Upload
gbdriver80
View
2.684
Download
10
Tags:
Embed Size (px)
DESCRIPTION
Citation preview
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