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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

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Fractals

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  1. Fractals Laura Wierschke Libby Welton

  2. 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

  3. Divergent Fractal

  4. 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 Mandelbrot Sets

  5. Convergent Fractal

  6. 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 What is a Fractal?

  7. Snowflake Fern Maple Leaf Coastlines Silhouette of tree FRACTALS IN NATURE Iterated Function System Fractals (IFS) Fern Koch’s Snowflake Maple Leaf

  8. L System Fractals

  9. Kleinian Group Fractals

  10. KLEINIAN FRACTAL

  11. JuliaBrot, Quaternion and Hypercomplex Fractals

  12. Circle and Sphere inversion fractals

  13. Hyperbolic Tessellation Fractals

  14. Hyperbolic Tessellation

  15. Strange Attractors

  16. 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, 2009http://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, 2009linas.org/math/dedekind/dedekind.html

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