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# Fractal

A fractal is an object or quantity that 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.

Illustrated above are the fractals known as the Gosper island, Koch snowflake, box fractal, Sierpiński sieve, Barnsley's fern, and Mandelbrot set.

Attractor, Backtracking, Barnsley's Fern, Box Fractal, Cactus Fractal, Cantor Dust, Cantor Set, Cantor Square Fractal, Carotid-Kundalini Fractal, Cesàro Fractal, Chaos Game, Circles-and-Squares Fractal, Coastline Paradox, Dendrite Fractal, Dragon Curve, Fat Fractal, Fatou Set, Fractal Dimension, Gosper Island, H-Fractal, Hénon Map, Iterated Function System, Julia Set, Kaplan-Yorke Map, Koch Antisnowflake, Koch Snowflake, Lévy Fractal, Lévy Tapestry, Lindenmayer System, Lorenz Attractor, Mandelbrot Set, Mandelbrot Tree, Menger Sponge, Minkowski Sausage, Mira Fractal, Newton's Method, Pentaflake, Peano Curve, Peano-Gosper Curve, Pythagoras Tree, Rabinovich-Fabrikant Equation, Rep-Tile, San Marco Fractal, Self-Similarity, Siegel Disk Fractal, Sierpiński Carpet, Sierpiński Curve, Sierpiński Sieve, Star Fractal, Strange Attractor, Zaslavskii Map Explore this topic in the MathWorld classroom

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## References

Barnsley, M. F. and Rising, H. Fractals Everywhere, 2nd ed. Boston, MA: Academic Press, 1993.Bogomolny, A. "Fractal Curves and Dimension." http://www.cut-the-knot.org/do_you_know/dimension.shtml.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. (Ed.). Classics on Fractals. Reading, MA: Addison-Wesley, 1993.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.le Méhaute, A. Fractal Geometries: Theory and Applications. Boca Raton, FL: CRC Press, 1992.Mandelbrot, B. B. Fractals: Form, Chance, & Dimension. San Francisco, CA: W. H. Freeman, 1977.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.; Jürgens, H.; and Saupe, D. Fractals for the Classroom, Part 1: Introduction to Fractals and Chaos. 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.Peitgen, H.-O. and Saupe, D. (Eds.). The Science of Fractal Images. New York: Springer-Verlag, 1988.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.Taylor, M. C. and Louvet, J.-P. "sci.fractals FAQ." http://www.faqs.org/faqs/sci/fractals-faq/.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. "Books about Fractals." http://www.ericweisstein.com/encyclopedias/books/Fractals.html.Yamaguti, M.; Hata, M.; and Kigami, J. Mathematics of Fractals. Providence, RI: Amer. Math. Soc., 1997.

Fractal

## Cite this as:

Weisstein, Eric W. "Fractal." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Fractal.html