Date post: | 15-Dec-2014 |
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Fractals
Joe CzuprynMTH 491
I. Introduction
II. Dimension
III. Brief history
IV. Specific fractals and their properties
V. Appearances and applications
Presentation Outline
Introduction
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.)
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
Dimension Of A Line
a b c
Dimension Of A Square
Dimension Of A Cube
1600s - Gottfried Leibniz
1883 - Georg Cantor
1904 – Helge von Koch
1915 – Vaclav Sierpinski
Early 1900s – Gaston Julia and Pierre
Fatou
Early History
Polish-born, French mathematician
Fractals: Form, Chance and Dimension (1975)
The Fractal Geometry of Nature (1982)
Benoit Mandelbrot
In the nth step, 3(n-1) triangles will be removed.
Sierpinski Triangle
Sierpinski Triangle - Dimension
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
Koch Snowflake - Dimension
Area of an equilateral triangle =
Formula for a geometric series with common ratio r:
Koch snowflake – Area
Koch snowflake – Area (cont.)
Koch snowflake – Area (cont.)
Using geometric series
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
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.)
Trees and plants
In the human body: Blood vessels Alveoli in the lungs
Naturally Occurring Fractals
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
Fractal patterns exist in a healthy human heartbeat
May give doctors a way to detect small tumors/early stages of cancer
Fractals And Medicine