


 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 reallife
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 912) by M.A.
Connors
 http://www.math.umass.edu/~mconnors/fractal/fractal.html
 This website explores fractal dimensions of
strictly selfsimilar fractals, from Cantor Dust to
the Fractal Skewed Web. Includes a teacher
information section.
 Related Websites:
 2) FractED (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
(Linkssite)
 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 58) http://math.rice.edu/~lanius/frac/



 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 wellknown aspects of chaos theory are
fractal images. Fractals appear to be complex and
hardtocomprehend 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.cygnussoftware.com/gallery/stampindex.htm
 5) Linas' Art Gallery by L. Vepstas http://www.linas.org/artgallery/
 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 612) by D.
Buckeye
 http://explorer.scrtec.org/explorer/explorerdb/rsrc/81001304081ED7D4C.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/explorerdb/rsrc/82099443781ED7D4C.2.PDF
 3) Fractals  Computer Generated 3
 http://explorer.scrtec.org/explorer/explorerdb/rsrc/82099443381ED7D4C.2.PDF
 3) Fractals  Koch and Canter (Grades
612)
 http://explorer.scrtec.org/explorer/explorerdb/rsrc/81001304581ED7D4C.2.PDF
 4) Fractals  Paper Folding (Grades 612)
 http://explorer.scrtec.org/explorer/explorerdb/rsrc/82099442881ED7D4C.2.PDF

 Infinity,
SelfSimilarity and Recursion (Grade
8)
 http://www.shodor.org/interactivate/lessons/frac1.html
 This lesson introduces learners to line
deformation fractals, infinity, selfsimilarity 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 712) 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

selfsimilar

dimension

recursion

Mandelbrot set

nature

infinity

fraction

mathematics

art

repetition

tessellation

order

random

beauty

Julia set

computation

parameter

generation

polynomial equation

xcoordinate

ycoordinate

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.
