FRACTALS
Laura Wierschke
Libby Welton
HISTORY OF FRACTALS: JULIA SETS
Gaston Julia (1873-1978): French mathematician who worked with fractals
Made fractals that were named after him called the Julia Sets
Two types Connected sets Cantor sets
Had disadvantage to Mandelbrot No computers
DIVERGENT FRACTAL
MANDELBROT SETS
Benoit Mandelbrot (1924-present): Polish mathematician who studied fractals
Able to use computers Found a simpler equation to the Julia sets
that included all Julia Sets These sets called Mandelbrot sets Julian and Mandelbrot worked with non-
Euclidean geometry Made fractals that could easily represent things
like snowflakes and coastlines- something not easily done with Euclidean geometry
CONVERGENT FRACTAL
WHAT IS A FRACTAL?
Self-similar figure that repeats over and over in infinite iterations Iteration: Every time the pattern is repeated Axiom: Beginning of fractal Recursion: the rule at which the fractal is
repeated Magnifying a fractal will give a smaller, but
similar fractal Graphed on complex number plane
X-axis is real numbers Y-axis is complex numbers
FRACTALS IN NATURE Iterated Function System Fractals (IFS)
Snowflake Fern Maple Leaf Coastlines Silhouette of tree
Koch’s Snowflake
Fern
Maple Leaf
L SYSTEM FRACTALS
KLEINIAN GROUP FRACTALS
KLEINIAN FRACTAL
JULIABROT, QUATERNION AND HYPERCOMPLEX FRACTALS
Circle and Sphere inversion fractals
Hyperbolic Tessellation Fractals
Hyperbolic Tessellation
STRANGE ATTRACTORS
WORKS CITED Apollonian Gasket. May 31, 2009. Mathworld Team. June 2, 2009. mathworld.wolfram.com/ApollonianGasket.html
Chalk River Graphics. Castle One. 2008. June 2, 2009 http://www.fractalpalace.com/Details-CK1.php
Chalk River Graphics. Centipedius Kleinianus I. 2008. June 2, 2009 http://www.fractalpalace.com/Details-CK1.php
Chalk River Graphics. Eggs Hyperbolic .2008. June 2, 2009. http://www.fractalpalace.com/Details-EH.php
Chalk River Graphics. Hyperbolic Tessallation I. 2008. June 2, 2009. http://www.fractalpalace.com/Details-HT1.php
Chalk River Graphics. Pizza Bug .2008. June 2, 2009. http://www.fractalpalace.com/Details-EH.php
Circle and Sphere Inversion Fractals. June 2, 2009 http://www.hiddendimension.com/CircleInversionFractals.html
“Convergant Fractals.” Mathematics of Convergent Fractals . June 2, 2009 http://www.hiddendimension.com/Convergent_Fractals_Main.html
"Fractal Mathematics Main page." Hidden Dimension Galleries. 03 June 2009 <http://www.hiddendimension.com/Mathematics_Main.html>.
"Fractals: An Introductory Lesson." Arcytech Main Page. 03 June 2009 <http://www.arcytech.org/java/fractals/>.
“JuliaBrot, Quaternion and Hypercomplex Fractals.”Mathematics of JuliaBrot, Quaternion and Hypercomplex Fractals. June 2, 2009 http://www.hiddendimension.com/JuliaBrot_Fractals_Main.html
“Kleinian Group.” Kleinian Group Fractals. June 2, 2009. http://www.hiddendimension.com/KleinianGroup_Fractals_Main.html
L-System Fractals. August 27, 2008. Soltutorial. June 2, 2009. sol.gfxile.net/lsys.html
McWorter, William. Fractint L-System True Fractals. January 1997. June 2, 2009. http://spanky.triumf.ca/www/FractInt/LSYS/truefractal.html
Morrison, Andy. June 2, 2009 http://www.dannyburk.com/red_maple_leaf_4x5.htm
Seirpinski. Seirpinski’s Triangle. November 27, 1995. Chaos. June 2, 2009. www.zeuscat.com/andrew/chaos/sierpinski.html
Strange Attractors. 2009. Fractal Science Kit. June 2, 2009 www.fractalsciencekit.com/types/orbital.htm
Thelin, Johan. Attracting Fractals. June 2, 2009http://www.thelins.se/johan/2008/07/attracting-fractals.html
Vepstas, Linas . The Mandelbrot Set as a Modular Form. 30 May 2005. June 2, 2009 linas.org/math/dedekind/dedekind.html