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
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
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.)
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21. Alveoli in the lungsNaturally 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
