Fractals and Linear Algebra. MAA Indiana Spring Meeting PDF Free Download

Fractals and Linear Algebra MAA Indiana Spring Meeting 2004

Creation of a Fractal

Fractal

Fractal An object composed of smaller copies of itself

Fractal An object composed of smaller copies of itself Iterated Function System

Fractal An object composed of smaller copies of itself Iterated Function System A process of repeatedly replacing shapes with other shapes

First Transformations T1 T 1 T2 T 2 T3 T 3

Describe (in words) the action performed by T1, T2, and T3

Describe (in words) the action performed by T1, T2, and T3 Draw images of the following

Describe (in words) the action performed by T1, T2, and T3 Draw images of the following T 3 T 1 I 2 T 2 T ( 1 I 2 ) T 1 T ( 2 I 2 ) T 2 T ( 3 I 2 ) ( ) ( ) = T 3 T ( 1 I 2 ) T 1 T 3 T ( 2 I 2 ) T 3 T 3 T ( 3 I 2 )

Result

Resulting Fractal

Trickier Transformations

Trickier Transformations T3 T 3 T1 T 1 T 2 T2

T3 Compositions T 3 T1 T 1 T 2 T2

T3 Compositions T 3 T1 T 1 T 2 T2 T 1 T ( 1 I 2 ) P P

T3 Compositions T 3 T1 T 1 T 2 T2 T 1 T ( 1 I 2 ) P P T 2 T ( 2 I 2 ) P

Draw images of the following ( ) ( ) T 3 T 1 I 2 T 3 T 2 I 2 T 2 T ( 1 I 2 )

Draw images of the following ( ) ( ) T 3 T 1 I 2 T 3 T 2 I 2 T 2 T ( 1 I 2 ) If this is the first iteration of three transformations T 3 draw next two iterations of the complete set of transformations indicating the orientation of the squares T 1 T 2

Resulting Fractal

Resulting Fractal T 3 T 1 T 2

Resulting Fractal

Resulting Fractal T 3 T 1 T 2

Working Backwards

?? Working Backwards

Consider the transformations shown to the right T1 T2 The figure below is the third iterate Identify T2 T1 T2 T2 T2 T2

Sierpinski(ish) Fractals Each of the following fractals was created using three transformations. Sketch the orientation of the P s

Formulas What are the formulas for these three transformations? What are the formulas for these three transformations? T 3 T 1 T 2

Formulas

Formulas T ( x, y) = ( ax + by + e, cx + dy + e)

Formulas T ( x, y) = ( ax + by + e, cx + dy + e) T x y a b = c d x y + e f

Matrix Transformations

Matrix Transformations A transformation is a rule for mapping one space to another space.

Matrix Transformations A transformation is a rule for mapping one space to another space. T:R 2 R 2 T ( x ) = A x T x y = a b c d x y

What is the image of the unit square under the following matrix transformations?

What is the image of the unit square under the following matrix transformations? T x y = 0.25 0 0 0.5 x y

What is the image of the unit square under the following matrix transformations? y T x y = 0.25 0 0 0.5 x y ( 0, 0.5 ) ( 0.25, 0)

y ( 0, 0.5 ) ( 0.25, 0) What is the image of the unit square under the following matrix transformations? T x y = 0.25 0 0 0.5 x y T x y = 0 1 1 0 x y

What is the image of the unit square under the following matrix transformations? y T x y = 0.25 0 0 0.5 x y ( 0, 0.5 ) ( 0.25, 0) y T x y = 0 1 1 0 x y (0, 1) (0, 1) x

y ( 0, 0.5 ) ( 0.25, 0) What is the image of the unit square under the following matrix transformations? T x y = 0.25 0 0 0.5 x y T x y = 0 1 1 0 x y T x y = 0 1 1 0 x y y x (0, 1) (0, 1)

What is the image of the unit square under the following matrix transformations? y T x y = 0.25 0 0 0.5 x y ( 0, 0.5 ) ( 0.25, 0) y T x y = 0 1 1 0 x y (0, 1) (0, 1) x T x y = 0 1 1 0 x y (0, 1) y (1, 0) x

What are the matrices that transform the unit square into the following figures? y y y (0, 1) (0, 1) (1, 0) x (1, 0) x (0, 1) x Reflection across the x axis (0, 1) y Reflection across the y axis Reflection across the line y=-x ( 1, 0) (0, k) (1, 0) x Contraction in y

y y (0, 1) (1, 0) x (1, 0) x Reflection across the x axis (0, 1)

y y (0, 1) (1, 0) x (1, 0) x 1 0 0 1 Reflection across the x axis (0, 1)

y y (0, 1) (1, 0) x (1, 0) x 1 0 0 1 Reflection across the x axis (0, 1) y y (0, 1) (0, 1) (1, 0) Reflection across the y axis x (0, 1) x

y y (0, 1) (1, 0) x (1, 0) x 1 0 0 1 Reflection across the x axis (0, 1) y y (0, 1) (1, 0) x (0, 1) (0, 1) x 1 0 0 1 Reflection across the y axis

y y (0, 1) (0, k) (1, 0) x (1, 0) x Contraction in y

(0, 1) y (1, 0) x (0, k) y (1, 0) x 1 0 0 k Contraction in y

(0, 1) y (1, 0) x (0, k) y (1, 0) x 1 0 0 k Contraction in y (0, 1) y ( 1, 0) (1, 0) x Reflection across the line y=-x

(0, 1) y (1, 0) x (0, k) y (1, 0) x 1 0 0 k Contraction in y y (0, 1) (1, 0) x ( 1, 0) 0 1 1 0 Reflection across the line y=-x

A matrix transforms the unit square into a parallelogram whose sides are determined by the column vectors. The matrix a c b d transforms the unit square into the parallelogram with vertices (0,0) (a,c) (b,d) (a+b,c+d) (0 0) ( b, d ) ( a, c) ( a+b, c+d )

Composition

Composition Draw the result of the transformation that

Composition Draw the result of the transformation that Reflects the unit square through the line x=y and then

Composition Draw the result of the transformation that Reflects the unit square through the line x=y and then Reflects the result across the x axis

Composition Draw the result of the transformation that Reflects the unit square through the line x=y and then Reflects the result across the x axis (0, 1) y (1, 0) x

Composition Draw the result of the transformation that Reflects the unit square through the line x=y and then Reflects the result across the x axis y y (0, 1) (0, 1) (1, 0) x (1, 0) x

Composition Draw the result of the transformation that Reflects the unit square through the line x=y and then Reflects the result across the x axis y y y (0, 1) (0, 1) (1, 0) x (1, 0) x (1, 0) x (0, 1)

Composition

Composition reflection in line y=x reflection across x axis composition

Composition y y reflection in line y=x (0, 1) (1, 0) x (0, 1) (1, 0) x reflection across x axis composition

Composition y y reflection in line y=x reflection across x axis composition (0, 1) (0, 1) y (1, 0) (1, 0) x x (0, 1) y (1, 0) (1, 0) x x (0, 1)

Composition y y reflection in line y=x reflection across x axis composition (0, 1) (0, 1) y (1, 0) (1, 0) x x (0, 1) y (1, 0) (1, 0) x x 1 0 0 1 0 1 (0, 1) 0 1 1 0 1 0

Composition y y reflection in line y=x reflection across x axis composition (0, 1) (0, 1) y (1, 0) (1, 0) x x (0, 1) y (1, 0) (1, 0) x x 1 0 0 1 0 1 0 1 1 0 1 0 0 1 1 0 (0, 1)

Composition

Composition Draw the result of the transformation that

Composition Draw the result of the transformation that Reflects the unit square across the x axis and then

Composition Draw the result of the transformation that Reflects the unit square across the x axis and then Reflects the result through the line x=y

Composition Draw the result of the transformation that Reflects the unit square across the x axis and then Reflects the result through the line x=y (0, 1) y (1, 0) x

Composition Draw the result of the transformation that Reflects the unit square across the x axis and then Reflects the result through the line x=y (0, 1) y y (1, 0) x (1, 0) x (0, 1)

Composition Draw the result of the transformation that Reflects the unit square across the x axis and then Reflects the result through the line x=y (0, 1) y y (1, 0) x (1, 0) x (-1,0) (0, 1)

Composition Draw the result of the transformation that Reflects the unit square across the x axis and then Reflects the result through the line x=y (0, 1) y y (1, 0) x (1, 0) x (-1,0) (0, 1) What is the matrix that performs this transformation?

Composition

Composition reflection across x axis reflection in line y=x composition

Composition y y reflection across x axis (0, 1) (1, 0) x (1, 0) x reflection in line y=x (0, 1) composition

Composition y y reflection across x axis (0, 1) (1, 0) x (1, 0) x reflection in line y=x composition y (0, 1) (1, 0) x (-1,0) (0, 1)

Composition y y reflection across x axis (0, 1) (1, 0) x (1, 0) x reflection in line y=x composition y (0, 1) 1 0 1 0 0 1 0 1 0 1 1 0 (0, 1) (1, 0) x (-1,0)

Composition y y reflection across x axis (0, 1) (1, 0) x (1, 0) x reflection in line y=x composition y (0, 1) 1 0 1 0 0 1 0 1 0 1 1 0 (0, 1) (1, 0) x (-1,0) 0 1 1 0

Composition as Matrix Multiplication

Composition as Matrix Multiplication Reflect the unit square through the line x=y and then reflect the result across the x axis Reflect the unit square across the x axis and the reflect the result through the line x=y

Composition as Matrix Multiplication Reflect the unit square through the line x=y and then reflect the result across the x axis 1 0 0 1 0 1 1 0 = 0 1 1 0 Reflect the unit square across the x axis and the reflect the result through the line x=y

Inverses

Inverses What is the matrix that takes the unit square to the parallelogram?

Inverses What is the matrix that takes the unit square to the parallelogram? What is the matrix that takes the parallelogram to the unit square?

Inverses What is the matrix that takes the unit square to the parallelogram? What is the matrix that takes the parallelogram to the unit square? (0, 1) y (6, 6) (2, 4) (4, 2) (1, 0) x (0 0)

Inverses What is the matrix that takes the unit square to the parallelogram? (0, 1) y (6, 6) (2, 4) (4, 2) (1, 0) x (0 0) What is the matrix that takes the parallelogram to the unit square?

Inverses What is the matrix that takes the unit square to the parallelogram? (0, 1) y 4 2 2 4 (2, 4) (4, 2) (6, 6) (1, 0) x (0 0) What is the matrix that takes the parallelogram to the unit square?

Inverses What is the matrix that takes the unit square to the parallelogram? (0, 1) y (1, 0) x 1 12 4 2 2 4 4 2 2 4 (2, 4) (0 0) (4, 2) (6, 6) What is the matrix that takes the parallelogram to the unit square?

The matrix a c b d ( b, d ) transforms the unit square into the (0 0) parallelogram with vertices (0,0) (a,c) (b,d) (a+b,c+d) ( a, c) ( a+b, c+d )

The matrix a c b d ( b, d ) transforms the unit square into the (0 0) parallelogram with vertices (0,0) (a,c) (b,d) (a+b,c+d) The matrix 1 ad bc d c b a transforms the parallelogram back into the unit square ( a, c) ( a+b, c+d )

The matrix 2 3 1 4 Determinants transforms the unit square to the parallelogram shown.

The matrix 2 3 1 4 Determinants transforms the unit square to the parallelogram shown. What is the area of the parallelogram?

The matrix 2 3 1 4 Determinants transforms the unit square to the parallelogram shown. What is the area of the parallelogram? Area = 5

The magnification factor of 5 area of the image region area of the original region remains the same for all geometric figures transformed by the matrix 2 3 1 4

The matrix a c b d transforms the unit square into the parallelogram with (0 0) ( b, d ) vertices (0,0) (a,c) (b,d) (a+b,c+d) ( a, c) ( a+b, c+d )

The matrix a c b d transforms the unit square into the parallelogram with (0 0) ( b, d ) vertices (0,0) (a,c) (b,d) (a+b,c+d) ( a, c) ( a+b, c+d ) Its area is...

The matrix a c b d transforms the unit square into the parallelogram with (0 0) ( b, d ) vertices (0,0) (a,c) (b,d) (a+b,c+d) ( a, c) ( a+b, c+d ) Its area is... ad-bc

The matrix a c b d transforms the unit square into the parallelogram with (0 0) ( b, d ) vertices (0,0) (a,c) (b,d) (a+b,c+d) ( a, c) ( a+b, c+d ) Its area is... ad-bc which is the absolute value of the determinant of the matrix.

If we have an affine transformation T x y = a b c d x y + e f and transform a region S in the plane the area of the transformed region T(S) is the area of S times ad-bc

If we have an affine transformation T x y = a b c d x y + e f and transform a region S in the plane the area of the transformed region T(S) is the area of S times ad-bc The translation vector does not affect the area

Eigenvalues and Eigenvectors The matrix 1 1 0 2 sends the unit square to the parallelogram shown with area of 2.

There is a better way of viewing the area relationships...

There is a better way of viewing the area relationships... Under 1 1 0 2

Under 1 1 0 2 1 0 1 0 1 1 2 2 There is a better way of viewing the area relationships...

The matrix 3 1 1 3 sends the unit square to the parallelogram shown.

Again there is a better way of viewing this

Again there is a better way of viewing this Under 3 1 1 3

Again there is a better way of viewing this Under 3 1 1 3 1 1 2 2 = 2 1 1 1 1 4 4 = 4 1 1

Again there is a better way of viewing this Under 3 1 1 3 1 1 2 2 = 2 1 1 1 1 4 4 = 4 1 1 With this view, the scaling relationships are clear.

Eigenvectors are vectors that point in the same (or opposite) direction before and after multiplication by the matrix.

Eigenvectors are vectors that point in the same (or opposite) direction before and after multiplication by the matrix. Multiplication may change the length of the vector

Eigenvectors are vectors that point in the same (or opposite) direction before and after multiplication by the matrix. Multiplication may change the length of the vector The multiplication factors associated with each eigenvector is the eigenvalue for that eigenvector.

For 1 1 0 2 1 0 1 0 1 1 2 2

For 1 1 0 2 1 0 1 0 1 1 2 2 is an eigenvector with eigenvalue 1 1 0

For 1 1 0 2 1 0 1 0 1 1 2 2 is an eigenvector with eigenvalue 1 1 0 is an eigenvector with eigenvalue 2 1 1

For 3 1 1 3 1 1 2 2 = 2 1 1 1 4 1 4 = 4 1 1

For 3 1 1 3 1 1 2 2 = 2 1 1 1 4 1 4 = 4 1 1 1 1 is an eigenvector with eigenvalue 2

For 3 1 1 3 1 1 2 2 = 2 1 1 1 4 1 4 = 4 1 1 1 1 1 1 is an eigenvector with eigenvalue 2 is an eigenvector with eigenvalue 4

Rotation almost always implies no real eigenvalues

Rotation almost always implies no real eigenvalues Are there rotations with real eigenvalues?

Rotation almost always implies no real eigenvalues Are there rotations with real eigenvalues? Shrinking or stretching without rotating or flipping means there are two or there are infinitely many eigenvectors Flipping gives both a positive and negative eigenvalue and two sets of eigenvectors

Fractal Dimension An object of Dimension D can be thought of as composed of n copies, each copy scaled down by a factor of r

Fractal Dimension An object of Dimension D can be thought of as composed of n copies, each copy scaled down by a factor of r A 1 inch square is made up of 4 half inch squares

Fractal Dimension An object of Dimension D can be thought of as composed of n copies, each copy scaled down by a factor of r A 1 inch square is made up of 4 half inch squares A 1 inch cube is made up of 64 quarter inch cubes

Dimension of Sierpinski Triangle The Sierpinski Triangle is composed of 3 half-sized version of itself It s fractal dimension is D = log3 log( 1 ) = log3 2 log2 1.584

Transformations of the form r cosθ rsinθ r sinθ rcosθ rcosθ r sinθ rsinθ r cosθ map the unit square to an rxr square Transformations of the form a b b a a b b a map the unit square to an rxr square where r = det M

Transformations of the form a b b a a b b a are called similitudes A fractal that is generated by n similitudes with scale factors r1,r2,r3,...,rn has dimension D where 1 = r 1 D + r 2 D + r 3 D + + r n D

Final Project

Final Project What are the matrices that transform the unit square to...

Acknowledgments Annalisa Crannell (Franklin and Marshall College) Viewpoints (Workshop on Mathematics and Art) Peter Van Roy: Fractasketch http://www.info.ucl.ac.be/people/pvr/fracta.html Ron Kneusel: Fractal Lab Kit http://archives.math.utk.edu/software/mac/fractals/fractallabkit/ Denvir Consultancy: Fractal Generator http://www.denvirconsultancy.com/marshall/fractalgenerator.html

Fractals and Linear Algebra. MAA Indiana Spring Meeting 2004