Fractals

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Fractals

  1. 1. Fractals: A Brief OverviewHarlan J. Brothers, Director of TechnologyThe Country School Madison, CTThis presentation provides a broad and basic introduction to thesubject of fractal geometry.My thanks to Michael Frame at Yale University for the use ofmany of the photos and graphics that appear here. Hisfascinating and comprehensive treatment of the subject can befound at: http://classes.yale.edu/Fractals/ .
  2. 2. Familiar SymmetriesWe commonly recognize when shapes demonstrate symmetryunder the three familiar transformations of reflection, rotation,and translation. Reflection Rotation Translation
  3. 3. Scaling SymmetryFractals demonstrate a fourth type of symmetry; they possess“self-similarity.”Self-similar objects appear the same under magnification. Theyare, in some fashion, composed of smaller copies of themselves.This characteristic is often referred to as “scaling symmetry” or“scale invariance.” Sierpinski Gasket
  4. 4. Scaling SymmetryNot all self-similarity, however, is of a fractal nature. Objects likespirals and nested dolls that are self-similar around a single pointare NOT fractal. Not fractal Not fractal
  5. 5. FractalsIn the broadest sense, fractals can be divided into twocategories: objects that occur in Nature, and mathematical constructions.
  6. 6. Fractals in NatureNatural objects exhibit scalingsymmetry, but only over alimited range of scales. Theyalso tend to be “roughly” self-similar, appearing more or lessthe same at different scales ofmeasurement. Sometimesthis means that they arestatistically self-similar; that isto say, they have a distributionof elements that is similarunder magnification.
  7. 7. Fractals in Nature Trees Ferns
  8. 8. Fractals in Nature Mountains
  9. 9. Fractals in Nature Coastline and snow fields of Norway Waterfall
  10. 10. Fractals in Nature Clouds
  11. 11. Fractals in Nature Bacterial colony Lightening (courtesy E. Ben-Jacob)
  12. 12. Mathematical ConstructionsIn contrast to naturally occurring fractals, mathematical fractalscan possess an infinite range of scaling symmetry. The morecommon constructions also tend to be exactly self-similar. Koch Curve The Koch curve above is composed of exactly four copies of itself. Can you construct it from just two?
  13. 13. Mathematical ExamplesSierpinski Gasket Menger Sponge Mandelbrot Set Cantor Comb Koch Snowflake
  14. 14. Scale InvarianceThe fact that a fractal object is, in some sense, composed ofsmaller copies of itself, has interesting implications. One ofthese is that when we examine a fractal shape without a suitableframe of reference, it is often impossible to tell the scale ofmagnification at which it is being viewed.For natural phenomena, this translates to uncertainty withrespect to the distance, extent, or size of the object. We endwith two examples of scale invariance.What do the following two images look like to you?
  15. 15. Image 1
  16. 16. Image 2
  17. 17. Image 1 (another view)
  18. 18. Image 2 (another view)
  19. 19. DiscussionDepending on who you ask, the preceding images may look likesatellite or aerial photos, rock formations, or photomicrographs.This simply illustrates the fact that certain natural processes, likeerosion or the formation ice crystals, follow patterns that can berepeated at many scales of measurement. Without a frame ofreference, a photograph of a rock sitting one meter away caneffectively look the same as a boulder several meters away or acliff hundreds of meters distant.With a knowledgeable eye, one sees a natural world thatabounds in fractal shapes.
  20. 20. Author InformationHarlan J. BrothersDirector of TechnologyThe Country SchoolMadison, CT 06443Tel. (203) 421-3113E-mail: harlan@thecountryschool.org

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