Fractals and Linear Algebra. MAA Indiana Spring Meeting 2004
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1 Fractals and Linear Algebra MAA Indiana Spring Meeting 2004
2 Creation of a Fractal
3 Fractal
4 Fractal An object composed of smaller copies of itself
5 Fractal An object composed of smaller copies of itself Iterated Function System
6 Fractal An object composed of smaller copies of itself Iterated Function System A process of repeatedly replacing shapes with other shapes
7 First Transformations T1 T 1 T2 T 2 T3 T 3
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9 Describe (in words) the action performed by T1, T2, and T3
10 Describe (in words) the action performed by T1, T2, and T3 Draw images of the following
11 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 )
12 Result
13 Result
14 Resulting Fractal
15 Resulting Fractal
16 Resulting Fractal
17 Resulting Fractal
18 Trickier Transformations
19 Trickier Transformations T3 T 3 T1 T 1 T 2 T2
20 T3 Compositions T 3 T1 T 1 T 2 T2
21 T3 Compositions T 3 T1 T 1 T 2 T2 T 1 T ( 1 I 2 ) P P
22 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
23 Draw images of the following ( ) ( ) T 3 T 1 I 2 T 3 T 2 I 2 T 2 T ( 1 I 2 )
24 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
25 Resulting Fractal
26 Resulting Fractal T 3 T 1 T 2
27 Resulting Fractal T 3 T 1 T 2
28 Resulting Fractal T 3 T 1 T 2
29 Resulting Fractal
30 Resulting Fractal T 3 T 1 T 2
31 Resulting Fractal T 3 T 1 T 2
32 Resulting Fractal T 3 T 1 T 2
33 Working Backwards
34 Working Backwards
35 Working Backwards
36 ?? Working Backwards
37 Consider the transformations shown to the right T1 T2 The figure below is the third iterate Identify T2 T1 T2 T2 T2 T2
38 Sierpinski(ish) Fractals Each of the following fractals was created using three transformations. Sketch the orientation of the P s
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46 Formulas What are the formulas for these three transformations? What are the formulas for these three transformations? T 3 T 1 T 2
47 Formulas
48 Formulas T ( x, y) = ( ax + by + e, cx + dy + e)
49 Formulas T ( x, y) = ( ax + by + e, cx + dy + e) T x y a b = c d x y + e f
50 Matrix Transformations
51 Matrix Transformations A transformation is a rule for mapping one space to another space.
52 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
53 What is the image of the unit square under the following matrix transformations?
54 What is the image of the unit square under the following matrix transformations? T x y = x y
55 What is the image of the unit square under the following matrix transformations? y T x y = x y ( 0, 0.5 ) ( 0.25, 0)
56 y ( 0, 0.5 ) ( 0.25, 0) What is the image of the unit square under the following matrix transformations? T x y = x y T x y = x y
57 What is the image of the unit square under the following matrix transformations? y T x y = x y ( 0, 0.5 ) ( 0.25, 0) y T x y = x y (0, 1) (0, 1) x
58 y ( 0, 0.5 ) ( 0.25, 0) What is the image of the unit square under the following matrix transformations? T x y = x y T x y = x y T x y = x y y x (0, 1) (0, 1)
59 What is the image of the unit square under the following matrix transformations? y T x y = x y ( 0, 0.5 ) ( 0.25, 0) y T x y = x y (0, 1) (0, 1) x T x y = x y (0, 1) y (1, 0) x
60 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
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62 y y (0, 1) (1, 0) x (1, 0) x Reflection across the x axis (0, 1)
63 y y (0, 1) (1, 0) x (1, 0) x Reflection across the x axis (0, 1)
64 y y (0, 1) (1, 0) x (1, 0) x Reflection across the x axis (0, 1) y y (0, 1) (0, 1) (1, 0) Reflection across the y axis x (0, 1) x
65 y y (0, 1) (1, 0) x (1, 0) x Reflection across the x axis (0, 1) y y (0, 1) (1, 0) x (0, 1) (0, 1) x Reflection across the y axis
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67 y y (0, 1) (0, k) (1, 0) x (1, 0) x Contraction in y
68 (0, 1) y (1, 0) x (0, k) y (1, 0) x k Contraction in y
69 (0, 1) y (1, 0) x (0, k) y (1, 0) x k Contraction in y (0, 1) y ( 1, 0) (1, 0) x Reflection across the line y=-x
70 (0, 1) y (1, 0) x (0, k) y (1, 0) x k Contraction in y y (0, 1) (1, 0) x ( 1, 0) Reflection across the line y=-x
71 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 )
72 Composition
73 Composition Draw the result of the transformation that
74 Composition Draw the result of the transformation that Reflects the unit square through the line x=y and then
75 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
76 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
77 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
78 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
79 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
80 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
81 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)
82 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) What is the matrix that performs this transformation?
83 Composition
84 Composition reflection in line y=x reflection across x axis composition
85 Composition y y reflection in line y=x (0, 1) (1, 0) x (0, 1) (1, 0) x reflection across x axis composition
86 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)
87 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)
88 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)
89 Composition
90 Composition Draw the result of the transformation that
91 Composition Draw the result of the transformation that Reflects the unit square across the x axis and then
92 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
93 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
94 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
95 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
96 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)
97 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)
98 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)
99 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?
100 Composition
101 Composition reflection across x axis reflection in line y=x composition
102 Composition y y reflection across x axis (0, 1) (1, 0) x (1, 0) x reflection in line y=x (0, 1) composition
103 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)
104 Composition y y reflection across x axis (0, 1) (1, 0) x (1, 0) x reflection in line y=x composition y (0, 1) (0, 1) (1, 0) x (-1,0)
105 Composition y y reflection across x axis (0, 1) (1, 0) x (1, 0) x reflection in line y=x composition y (0, 1) (0, 1) (1, 0) x (-1,0)
106 Composition as Matrix Multiplication
107 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
108 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
109 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 =
110 Inverses
111 Inverses What is the matrix that takes the unit square to the parallelogram?
112 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?
113 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)
114 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?
115 Inverses What is the matrix that takes the unit square to the parallelogram? (0, 1) y (2, 4) (4, 2) (6, 6) (1, 0) x (0 0) What is the matrix that takes the parallelogram to the unit square?
116 Inverses What is the matrix that takes the unit square to the parallelogram? (0, 1) y (1, 0) x (2, 4) (0 0) (4, 2) (6, 6) What is the matrix that takes the parallelogram to the unit square?
117 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 )
118 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 )
119 The matrix Determinants transforms the unit square to the parallelogram shown.
120 The matrix Determinants transforms the unit square to the parallelogram shown. What is the area of the parallelogram?
121 The matrix Determinants transforms the unit square to the parallelogram shown. What is the area of the parallelogram? Area = 5
122 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
123 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 )
124 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...
125 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
126 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.
127 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
128 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
129 Eigenvalues and Eigenvectors The matrix sends the unit square to the parallelogram shown with area of 2.
130 There is a better way of viewing the area relationships...
131 There is a better way of viewing the area relationships... Under
132 Under There is a better way of viewing the area relationships...
133 Under There is a better way of viewing the area relationships...
134 The matrix sends the unit square to the parallelogram shown.
135 Again there is a better way of viewing this
136 Again there is a better way of viewing this Under
137 Again there is a better way of viewing this Under = = 4 1 1
138 Again there is a better way of viewing this Under = = With this view, the scaling relationships are clear.
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140 Eigenvectors are vectors that point in the same (or opposite) direction before and after multiplication by the matrix.
141 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
142 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.
143 For
144 For is an eigenvector with eigenvalue 1 1 0
145 For is an eigenvector with eigenvalue is an eigenvector with eigenvalue 2 1 1
146 For = = 4 1 1
147 For = = is an eigenvector with eigenvalue 2
148 For = = is an eigenvector with eigenvalue 2 is an eigenvector with eigenvalue 4
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150 Rotation almost always implies no real eigenvalues
151 Rotation almost always implies no real eigenvalues Are there rotations with real eigenvalues?
152 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
153 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
154 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 Shear transformations correspond to a deficient eigenspace
155 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
156 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
157 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
158 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 n = r D
159 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 n = r D D = logn logr
160 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 log
161 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
162 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
163 Final Project
164 Final Project What are the matrices that transform the unit square to...
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177 Acknowledgments Annalisa Crannell (Franklin and Marshall College) Viewpoints (Workshop on Mathematics and Art) Peter Van Roy: Fractasketch Ron Kneusel: Fractal Lab Kit Denvir Consultancy: Fractal Generator
MATH 423 Linear Algebra II Lecture 20: Geometry of linear transformations. Eigenvalues and eigenvectors. Characteristic polynomial.
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