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Joe Czupryn
MTH 491
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I. Introduction
II. Dimension
III. Brief history
IV. Specific fractals and their properties
V. Appearances and applications
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applications
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ƒí Self- similarity – when broken into smaller and smaller pieces, the new pieces look exactly the same as the original
ƒí Dimension - how much an object fills a space
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ƒí S represents the scaling factor and is always a natural number.
ƒí N represents the number of smaller, self- similar figures ( for a scaling factor S) needed to create the larger figure.
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a
b
c
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c
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c
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ƒí 1600s - Gottfried Leibniz
ƒí 1883 - Georg Cantor
ƒí 1904 – Helge von Koch
ƒí 1915 – Vaclav Sierpinski
ƒí Early 1900s – Gaston Julia and Pierre Fatou
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ƒí Polish- born, French mathematician
ƒí Fractals: Form, Chance and Dimension ( 1975)
ƒí The Fractal Geometry of Nature ( 1982)
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ƒí In the n th step, 3 ( n- 1) triangles will be removed.
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removed.
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ƒí ä will be used to refer to the side length of the equilateral triangle.
ƒí In the n th iteration, 3 * 4 ( n- 1) triangles are added.
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added.
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ƒí Area of an equilateral triangle =
ƒí Formula for a geometric series with common ratio r:
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r:
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Using geometric series
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ƒí Each iteration increased the length of a side to ( 4/ 3) its original length.
ƒí Thus, for the n th iteration, the overall perimeter is increasing by ( 4/ 3) n .
Divergent
Sequence
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ƒí The perimeter is then considered to be infinite!
ƒí How does this apply to Mandelbrot’s “ How long is the Coast Line of Britain?” problem?
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ƒí Trees and plants
ƒí In the human body:
„X Blood vessels
„X Alveoli in the lungs
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ƒí Used by Boeing to generate some of the first 3- D computer generated images
ƒí Currently being used to make antennas smaller in cell phones
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ƒí Fractal patterns exist in a healthy human heartbeat
ƒí May give doctors a way to detect small tumors/ early stages of cancer
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