Lesson 11'6 Fractals

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Lesson 11.6Fractals

• By Mr. Wheeler, Ms. Simard, Ms. Carpenter, and
  Ms. Wahlen

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Vocabulary

• Fractal- A geometric object that exhibits some
  type of self-similarity
• Self-similarity- The property, possessed by
  fractals, that every subdivision of the fractal
  has a structure similar to the structure as a
  whole
• Iteration- The repetitive process of the same rule

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Key Concepts

• Fractals can be created by repeating the same
  process over and over.
• The Menger sponge, is created by starting with a
  certain shape (a cube) and changing it according
  to a certain rule (removing a part of the cube).
• The same rule is applied to the newly changed
  shape and the process continued in an iteration.

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Menger Sponge

• The structure of the object always looks the
  same, whether seen in a highly magnified view,
  from a moderate distance, or from far away.
• If you were to cut off a small cube-shaped
  portion and examine it, the piece would be a
  miniature copy of the entire sponge.

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Sierpenski Gasket

• The Sierpenski gasket is created by repeatedly
  applying a single rule to an initial shape.
• Rules
• Start with a solid triangle and find the midpoint
  of each side.
• Iteration 1- Connect the midpoints of the sides
  to form four congruent isosceles triangles in the
  interior of the original. Remove the center
  triangle.
• Each new iteration is performed on the remaining
  triangles.

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Fractal Dimension

• Take a 1 x 1 x 1 cube and double its length,
  width, and height.
• How many copies of the original size cube do you
  get?
• Doubling the side creates eight copies.

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Fractal Dimension

• The dimension of the figure is the exponent of
  the number of sides.
• When you double the sides and create a similar
  figure, you write the number of copies as a power
  of 2 and the exponent will be the dimension.

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Examples in Nature

• In a river and its tributaries, every tributary
  has its own tributaries so that it has the same
  design as the river but on a smaller scale
• The branching of trees and their roots
• Blood vessels, nerves, and bronchioles in the
  human body
• The distribution of galaxies in the universe

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Bibliography

• Schultz, James E., Kathleen A. Hollowell, Wade
  Ellis, and Paul A. Kennedy. Holt Geometry. Holt,
  Rinehart, and Winston, 2004. 738-745.
• Lanius, Cynthia. "Fractals." 2007. 16 May 2007
  lthttp//math.rice.edu/lanius/frac/gt.
• "fractal." McGraw-Hill Encyclopedia of Science
  and Technology. The McGraw-Hill Companies, Inc.,
  2005. Answers.com 16 May. 2007.
  http//www.answers.com/topic/fractal

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The End