Fractals

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