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Using Linear Algebra Techniques to Generate Fractals

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Olivia Califano

on 29 January 2014

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Transcript of Using Linear Algebra Techniques to Generate Fractals

Using Linear Algebra Techniques to Generate Fractals
What is a Fractal?

A fractal is an irregular geometric object that is infinitely complex; "that is, upon enlarging a portion of the fractal image, more complexity is revealed, and the same is true for a portion of that image, and so on ad infinitum"

Koch Curve
Koch Curve Structure
Fractal Template
Concepts
The Koch Curve is generated by the following four similitude transformations:
The First Transformation:
The Second Transformation:
The Third Transformation:
The Fourth Transformation:
Applying the four transformations to the initial image:
Applying the four affine transformations to K2:
Affine Transformation:


Rotation Matrix:


Similitude Transformation:

T1 acts on all five vertices individually:
First Iteration of Transformation 1:
First Iteration of Transformation 2:
First Iteration of Transformation 4:
First Iteration: Applying the four Koch Curve Transformations to all vertices of K1
Initial Image: K1
Result: K2
Initial Image: K1
T1
T2
T3
T4


In the previous slides, we performed each transformation on an entire image (the whole set of points at once). However, affine transformations operate on individual points within a set .
Applying the four affine transformations to K3:
K3
K4
T1
T2
T3
T4
Clarification
a
T1[b]
T1[d]
T1[c]
T1[e]
b
c
d
e
a
b
c
d
e
a
b
c
d
e
a
b
c
d
e
(1/3,0)
(1/3,√3/2)
(1/6,√3/18)
(2/3,0)
(2/9,0)
(1,0)
(1/3,0)
First Iteration of Transformation 3:
Initial Image: K1
a
b
c
d
e
T1[a]
T1[b]
T1[c]
T1[d]
T1[e]
T2[a]
T2[b]
T2[c]
T2[d]
T2[e]
T3[a]
T3[b]
T3[c]
T3[d]
T3[e]
T4[a]
T4[b]
T4[c]
T4[d]
T4[e]
K2
Bibliography
http://www.saintjoe.edu/~karend/m244/affine-112.pdf

http://online.redwoods.edu/instruct/darnold/laproj/Fall97/Woody/fractals.pdf

http://home2.fvcc.edu/~dhicketh/LinearAlgebra/studentprojects/fall2004/jesseklang/Fractalsproject.htm

http://www.math.utah.edu/~korevaar/ACCESS2008/ClassicalFractals.pdf
http://online.redwoods.edu/instruct/darnold/laproj/Fall97/Woody/fractals.pdf
http://www.math.utah.edu/~korevaar/ACCESS2008/ClassicalFractals.pdf

(http://www.math.ubc.ca/~cass/courses/m308-03b/projects-03b/skinner/introduction.htm)
K3
K2
T1
T2
T3
T4
A set S is self-similar if it can be divided into N congruent subsets, each of which when magnified by a constant factor M yields the entire set S.

The fractal dimension of a self-similar set S is:
1
2
3
4
N=4
M=3
1/3
1/3
1/3
1/3
3^3 congruent pieces
3^2 congruent pieces
Hausdorff dimension
N = 9
M = 3
D = log(9)/log(3) = 2
N = 27
M=3
D = log(27)/log(3) = 3
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