Fractals and Iterated Affine Transformations Dr Scott Schaefer

Fractals and Iterated Affine Transformations Dr. Scott Schaefer 1

What are Fractals? 2/193

What are Fractals? 3/193

What are Fractals? 4/193

What are Fractals? 5/193

What are Fractals? n Recursion made visible n A self-similar shape created from a set of contractive transformations 6/193

Contractive Transformations n A transformation F(X) is contractive if, for all compact sets Transformations on sets? n Distance between sets? n 7/193

Transformations on Sets n Given a transformation F(x), n In other words, F(X) means apply the transformation to each point in the set X 8/193

Hausdorff Distance 9/193

Hausdorff Distance 10/193

Hausdorff Distance 11/193

Hausdorff Distance 12/193

Hausdorff Distance 13/193

Hausdorff Distance 14/193

Hausdorff Distance 15/193

Hausdorff Distance 16/193

Hausdorff Distance 17/193

Hausdorff Distance 18/193

Contractive Transformations n A transformation F(X) is contractive if, for all compact sets 19/193

Iterated Affine Transformations n Special class of fractals where each transformation is an affine transformation 20/193

Rendering Fractals Given starting set X 0 n Attractor is 21/193

Rendering Fractals Given starting set X 0 n Attractor is 22/193

Rendering Fractals Given starting set X 0 n Attractor is 23/193

Rendering Fractals Given starting set X 0 n Attractor is 24/193

Rendering Fractals Given starting set X 0 n Attractor is 25/193

Rendering Fractals 26/193

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Contractive Transformations n A transformation F(X) is contractive if, for all compact sets n A set of transformations has a unique attractor if all transformations are contractive u That attractor is independent of the starting shape!!! 50/193

Rendering Fractals 51/193

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Rendering Fractals 64/193

Finding Transformations 65/193

Finding Transformations 66/193

Finding Transformations 67/193

Finding Transformations 68/193

Finding Transformations 69/193

Fractal Tennis Start with any point x For ( i=1; i<100; i++ ) x = Mrandom*x For ( i=1; i<100000; i++ ) draw(x) x = Mrandom*x 70/193

Fractal Tennis Start with any point x For ( i=1; i<100; i++ ) x = Mrandom*x For ( i=1; i<100000; i++ ) draw(x) x = Mrandom*x Gets a point on the fractal 71/193

Fractal Tennis Start with any point x For ( i=1; i<100; i++ ) x = Mrandom*x For ( i=1; i<100000; i++ ) draw(x) x = Mrandom*x Creates new points on the fractal 72/193

Fractal Tennis – Example 25, 000 Points 73/193

Fractal Tennis – Example 50, 000 Points 74/193

Fractal Tennis – Example 75, 000 Points 75/193

Fractal Tennis – Example 100, 000 Points 76/193

Fractal Tennis – Example 125, 000 Points 77/193

Fractal Tennis – Example 150, 000 Points 78/193

Fractal Tennis – Example 175, 000 Points 79/193

Fractal Tennis – Example 200, 000 Points 80/193

Modified Fractal Tennis 81/193

Modified Fractal Tennis 25, 000 Points 82/193

Modified Fractal Tennis 50, 000 Points 83/193

Modified Fractal Tennis 75, 000 Points 84/193

Modified Fractal Tennis 100, 000 Points 85/193

Modified Fractal Tennis 125, 000 Points 86/193

Modified Fractal Tennis 150, 000 Points 87/193

Modified Fractal Tennis 175, 000 Points 88/193

Modified Fractal Tennis 200, 000 Points 89/193

Modified Fractal Tennis n Weight probabilities based on area 200, 000 Points 90/193

Modified Fractal Tennis n Weight probabilities based on area 25, 000 Points 91/193

Modified Fractal Tennis n Weight probabilities based on area 50, 000 Points 92/193

Modified Fractal Tennis n Weight probabilities based on area 75, 000 Points 93/193

Modified Fractal Tennis n Weight probabilities based on area 100, 000 Points 94/193

Modified Fractal Tennis n Weight probabilities based on area 125, 000 Points 95/193

Modified Fractal Tennis n Weight probabilities based on area 150, 000 Points 96/193

Modified Fractal Tennis n Weight probabilities based on area 175, 000 Points 97/193

Modified Fractal Tennis n Weight probabilities based on area 200, 000 Points 98/193

Condensation Sets n Given a condensation set C 99/193

Condensation Set Example 100/193

Condensation Sets – Example Condensation Set 101/193

Condensation Sets – Example 102/193

Condensation Sets – Example 103/193

Condensation Sets – Example 104/193

Condensation Sets – Example 105/193

Condensation Sets – Example 106/193

Condensation Sets – Example 107/193

Condensation Sets – Example 108/193

Condensation Sets – Example 109/193

Condensation Sets – Example 110/193

Condensation Sets – Example 111/193

Condensation Sets – Example 112/193

Condensation Sets – Example 113/193

Condensation Sets – Example 114/193

Condensation Sets – Example 115/193

Condensation Sets – Example 116/193

Condensation Sets – Example 117/193

Condensation Sets – Example 118/193

Condensation Sets – Example 119/193

Condensation Sets – Example 120/193

Condensation Sets – Example 121/193

Rendering Fractals 122/193

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Condensation Set Example 130/193

Condensation Set Example 131/193

Condensation Set Example 132/193

Condensation Set Example 133/193

Condensation Set Example 134/193

Condensation Set Example 135/193

Condensation Set Example 136/193

Condensation Set Example 137/193

Condensation Set Example 138/193

Condensation Set Example 139/193

Condensation Set Example 140/193

Fractal Dimension n Fractal dimension = 141/193

Fractal Dimension n Fractal dimension = 142/193

Fractal Dimension n Fractal dimension = 143/193

Fractal Dimension n Fractal dimension = 144/193

Fractal Dimension n Fractal dimension = 145/193

Fractal Dimension n Fractal dimension = 146/193

Fractal Dimension n Fractal dimension = 147/193

Fractal Dimension n Fractal dimension = 148/193

Fractal Dimension n Fractal dimension = 149/193

Fractal Dimension n Fractal dimension = 150/193

Fractal Dimension n Fractal dimension = 151/193

Fractal Dimension n Fractal dimension = 152/193

Fractal Dimension n Fractal dimension = 153/193

Fractal Dimension n Fractal dimension = 154/193

Fractal Dimension n Fractal dimension = 155/193

Fractal Dimension n Fractal dimension = 156/193

Fractal Dimension n Fractal dimension = 157/193

Fractal Dimension n Fractal dimension = 158/193

Fractal Dimension n Fractal dimension = 159/193

Fractal Dimension n Fractal dimension = 160/193

Fractal Dimension n Fractal dimension = 161/193

Fractal Dimension n Fractal dimension = 162/193

Fractal Dimension n Fractal dimension = 163/193

Fractal Dimension n Fractal dimension = 164/193

Fractal Dimension n Fractal dimension = 165/193

Fractal Dimension n Fractal dimension = 166/193

Fractal Dimension n Fractal dimension = 167/193

Fractal Dimension n Fractal dimension = 168/193

Fractal Dimension n Fractal dimension = 169/193

Fractal Dimension n Fractal dimension = 170/193

Fractal Dimension n Fractal dimension = 171/193

Fractal Dimension n Fractal dimension = 172/193

Fractal Dimension n Fractal dimension = 173/193

Fractal Dimension n Fractal dimension = 174/193

Fractal Dimension n Fractal dimension = 175/193

Fractal Dimension n Fractal dimension = 176/193

Fractal Dimension n Fractal dimension = 177/193

Fractal Dimension n Fractal dimension = 178/193

Weird Fractal Properties n Fractal curves can have infinite length!!! 179/193

Weird Fractal Properties n Fractal curves can have infinite length!!! 180/193

Weird Fractal Properties n Fractal curves can have infinite length!!! 181/193

Weird Fractal Properties n Fractal curves can have infinite length!!! 182/193

Weird Fractal Properties n Fractal curves can have infinite length!!! 183/193

Weird Fractal Properties n Fractal curves can have infinite length!!! 184/193

Weird Fractal Properties n Fractal curves can have infinite length!!! 185/193

Weird Fractal Properties Fractal curves can have infinite length!!! n But only enclose finite area? n 186/193

Weird Fractal Properties Fractal curves can have infinite length!!! n But only enclose finite area? n 187/193

Weird Fractal Properties Fractal curves can have infinite length!!! n But only enclose finite area? n 188/193

Weird Fractal Properties Fractal curves can have infinite length!!! n But only enclose finite area? n 189/193

Weird Fractal Properties Fractal curves can have infinite length!!! n But only enclose finite area? n 190/193

Weird Fractal Properties Fractal curves can have infinite length!!! n But only enclose finite area? n 191/193

Weird Fractal Properties Fractal curves can have infinite length!!! n But only enclose finite area? n 192/193

Weird Fractal Properties 193/193