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Fractals presentation

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  • 1. Fractals
    Joe Czupryn
    MTH 491
  • 2. Introduction
    Dimension
    Brief history
    Specific fractals and their properties
    Appearances and applications
    Presentation Outline
  • 3. Introduction
  • 4. Self-similarity – when broken into smaller and smaller pieces, the new pieces look exactly the same as the original
    Dimension - how much an object fills a space
    Introduction (cont.)
  • 5. S represents the scaling factor and is always a natural number.
    N represents the number of smaller, self-similar figures (for a scaling factor S) needed to create the larger figure.
    Dimension
  • 6. Dimension Of A Line
    b
    c
    a
  • 7. Dimension Of A Square
  • 8. Dimension Of A Cube
  • 9. 1600s - Gottfried Leibniz
    1883 - Georg Cantor
    1904 – Helge von Koch
    1915 – Vaclav Sierpinski
    Early 1900s – Gaston Julia and Pierre Fatou
    Early History
  • 10. Polish-born, French mathematician
    Fractals: Form, Chance and Dimension (1975)
    The Fractal Geometry of Nature (1982)
    Benoit Mandelbrot
  • 11. In the nth step, 3(n-1) triangles will be removed.
    Sierpinski Triangle
  • 12. Sierpinski Triangle - Dimension
  • 13. δ will be used to refer to the side length of the equilateral triangle.
    In the nth iteration, 3 * 4(n-1) triangles are added.
    Koch Snowflake
  • 14. Koch Snowflake - Dimension
  • 15. Koch snowflake – Area
    Area of an equilateral triangle =
    Formula for a geometric series with common ratio r:
  • 16. Koch snowflake – Area (cont.)
  • 17. Koch snowflake – Area (cont.)
    Using geometric series
  • 18. Each iteration increased the length of a side to (4/3) its original length.
    Thus, for the nth iteration, the overall perimeter is increasing by (4/3)n.
    Koch Snowflake - Perimeter
    Divergent
    Sequence
  • 19. The perimeter is then considered to be infinite!
    How does this apply to Mandelbrot’s “How long is the Coast Line of Britain?” problem?
    Koch Snowflake – Perimeter (cont.)
  • 20. Trees and plants
    In the human body:
    • Blood vessels
    • 21. Alveoli in the lungs
    Naturally Occurring Fractals
  • 22. Used by Boeing to generate some of the first 3-D computer generated images
    Currently being used to make antennas smaller in cell phones
    Fractals And Technology
  • 23. Fractal patterns exist in a healthy human heartbeat
    May give doctors a way to detect small tumors/early stages of cancer
    Fractals And Medicine

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