**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