Fractal

An object or quantity which displays Self-Similarity, in a somewhat technical sense, on all scales. The object need not exhibit exactly the same structure at all scales, but the same ``type'' of structures must appear on all scales. A plot of the quantity on a log-log graph versus scale then gives a straight line, whose slope is said to be the Fractal Dimension. The prototypical example for a fractal is the length of a coastline measured with different length Rulers. The shorter the Ruler, the longer the length measured, a Paradox known as the Coastline Paradox.

See also Backtracking, Barnsley's Fern, Box Fractal, Butterfly Fractal, Cactus Fractal, Cantor Set, Cantor Square Fractal, Carotid-Kundalini Fractal, Cesàro Fractal, Chaos Game, Circles-and-Squares Fractal, Coastline Paradox, Dragon Curve, Fat Fractal, Fatou Set, Flowsnake Fractal, Fractal Dimension, H-Fractal, Hénon Map, Iterated Function System, Julia Fractal, Kaplan-Yorke Map, Koch Antisnowflake, Koch Snowflake, Lévy Fractal, Lévy Tapestry, Lindenmayer System, Mandelbrot Set, Mandelbrot Tree, Menger Sponge, Minkowski Sausage, Mira Fractal, Newton's Method, Pentaflake, Pythagoras Tree, Rabinovich-Fabrikant Equation, San Marco Fractal, Sierpinski Carpet, Sierpinski Curve, Sierpinski Sieve, Star Fractal, Zaslavskii Map

References

Fractals

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Bogomolny, A. ``Fractal Curves and Dimension.'' http://www.cut-the-knot.com/do_you_know/dimension.html.

Brandt, C.; Graf, S.; and Zähle, M. (Eds.). Fractal Geometry and Stochastics. Boston, MA: Birkhäuser, 1995.

Bunde, A. and Havlin, S. (Eds.). Fractals and Disordered Systems, 2nd ed. New York: Springer-Verlag, 1996.

Bunde, A. and Havlin, S. (Eds.). Fractals in Science. New York: Springer-Verlag, 1994.

Devaney, R. L. Complex Dynamical Systems: The Mathematics Behind the Mandelbrot and Julia Sets. Providence, RI: Amer. Math. Soc., 1994.

Devaney, R. L. and Keen, L. Chaos and Fractals: The Mathematics Behind the Computer Graphics. Providence, RI: Amer. Math. Soc., 1989.

Edgar, G. A. Classics on Fractals. Reading, MA: Addison-Wesley, 1994.

Eppstein, D. ``Fractals.'' http://www.ics.uci.edu/~eppstein/junkyard/fractal.html.

Falconer, K. J. The Geometry of Fractal Sets, 1st pbk. ed., with corr. Cambridge, England Cambridge University Press, 1986.

Feder, J. Fractals. New York: Plenum Press, 1988.

Giffin, N. ``The Spanky Fractal Database.'' http://spanky.triumf.ca/www/welcome1.html.

Hastings, H. M. and Sugihara, G. Fractals: A User's Guide for the Natural Sciences. New York: Oxford University Press, 1994.

Kaye, B. H. A Random Walk Through Fractal Dimensions, 2nd ed. New York: Wiley, 1994.

Lauwerier, H. A. Fractals: Endlessly Repeated Geometrical Figures. Princeton, NJ: Princeton University Press, 1991.

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Mandelbrot, B. B. The Fractal Geometry of Nature. New York: W. H. Freeman, 1983.

Massopust, P. R. Fractal Functions, Fractal Surfaces, and Wavelets. San Diego, CA: Academic Press, 1994.

Pappas, T. ``Fractals--Real or Imaginary.'' The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, pp. 78-79, 1989.

Peitgen, H.-O.; Jürgens, H.; and Saupe, D. Chaos and Fractals: New Frontiers of Science. New York: Springer-Verlag, 1992.

Peitgen, H.-O. and Richter, D. H. The Beauty of Fractals: Images of Complex Dynamical Systems. New York: Springer-Verlag, 1986.

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Pickover, C. A. (Ed.). The Pattern Book: Fractals, Art, and Nature. World Scientific, 1995.

Pickover, C. A. (Ed.). Fractal Horizons: The Future Use of Fractals. New York: St. Martin's Press, 1996.

Rietman, E. Exploring the Geometry of Nature: Computer Modeling of Chaos, Fractals, Cellular Automata, and Neural Networks. New York: McGraw-Hill, 1989.

Russ, J. C. Fractal Surfaces. New York: Plenum, 1994.

Schroeder, M. Fractals, Chaos, Power Law: Minutes from an Infinite Paradise. New York: W. H. Freeman, 1991.

Sprott, J. C. ``Sprott's Fractal Gallery.'' http://sprott.physics.wisc.edu/fractals.htm.

Stauffer, D. and Stanley, H. E. From Newton to Mandelbrot, 2nd ed. New York: Springer-Verlag, 1995.

Stevens, R. T. Fractal Programming in C. New York: Henry Holt, 1989.

Takayasu, H. Fractals in the Physical Sciences. Manchester, England: Manchester University Press, 1990.

Tricot, C. Curves and Fractal Dimension. New York: Springer-Verlag, 1995.

Triumf Mac Fractal Programs. http://spanky.triumf.ca/pub/fractals/programs/MAC/.

Vicsek, T. Fractal Growth Phenomena, 2nd ed. Singapore: World Scientific, 1992.

Weisstein, E. W. ``Fractals.'' Mathematica notebook Fractal.m.

Yamaguti, M.; Hata, M.; and Kigami, J. Mathematics of Fractals. Providence, RI: Amer. Math. Soc., 1997.