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 3) Fractals by C. Lanius http://math.rice.edu/~lanius/frac/ 4) Fractals from Math Archives (Links-site) http://archives.math.utk.edu/topics/fractals.html   Fantastic Fractals by J. Sheu and M. Turvey http://www.techlar.com/fractals/websys.exe?file=index.html This comprehensive site on fractals first began as a ThinkQuest project. It still includes a kid's section that uses audio clips to explain the basics of fractals in an interesting manner, and the rest of the site has just about any fact or explanation a high school student might need on the topic. Also has quizzes, downloadable files, links, and more. Another Comprehensive Fractal Website: 2) Amazing Seattle Fractals! http://www.fractalarts.com/ASF/index.html   Fractals by S. Alejandre from Math Forum http://mathforum.org/alejandre/workshops/fractal/fractal3.html This is an introduction to fractals. It also has some good links.   Fractals: An Introduction  http://www.dcet.k12.de.us/teach/quest/shari.htm This website provides very simple methods to construct fractals, to do and practice some math, to appreciate the beauty of fractals, and finally to have fun with math. Related Website: 2) Fractals (Grades 5-8) http://math.rice.edu/~lanius/frac/   After exploring lots of fractal websites (below), complete one or more of these fractal projects.   Complete a Fractals WebQuest. Follow or adapt the directions found at one of these sites.  1) Defining Chaos Through Fractals (Grades 7-12) http://www.warrennet.org/warrentech/webquest/project.html 2) Fractal Chaos Webquest (Grades 9-12) by S. Hallar http://warrensburg.k12.mo.us/webquest/fractals/index.htm 3) Fractals http://staff.harrisonburg.k12.va.us/~dflick/fractal/fractals.htm 4) Fractals: Math or Art? http://www.dcet.k12.de.us/teach/quest/shari.htm 5) To Fractal or Not to Fractal--that is the Quest-ion. http://www.manteno.k12.il.us/webquest/middle/Math/Fractals/fractalquest.htm   Make Your Own Fractals. You might want to start at The Sierpinski Triangle, The Jurassic Park Fractal, and The Koch Snowflake at Cynthia Lanius's Fractals website. Then go to one of the online fractal generation programs. The IFSoft Home Page has two Java applets: fractalina, a program for making fractals; and franimate!, a program for animating fractals. You also can try Make Your Own Julia Set at The Fractory. Follow up by using the 'Make-Your-Own Mandelbrot Set" at the same website. Compare the fractals that you make. Try using the same parameters. Read the 'learn more' and 'how the fractals are generated' sections.   Create Some Fractal Music. Visit Fractal Music to begin learning about fractal music. Then go to the program set at Create Your Own Chaos and compose some of your own fractal music.   Play An Online Fractal Game. Go to the Leap Fractal Game and follow the provided rules. This is an interactive fractal game that gives the player a choice of three different games and each game can be played at four different skill levels - - Requires Java.    Websites By Kids for Kids Fractals and Chaos (1999 ThinkQuest Internet Challenge) http://library.advanced.org/26688/ This website will describe certain basic concepts of chaos theory including fractal math.   Fractals Unleashed (1999 ThinkQuest Internet Challenge) http://library.advanced.org/26242/full/index.html This is a comprehensive website about fractals.   Fractal World by E. Hanson, Lock Haven High School http://www.kcsd.k12.pa.us/~projects/fractal/ You have the choice of three fractal topic pages plus "Fractal Links" to other related sites. In "What is a Fractal?" you find basic information on the formulas and origins of fractal geometry. Under "What is a Fractal? 2" you access more complicated information like the Mandelbrot function and imaginary numbers. Finally, examine some computer generated fractals under "Sample Pictures".   Fractory (1996 ThinkQuest Internet Challenge) http://tqd.advanced.org/3288/ This page will help you learn about fractals: what they are and how to design them.   InterFACE: Internet Fractal and Chaos Education (1997 ThinkQuest Internet Challenge) http://library.advanced.org/11679/ InterFACE offers a series of lessons and quizzes on fractals and chaos theory. You can hear a fractal symphony, see a fractal movie, send a fractal postcard, and download two freeware programs.   Making Order Out of Chaos (1997 ThinkQuest Internet Challenge) http://library.advanced.org/12170/ The most well-known aspects of chaos theory are fractal images. Fractals appear to be complex and hard-to-comprehend at first glance, but the underlying code is very simple.    Our Fractal Universe: Mandelbrot and More (1999 ThinkQuest Internet Challenge) http://library.advanced.org/25810/ This page is a study of fractals, both simple and complex and the basic mathematical concepts behind them.    More Fractal Websites  Fractal Music: The Sound of Chaos  http://www.discovery.com/stories/technology/fractals/fractals.html Nowadays, the mathematics behind fractals is better known as chaos. An amateur musician creates his music using nothing more than a desktop computer and a mathematical entity called the Mandelbrot set, a simple equation better known for generating complex and haunting fractal images. (Requires Real Audio program, free download link provided) Other Fractal Music Site: 1) Fractal Music http://www.ozemail.com.au/~davesag/snd.html    Fractal Microscope at the National Center for Supercomputing Applications (NCSA)  http://archive.ncsa.uiuc.edu/Edu/Fractal/Fractal_Home.html This site explains about the bizarrely beautiful images known as fractals.   Fractal of the Web from Chaffey High School http://www.chaffey.org/fractals/index.html Here you find fractal information, programs, articles and other fractal related pages on the Internet. Other School Webpages for Fractals: 2) Chaos from Minnesota Tech Prep and Intermediate School District 287 http://www.chaosclub.com/index.html   Fractals, Chaos by P. Bourke http://astronomy.swin.edu.au/~pbourke/fractals/ This fractals site contains images and information for the more advanced learner. Another Fractal Site for Advanced Level Learners: 2) Fractals and Scale by D.G. Green http://life.csu.edu.au/complex/tutorials/tutorial3.html   Gallery of Linear Fractals   http://www.cs.uregina.ca/~hepting/art/gallery/ This fractal gallery is divided into five parts (use either text or images to navigate to them) containing over 25 images. Related Website: 2) Presentations at fractals http://www.cs.uregina.ca/~hepting/art/present/   Mandelbrot Set  http://www.mindspring.com/~chroma/mandelbrot.html This is the website of an applet that lets you explore the Mandelbrot set. Similar Websites: 2) Julia and Mandelbrot Set Explorer http://aleph0.clarku.edu/~djoyce/julia/explorer.html 3) Java Julia Set Generator http://www.unca.edu/~mcmcclur/java/Julia/ 4) Interactive Fractal Generation Using Iterative Function Systems http://www.bridgewater.edu/departments/physics/ISAW/FracMain.html   Spanky Fractal Database  http://spanky.triumf.ca/ Here you can explore a huge resource of fractal images, software and information. Other Fractal Images: 2) Dance of Chance http://polymer.bu.edu/museum/ 3) Fractal Domains Gallery by D.C. De Mars http://www.fractaldomains.com/gallery/ 4) Fractal eXtreme http://www.cygnus-software.com/gallery/stampindex.htm 5) Linas' Art Gallery by L. Vepstas http://www.linas.org/art-gallery/ 6) Fractal Galleries http://www.glyphs.com/art/fractals/ 7) Sprott's Fractal Gallery http://sprott.physics.wisc.edu/fractals.htm   Websites for Teachers Fractals - Computer Generated (Grades 6-12) by D. Buckeye http://explorer.scrtec.org/explorer/explorer-db/rsrc/810013040-81ED7D4C.2.PDF This lesson includes a BASIC program that generates a right triangle fractal. Other Fractal Lessons by D. Buckeye: 2) Fractals - Computer Generated 2 http://explorer.scrtec.org/explorer/explorer-db/rsrc/820994437-81ED7D4C.2.PDF 3) Fractals - Computer Generated 3 http://explorer.scrtec.org/explorer/explorer-db/rsrc/820994433-81ED7D4C.2.PDF 3) Fractals - Koch and Canter (Grades 6-12) http://explorer.scrtec.org/explorer/explorer-db/rsrc/810013045-81ED7D4C.2.PDF 4) Fractals - Paper Folding (Grades 6-12) http://explorer.scrtec.org/explorer/explorer-db/rsrc/820994428-81ED7D4C.2.PDF   Infinity, Self-Similarity and Recursion (Grade 8)  http://www.shodor.org/interactivate/lessons/frac1.html This lesson introduces learners to line deformation fractals, infinity, self-similarity and recursion and leads them in practice of fraction, pattern recognition, perimeter and area skills. Other Lesson Plans in the Series: 2) Chaos http://www.shodor.org/interactivate/lessons/frac5.html 3) Fractals and the Chaos Game http://www.shodor.org/interactivate/lessons/frac3.html 4) Properties of Fractals http://www.shodor.org/interactivate/lessons/frac4.html 5) Introduction to Fractals: Geometric Fractals http://www.shodor.org/interactivate/lessons/frac2.html 6) Irregular Fractals http://www.shodor.org/interactivate/lessons/frac7.html 7) Mandelbrot Set http://www.shodor.org/interactivate/lessons/frac9.html 8) Pascal's Triangle http://www.shodor.org/interactivate/lessons/frac6.html   Internet Fractal Lesson (Grades 7-12) by D. Lewis http://www.challenge.state.la.us/edres/lessons/HighSchool/lesson11.htm This lesson has students studying fractals using the World Wide Web and generating their own fractals.   fractal pattern geometry chaos self-similar dimension recursion Mandelbrot set nature infinity fraction mathematics art repetition tessellation order random beauty Julia set computation parameter generation polynomial equation x-coordinate y-coordinate pixel iteration function color model Sierpinski triangle fractal dimension Koch snowflake Peano curve Lorenz attractor     Created by Annette Lamb and Larry Johnson 12/99. Updated by Nancy Smith, 6/02.