals - Page 01 Click the thumbnails to view the corresponding fractals. arach.gif THUMBNAIL bay.gif Click the thumbnails to view the corresponding fractals http://www.iconbazaar.com/fractals/index.html

This site relies on advertising revenue to stay online. Please help us by clicking a sponsor's ad. Fractals Contents Page 01 Page 04 Page 02 Page 05 ... Conditions of Use Permission is granted for limited personal use of up to twenty (20) images from ALL collections. ALL commercial or institutional use requires licensure Redistribution of any image found at this web site in any format is strictly prohibited. Last updated: August 5, 2007. All images 1994 - 2004 Randy D. Ralph, Christopher E. Ralph, IconBAZAAR LLC. Other Resources: NetStrider Design My Realty Guide Web Hosting Web Directory ... NEXT Click the thumbnails to view the corresponding Fractals THUMBNAIL THUMBNAIL THUMBNAIL THUMBNAIL ... THUMBNAIL Click the thumbnails to view the corresponding Fractals UP BACK NEXT This site uses advertising revenues to stay online. Please click a sponsor's advertisement during your visit

14. Fractals www.worldscinet.com/fractals/fractals.shtml Similar pages Order in the Chaos - Synchronicity through fractals http://www.worldscinet.com/fractals/fractals.shtml

News New Journals Browse Journals Search ... Fractals Fractals Complex Geometry, Patterns, and Scaling in Nature and Society The investigation of phenomena involving complex geometry, patterns and scaling has gone through a spectacular development in the past decades. For this relatively short time, geometrical and/or temporal scaling have been shown to represent the common aspects of many processes occurring in an unusually diverse range of fields including physics, mathematics, biology, chemistry, economics, technology and human behavior. As a rule, the complex nature of a phenomenon is manifested in the underlying intricate geometry which in most of the cases can be described in terms of objects with non-integer (fractal) dimension. In other cases, the distribution of events in time or various other quantities show specific scaling behavior, thus providing a better understanding of the relevant factors determining the given processes. More News An Appeal from the Honorary Editor to the Readers of Fractals Announcement: After 10 years of dedicated service to Fractals, Dr. Michael Shlesinger will retire from his role as Regional Managing Editor, but will continue to serve as a member of the Editorial Board. Dr. Shlesinger will be succeeded by current editorial board member Dr. Michael Frame of the Mathematics Department at Yale University.

15. The Fractal Microscope www.ncsa.uiuc.edu/Edu/Fractal/Fractal_Home.html Similar pages fractalsThe primary focus of the Community Learning Network (CLN) is to help K-12 teachers integrate Information Technology into their classrooms. http://www.ncsa.uiuc.edu/Edu/Fractal/Fractal_Home.html

16. Fractals Easier A fractal is a shape, often drawn by a computer, that repeats itself in a pattern. The design shapes usually reoccur in different sizes. http://42explore.com/fractal.htm

The Topic: Fractals Easier - A fractal is a shape, often drawn by a computer, that repeats itself in a pattern. The design shapes usually reoccur in different sizes. Harder - Fractals are endlessly repeating patterns that vary according to a set formula, a mixture of art and geometry. Fractals are any pattern that reveals greater complexity as it is enlarged A real-life example of fractals is ice crystals freezing on a glass window. You can see countless variations of the same pattern emerge in the crystals over time. Exploring Fractals (Grades 9-12) by M.A. Connors http://www.math.umass.edu/~mconnors/fractal/fractal.html This website explores fractal dimensions of strictly self-similar fractals, from Cantor Dust to the Fractal Skewed Web. Includes a teacher information section. Related Websites: 2) Fract-ED (Grades 8 and Above) http://www.ealnet.com/ealsoft/fracted.htm

17. Fractal Explorer: Mandelbrot And Julia Sets (by Fabio Cesari) Many people have probably been fascinated by the infinite complexity and beauty of fractals. I wrote this brief tutorial to explain, in simple terms, http://www.geocities.com/CapeCanaveral/2854/

var track_id=9988; var sc_project=1438278; var sc_invisible=1; var sc_partition=13; var sc_security="919ae89d"; Keywords: fractals, mandelbrot set, fractal, julia sets, quaternion, quaternion julia sets, Mandelbrot, Julia Your browser doesn't support frames. This site is best viewed with , or an equivalent browser that supports JavaScript and frames You can always access this no-frames version of this site. If you have troubles accessing it, please let me know Many people have probably been fascinated by the infinite complexity and beauty of fractals. I wrote this brief tutorial to explain, in simple terms, how the Mandelbrot set and Julia sets are generated. This document provides an informal introduction to these subjects, and is only intended to be a starting point to learn more about fractals and fractal geometry. You can contribute to the future development of this site by filling out the feedback form Comments and suggestions are very appreciated. Have fun! About complex numbers Mandelbrot set Julia sets Images gallery ... Quaternion Julia sets images gallery" Other pages: About the author Links Feedback form Sign my guestbook ... View my guestbook This page hosted by Get your own Free Home Page

18. Fractal Gallery The Beauty of fractals. Gallery of Fractal Images generated by FractInt. http://fractals.hauner.cz/

Adam Hauner Adam Hauner

19. What Are Fractals? A definition of fractals as a unique digital art form and in the broader sense of fractal geometry. http://fractalarts.com/SFDA/whatarefractals.html

Seattle Fractals Digital Art Experience the "Art Of Infinity" Site Menu Home What Are Fractals? Screensavers Art Prints Purchase Screensavers Special Offers Support Testimonials Artist Bio Contact Links What are fractals? Definitions: Fractals : A geometric pattern that is repeated (iterated) at ever smaller (or larger) scales to produce (self similar) irregular shapes and surfaces that cannot be represented by classical (Euclidian) geometry. Fractals are used especially in computer modeling of irregular patterns and structures found in nature. Iteration : The process of repeating a set of instructions a specified number of times or until a specific result is achieved. Click for larger image Relative to this art form, fractals are a unique art form using mathematical formulas to create art with an infinite diversity of form, detail, color and light. In simple terms, a fractal is a graphical image that represents the behavior of a mathematical equation. The formula used determines how each pixel in an image is formed and colored. (Pixels are the smallest display elements that make up the images you see on a computer monitor or television.) A typical fractal image contains millions of these pixels. These complex images of extraordinary beauty can arise out of fairly simple mathematical functions and then by selectively modifying these formulas, changing coloring algorithms etc. one can create unique compositions previously unseen to the human eye. In the broader scientific sense, they are essentially geometric shapes or forms that are represented in natural objects, from a fern leaf or tree, to a spider web or snowflake, to larger phenomena such as clouds or even galaxies in space.

20. OnOne Software - Genuine Fractals 5 Genuine fractals 5 simply produces higher quality and sharper results than traditional scaling methods in Photoshop® or in any other third party product. http://www.ononesoftware.com/detail.php?prodLine_id=2

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