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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