1. 1 Dr. Scott Schaefer Fractals and Iterated Affine Transformations

2. 2/193 What are Fractals?

3. 3/193 What are Fractals?

4. 4/193 What are Fractals?

5. 5/193 What are Fractals?

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

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

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

9. 9/193 Hausdorff Distance

10. 10/193 Hausdorff Distance

11. 11/193 Hausdorff Distance

12. 12/193 Hausdorff Distance

13. 13/193 Hausdorff Distance

14. 14/193 Hausdorff Distance

15. 15/193 Hausdorff Distance

16. 16/193 Hausdorff Distance

17. 17/193 Hausdorff Distance

18. 18/193 Hausdorff Distance

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

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

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

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

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

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

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

26. 26/193 Rendering Fractals

27. 27/193 Rendering Fractals

28. 28/193 Rendering Fractals

29. 29/193 Rendering Fractals

30. 30/193 Rendering Fractals

31. 31/193 Rendering Fractals

32. 32/193 Rendering Fractals

33. 33/193 Rendering Fractals

34. 34/193 Rendering Fractals

35. 35/193 Rendering Fractals

36. 36/193 Rendering Fractals

37. 37/193 Rendering Fractals

38. 38/193 Rendering Fractals

39. 39/193 Rendering Fractals

40. 40/193 Rendering Fractals

41. 41/193 Rendering Fractals

42. 42/193 Rendering Fractals

43. 43/193 Rendering Fractals

44. 44/193 Rendering Fractals

45. 45/193 Rendering Fractals

46. 46/193 Rendering Fractals

47. 47/193 Rendering Fractals

48. 48/193 Rendering Fractals

49. 49/193 Rendering Fractals

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

51. 51/193 Rendering Fractals

52. 52/193 Rendering Fractals

53. 53/193 Rendering Fractals

54. 54/193 Rendering Fractals

55. 55/193 Rendering Fractals

56. 56/193 Rendering Fractals

57. 57/193 Rendering Fractals

58. 58/193 Rendering Fractals

59. 59/193 Rendering Fractals

60. 60/193 Rendering Fractals

61. 61/193 Rendering Fractals

62. 62/193 Rendering Fractals

63. 63/193 Rendering Fractals

64. 64/193 Rendering Fractals

65. 65/193 Finding Transformations

66. 66/193 Finding Transformations

67. 67/193 Finding Transformations

68. 68/193 Finding Transformations

69. 69/193 Finding Transformations

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

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

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

73. 73/193 Fractal Tennis – Example 25,000 Points

74. 74/193 Fractal Tennis – Example 50,000 Points

75. 75/193 Fractal Tennis – Example 75,000 Points

76. 76/193 Fractal Tennis – Example 100,000 Points

77. 77/193 Fractal Tennis – Example 125,000 Points

78. 78/193 Fractal Tennis – Example 150,000 Points

79. 79/193 Fractal Tennis – Example 175,000 Points

80. 80/193 Fractal Tennis – Example 200,000 Points

81. 81/193 Modified Fractal Tennis

82. 82/193 Modified Fractal Tennis 25,000 Points

83. 83/193 Modified Fractal Tennis 50,000 Points

84. 84/193 Modified Fractal Tennis 75,000 Points

85. 85/193 Modified Fractal Tennis 100,000 Points

86. 86/193 Modified Fractal Tennis 125,000 Points

87. 87/193 Modified Fractal Tennis 150,000 Points

88. 88/193 Modified Fractal Tennis 175,000 Points

89. 89/193 Modified Fractal Tennis 200,000 Points

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

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

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

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

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

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

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

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

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

99. 99/193 Condensation Sets Given a condensation set C

100. 100/193 Condensation Set Example

101. 101/193 Condensation Sets – Example Condensation Set

102. 102/193 Condensation Sets – Example

103. 103/193 Condensation Sets – Example

104. 104/193 Condensation Sets – Example

105. 105/193 Condensation Sets – Example

106. 106/193 Condensation Sets – Example

107. 107/193 Condensation Sets – Example

108. 108/193 Condensation Sets – Example

109. 109/193 Condensation Sets – Example

110. 110/193 Condensation Sets – Example

111. 111/193 Condensation Sets – Example

112. 112/193 Condensation Sets – Example

113. 113/193 Condensation Sets – Example

114. 114/193 Condensation Sets – Example

115. 115/193 Condensation Sets – Example

116. 116/193 Condensation Sets – Example

117. 117/193 Condensation Sets – Example

118. 118/193 Condensation Sets – Example

119. 119/193 Condensation Sets – Example

120. 120/193 Condensation Sets – Example

121. 121/193 Condensation Sets – Example

122. 122/193 Rendering Fractals

123. 123/193 Rendering Fractals

124. 124/193 Rendering Fractals

125. 125/193 Rendering Fractals

126. 126/193 Rendering Fractals

127. 127/193 Rendering Fractals

128. 128/193 Rendering Fractals

129. 129/193 Rendering Fractals

130. 130/193 Condensation Set Example

131. 131/193 Condensation Set Example

132. 132/193 Condensation Set Example

133. 133/193 Condensation Set Example

134. 134/193 Condensation Set Example

135. 135/193 Condensation Set Example

136. 136/193 Condensation Set Example

137. 137/193 Condensation Set Example

138. 138/193 Condensation Set Example

139. 139/193 Condensation Set Example

140. 140/193 Condensation Set Example

141. 141/193 Fractal Dimension Fractal dimension =

142. 142/193 Fractal Dimension Fractal dimension =

143. 143/193 Fractal Dimension Fractal dimension =

144. 144/193 Fractal Dimension Fractal dimension =

145. 145/193 Fractal Dimension Fractal dimension =

146. 146/193 Fractal Dimension Fractal dimension =

147. 147/193 Fractal Dimension Fractal dimension =

148. 148/193 Fractal Dimension Fractal dimension =

149. 149/193 Fractal Dimension Fractal dimension =

150. 150/193 Fractal Dimension Fractal dimension =

151. 151/193 Fractal Dimension Fractal dimension =

152. 152/193 Fractal Dimension Fractal dimension =

153. 153/193 Fractal Dimension Fractal dimension =

154. 154/193 Fractal Dimension Fractal dimension =

155. 155/193 Fractal Dimension Fractal dimension =

156. 156/193 Fractal Dimension Fractal dimension =

157. 157/193 Fractal Dimension Fractal dimension =

158. 158/193 Fractal Dimension Fractal dimension =

159. 159/193 Fractal Dimension Fractal dimension =

160. 160/193 Fractal Dimension Fractal dimension =

161. 161/193 Fractal Dimension Fractal dimension =

162. 162/193 Fractal Dimension Fractal dimension =

163. 163/193 Fractal Dimension Fractal dimension =

164. 164/193 Fractal Dimension Fractal dimension =

165. 165/193 Fractal Dimension Fractal dimension =

166. 166/193 Fractal Dimension Fractal dimension =

167. 167/193 Fractal Dimension Fractal dimension =

168. 168/193 Fractal Dimension Fractal dimension =

169. 169/193 Fractal Dimension Fractal dimension =

170. 170/193 Fractal Dimension Fractal dimension =

171. 171/193 Fractal Dimension Fractal dimension =

172. 172/193 Fractal Dimension Fractal dimension =

173. 173/193 Fractal Dimension Fractal dimension =

174. 174/193 Fractal Dimension Fractal dimension =

175. 175/193 Fractal Dimension Fractal dimension =

176. 176/193 Fractal Dimension Fractal dimension =

177. 177/193 Fractal Dimension Fractal dimension =

178. 178/193 Fractal Dimension Fractal dimension =

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

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

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

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

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

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

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

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

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

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

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

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

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

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

193. 193/193 Weird Fractal Properties