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 Computer generated artwork, with thousands of downloadable images and a new fractal every day! 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. 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 807 images Mechanical Gallery 275 images Miscellaneous Gallery 233 images Limited Palette Gallery 161 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.

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. 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.

6. Fractals www.worldscinet.com/fractals/fractals.shtml African fractals Modern Computing and Indigenous Design - Google Books Resultby Ron Eglash - 1999 - Mathematics - 258 pages 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 four years of dedicated service to the Fractals journal, Dr. Janos Kertesz will retire from his role as Regional Managing Editor (Europe), but will continue to serve as a member of the Editorial Board. Dr. Kertesz will be succeeded by current editorial board member Dr. Miroslav M. Novak of the Faculty of CISM, Kingston University, England.

7. 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

8. 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

9. 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

10. 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.

11. 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

12. Fractals The primary focus of the Community Learning Network (CLN) is to help K12 teachers integrate Information Technology into their classrooms. http://www.cln.org/themes/fractals.html

Fractals Theme Page This "Theme Page" has links to information about Fractals. Students and teachers will find curricular resources (information, content...) to help them learn about this topic. In addition, there are also links to instructional materials (lesson plans) which will help teachers provide instruction in this theme. Please read our An Internet Fractal Lesson To study fractals using World Wide Web and generating your own f ractal. (Grades 7-12) Chaos Club Educational resources, online games and message chat areas for students and teachers exploring fractals. Be sure to check out the "tutorial" for a unit by Cynthia Lanius in which elementary and middle students utilize online resources to study properties of fractals either independently or with a teacher. Lessons can be downloaded and printed. Explorer The Explorer is a collection of information on educational resources (instructional software, lab activities, lesson plans, student created materials ...) that K-12 mathematics and science educators can download. Enter "Fractals" in the search engine and you'll see about a dozen resources. Exploring Fractals This site for secondary students explores several aspects of fractals. After an introduction to fractals, the author explores assigning dimensions, both non-integer and integer, and several ways of generating fractals.

13. 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://ozviz.wasp.uwa.edu.au/~pbourke/fractals/

Fractals, Chaos New Walk About: Exploring the ASKAP site Warping patch for Quartz Composer FAQ: Spherical mirror projection for hemispherical domes Representing and modelling geometry in SecondLife Announcements 2008 Australasian Planetarium Society Conference 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.

14. 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/

15. 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://www.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

16. FractalNet Home Page The Fractal Images Company 268 Arlington Road Kensington, CA 94707. Voice 510528-0258 Fax 510-528-0243 e-mail marketing@fractals.com. http://www.fractals.com/

Best experienced with Click here to start. ChiaraMail(tm) Version 2.0 now available ChiaraMail Version 2.0 is now available for download. Check out the coolest and most effective spam filter on the planet, at http://www.chiaramail.com WebReader 3.0 now available Version 3.0 of WebReader(tm) includes a number of significant new features and benefits; for an overview of functional enhancements in Version 3.0, check out new features and enhancements . Beginning December 7, 1999, you can download WebReader 3.0 at no cost. Before downloading WebReader, make sure you meet the minimal system requirements FractalNet Commercial Links Linda Sy Skin Care , offering high-quality skin care products developed by a practising dermatologist. Treat yourself or your spouse to the gift of beautiful, healthy skin with our line of products previously available only from dermatologists. Real Estate Services The Fractal Images Company 268 Arlington Road Kensington, CA 94707 Voice: 510-528-0258 Fax: 510-528-0243 e-mail: marketing@fractals.com

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 World Gallery Thumbnails : Cosmic Recursive Fractal Flames Or Flame Frac Fractal World Gallery Home. Fractal World Gallery contains a collection of Pure flame fractals, Thanks for viewing the Fractal World Gallery. http://www.enchgallery.com/fractals/fracthumbs.htm

Fractal World Gallery contains a collection of Pure flame fractals, fractal flame composites, fractals, etc: established 1998 Flame Fractals date from 1998 to the Present by Cory Enc h I mages from this gallery may only be used with artist 's permission Fractal software includes Frax Flame and Apophysis for cosmic recursive fractal flames. FAQ I CONTACT I PRINTS More art work by Cory Ench at www.enchgallery.com images in room 6 click on the thumbnails for full view fractal image images in room 6 images in room 5 images in room 4 images in room 3 images in room 2 120 images in room Other non fractal art by Cory Ench at home Thanks for viewing the Fractal World Gallery. Please go to next gallery room for more cosmic recursive flame fractals. Don't forget to sign our guestbook and give some feedback. Sign Guestbook

19. Fractals - Amazing Seattle Fractals! - Screensavers, Tutorials, Free Software Original fractal art, free fractal screensavers, free software, tutorials, fractal art galleries. http://www.fractalarts.com/ASF/

Amazing Seattle Fractals! Home Fractal Art Galleries Fractal Tutorials Fractal Of The Week ... Animation Theatre E-Mail: Fractal Software Guest Galleries Awards Will Engdahl Galleries ... About Welcome to Amazing Seattle Fractals! If you came to learn more about fractals, learn how to create your own fractal art, download free fractal software programs or just to view some fractal art galleries, you've come to the right place! These unique, digital art images are created from using mathematical formulas. The result of these formulas are images of an astounding diversity of form, detail, color and light. I believe you will agree as you view this contemporary digital art form, that these images truly are AMAZING! Caution! This is a graphics intensive site! 2007 News! Due to the amount of SPAM these days many e-mails I receive get eliminated before even opening. If you write please be sure to write something regarding fractals in the subject line or your e-mail may get deleted inadvertantly! Do not just put hi, N/A, hello etc. in the subject line as this is common in many SPAM e-mails and they get deleted before opening! Thanks! Fall 2007 - Three new galleries added of Xenodream and Tiera-zon fractals! Check them out and I hope you enjoy!

20. Fractal Curves And Dimension fractals burst into the open in early 1970s. Their breathtaking beauty captivated many a layman and a professional alike. Striking fractal images can often http://www.cut-the-knot.org/do_you_know/dimension.shtml

var MyPageLoc = document.location; var MyPageTitle = document.title; G o o g ... e Web CTK Sites for teachers Sites for parents Awards Interactive Activities ... Sites for parents Fractal Curves and Dimension Fractals burst into the open in early 1970s. Their breathtaking beauty captivated many a layman and a professional alike. Striking fractal images can often be obtained with very elementary means. However, the definition of fractals is far from being trivial and depends on a formal definition of dimension. It takes a few chapters of an Advanced Analysis book to rigorously define a notion of dimension. The important thing is that the notion is not unique and even more importantly, for a given set, various definitions may lead to different numerical results. When the results differ the set is called fractal. Or in the words of Benoit Mandelbrot [Ref 2], the father of fractals: A fractal is by definition a set for which the Hausdorff-Besicovitch dimension strictly exceeds the topological dimension. The topological dimension of a smooth curve is, as one would expect, one and that of a sphere is two which may seem very intuitive. However, the formal definition was only given in 1913 by the Dutch mathematician L. Brouwer (1881-1966). A (solid) cube has the topological dimension of three because in any decomposition of the cube into smaller bricks there always are points that belong to at least

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