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### Fractals presentation

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