Fractals. SFU-CMS Math Camp 2008 Randall Pyke;

Size: px

Start display at page:

Download "Fractals. SFU-CMS Math Camp 2008 Randall Pyke;"

Kelley Cain
5 years ago
Views:

1 Fractals SFU-CMS Math Camp 2008 Randall Pyke;

2 Benoit Mandelbrot, 1977 How long is the coast of Britain?

3 How long is the coast of Britain?

4 How long is the coast of Britain?

5 How long is the coast of Britain? Coast gets longer and longer as ruler shrinks.... How long is the coast of Britain (or any coast)?

6 Regular curves

7 measured length coast line circle

8 Zooming in on regular curves

9 Zooming in on regular curves All regular curves look like straight lines if you zoom in enough (and that's why their measured length does not get arbitrarily large)

10 Zooming in on regular curves All regular curves look like straight lines if you zoom in enough (and that's why their measured length does not get arbitrarily large) But there are curves that don t straighten up as you get closer...

11 Zooming in on fractal curves

12 The measured length of fractal curves gets longer and longer because they never straighten out.

13 Regular (Euclidean) geometry

14 Fractal geometry....

19 Self-Similarity The whole fractal

20 Self-Similarity Pick a small copy of it

21 Self-Similarity

22 Self-Similarity Rotate it

23 Self-Similarity Blow it up: You get the same thing!

24 Self-Similarity Now continue with that piece

25 Self-Similarity You can find the same small piece on this small piece

26 Self-Similarity

27 Self-Similarity

29 Four self-similar pieces Infinitely many self-similar pieces!

30 Self-similar : Made up of smaller copies of itself

31 Other self-similar objects: Solid square Solid triangle These are self-similar, but not complicated Fractals are self-similar and complicated

32 Fractal dust Fractal curves Fractal areas Fractal volumes

33 How to draw fractals?

34 How to draw fractals? Here are three ways:

35 How to draw fractals? Here are three ways: - Removing pieces

36 How to draw fractals? Here are three ways: - Removing pieces - Adding pieces

37 How to draw fractals? Here are three ways: - Removing pieces - Adding pieces - The Chaos Game

38 Removing pieces

39 Removing pieces

40 Or in 3-D

42 Now you try.

43 Start with a triangle: Now you try

44 Now you try Start with a triangle: Partition it into 4 smaller triangles:

45 Now you try Remove sub-triangle 3:

46 Now you try Remove sub-triangle 3: Now continue...

47 Partition each remaining triangle into 4 smaller triangles

48 Partition each remaining triangle into 4 smaller triangles

49 Partition each remaining triangle into 4 smaller triangles Now remove each 3 triangle

57 Another type: Remove triangles 1 and 2

67 Shapes produced with the 4 triangle partition;

68 Sierpinski triangle (remove centre triangle): Shapes produced with the 4 triangle partition; Sierpinski variation (remove corner triangle):

69 Sierpinski triangle (remove centre triangle): Shapes produced with the 4 triangle partition; Sierpinski variation (remove corner triangle):

70 For more variety, begin with the partition of the triangle into 16 smaller triangles; How many different shapes can you make?

71 Can do the same with a square

72 What is the recipe for this one?

73 Remove all number 5 squares

74 What is the recipe for this image?

77 Remove all 4, 6, and 8 squares

78 Adding pieces

79 Finally..

80 Try this: Instead of this generator Use this one (all sides 1/3 long)

82 The Chaos Game

83 Playing the Chaos game to draw fractals

84 Playing the Chaos game to draw fractals

85 Playing the Chaos game to draw fractals

86 Playing the Chaos game to draw fractals

87 Playing the Chaos game to draw fractals

88 Playing the Chaos game to draw fractals

89 Playing the Chaos game to draw fractals

100

101

102

103

104

105

106

107

108

109

110

111 Why the Chaos Game Works

112 Why the Chaos Game Works Addresses:

113 Why the Chaos Game Works Addresses:

114 Why the Chaos Game Works Addresses:

115 Address length 1 Address length 2

116 Address length 3

117 Address length 3

118 Address length 3

119 Address length 3

120 Address length 3

121 Address length 3

122 Address length 3

123 Address length 3

124 Address length 3

125 Address length 3

126 Address length 3

127 Address length 3

128 Address length 4

129 Address length 4

130 Address length 4

131 Address length 4

132 Address length 4

133 Address length 4

134 Address length 4

135 Address length 4

136 Address length 4

137 Address length 4

138 What is Sierpinski s Triangle?

139 What is Sierpinski s Triangle? All regions without a 4 in their address:

140 What is Sierpinski s Triangle? All regions without a 4 in their address: So we can draw the Sierpinski triangle if we put a dot in every address region that doesn t have a 4

141 All fractals have an address system you can use to label parts of the fractal

142 All fractals have an address system you can use to label parts of the fractal

143 All fractals have an address system you can use to label parts of the fractal

144 All fractals have an address system you can use to label parts of the fractal

145 Let s begin the Sierpinski Chaos game at the bottom left corner

146 Let s begin the Sierpinski Chaos game at the bottom left corner Address of this point is

147 Let s begin the Sierpinski Chaos game at the bottom left corner Address of this point is

148 Let s begin the Sierpinski Chaos game at the bottom left corner Address of this point is

149 Suppose first game number is a 2;

150 Suppose first game number is a 2; Address of this game point is

151 Suppose first game number is a 2; Address of this game point is

152 And if the next game number is a 3;

153 And if the next game number is a 3; Address of this game point is

154 And if the next game number is a 3; Address of this game point is

155 And if the next game number is a 3; Address of this game point is

156 Next game number is a 2;

157 Next game number is a 2; Address of previous game point is

158 Next game number is a 2; Address of previous game point is Address of this game point is

159 Next game number is a 2; Address of previous game point is Address of this game point is

160 Next game number is a 2; Address of previous game point is Address of this game point is

161 Next game number is a 1;

162 Next game number is a 1; Address of previous game point is Address of this game point is

163 Next game number is a 1; Address of previous game point is Address of this game point is

164 Next game number is a 1; Address of previous game point is Address of this game point is

165 So, if the game numbers are s1, s2, s3,, sk,, the addresses of the game points are game point 1: s1.. game point 2: s2s1. game point 3: s3s2s1.. game point k: sk.s3s2s1.

166 So, if the game numbers are s1, s2, s3,, sk,, the addresses of the game points are game point 1: s1.. game point 2: s2s1. game point 3: s3s2s1.. game point k: sk.s3s2s1. Which means we can put a game point in every address region of the Sierpinski triangle if the game numbers produce every pattern of 1 s, 2 s, and 3 s.

167 Here s one sequence of game numbers that produces every pattern of 1 s, 2 s, and 3 s;

168 Here s one sequence of game numbers that produces every pattern of 1 s, 2 s, and 3 s; first, all patterns of length 1

169 Here s one sequence of game numbers that produces every pattern of 1 s, 2 s, and 3 s; then all patterns of length 2

170 Here s one sequence of game numbers that produces every pattern of 1 s, 2 s, and 3 s; now all patterns of length

171 This sequence will contain every pattern of 1 s, 2, and 3 s. So if we play the Sierpinski chaos game with this game sequence, the game points will cover all regions of the fractal.

172 Another sequence that contains all patterns is a random sequence

173 Another sequence that contains all patterns is a random sequence Choose each game number randomly, eg., roll a die;

174 Another sequence that contains all patterns is a random sequence Choose each game number randomly, eg., roll a die; if a 1 or 2 comes up, game number is 1

175 Another sequence that contains all patterns is a random sequence Choose each game number randomly, eg., roll a die; if a 1 or 2 comes up, game number is 1 if a 3 or 4 comes up, game number is 2

176 Another sequence that contains all patterns is a random sequence Choose each game number randomly, eg., roll a die; if a 1 or 2 comes up, game number is 1 if a 3 or 4 comes up, game number is 2 if a 5 or 6 comes up, game number is 3

177 Another sequence that contains all patterns is a random sequence Choose each game number randomly, eg., roll a die; if a 1 or 2 comes up, game number is 1 if a 3 or 4 comes up, game number is 2 if a 5 or 6 comes up, game number is 3 These game numbers will also draw the fractal.

178 Adjusting probabilities Suppose we randomly choose 1, 2, 3, but we choose 3 2/3 of the time and 1 and 2 1/6 of the time each (roll a die: if 1 comes up choose 1, if 2 comes up choose 2, if 3,4,5 or 6 come up choose 3)..

179

180 We can calculate how many game points land in a particular address region by calculating how often that address (in reverse) occurs in the game sequence. (Homework!)

181 Are equal probabilities always the best?

182 Fern Fractal Game rules; place 4 pins, choose 1,2,3,4 randomly. Actions;. If the numbers 1,2,3,4 are chosen equally often;

183

184 Fern Fractal Game rules; place pins, choose 1,2,3,4 randomly. Actions;. If the numbers 1,2,3,4 are chosen with frequencies; 1; 2% 2,3; 14% 4; 70%

185

186 What probabilities to use? Sometimes it is very difficult to determine the best probabilities to draw the fractal.

187 Another chaos game

188

189 What if action 4 was; move ½ distance towards pin 4 only (i.e., no rotation)

190

191 Is this a fractal? (self-similar?)

192 Is this a fractal? (self-similar?) Yes 4 pieces, but there is overlap

193 Is this a fractal? (self-similar?) Yes 4 pieces, but there is overlap

194 Is this a fractal? (self-similar?) Yes 4 pieces, but there is overlap

195 Is this a fractal? (self-similar?)

196 Pseudo Fractals

197

198 Remove all 12 s from game numbers..

199 Remove all 12 s from game numbers.. Remember, addresses of game points are the reverse of the game numbers. So here no game points land in areas whose address contains a 21

200 Remove all 12 s from game numbers.. Remember, addresses of game points are the reverse of the game numbers. So here no game points land in areas whose address contains a 21

201 Remove all 12 s from game numbers.. Remember, addresses of game points are the reverse of the game numbers. So here no game points land in areas whose address contains a 21

202 Note; this is not a fractal (is not self-similar)

203

204 Back to adjusting probabilities

205 Back to adjusting probabilities

206 Back to adjusting probabilities Equal probabilities

207

208

209

210 Unequal probabilities;

211

212 Unequal probabilities (again);

213

214 Playing the chaos game with non-random sequences

215 Playing the chaos game with non-random sequences Sierpinski game. Initial game point at bottom left corner.

216 Playing the chaos game with non-random sequences Sierpinski game. Initial game point at bottom left corner. Game numbers; (i.e., 123 repeating).

217 Playing the chaos game with non-random sequences Sierpinski game. Initial game point at bottom left corner. Game numbers; (i.e., 123 repeating). Outcome?

218

219

220

221

222

223

224

225

226

227

228

229

230

231

232

233

234

235

236

237

238

239

240

241

242

243

244

245

246

247

248

249 Looks like the game points are converging to 3 distinct points;

250 Looks like the game points are converging to 3 distinct points;

251 Looks like the game points are converging to 3 distinct points; Why?

252 Consider the 3 points with the following addresses;

253 Playing with game sequence , we see that C A B C etc. (add game number to left end of address of game point)

254 Playing with game sequence , we see that C A B C etc. (add game number to left end of address of game point) That is, the game points cycle through these 3 points in this order.

255 Playing with game sequence , we see that C A B C etc. (add game number to left end of address of game point) That is, the game points cycle through these 3 points in this order. No matter where the first game point is, the game points will end up at A, B, and C.

256 Random Fractals Random midpoints to define triangle decomposition Remove random triangle at each iteration

257 Random fractal curves

258 Using fractals to create real life images

259 Fractal Clouds

260 Fractal Clouds Decide the self-similar pieces

261 Fractal Clouds Decide the self-similar pieces Generate the fractal

262 now blur a bit Create a realistic cloud!

263 Real Mountains

264 Fractal Mountains

265 Fractals in Nature

266

267

268 Fractal Art

269

270

271

272

273

274

275 Julia sets G. Julia, P. Fatou ca 1920

276

277

278

279

280 This presentation: For more information: Fractals

Grade 6 Math Circles. Math Jeopardy

Grade 6 Math Circles. Math Jeopardy Faculty of Mathematics Waterloo, Ontario N2L 3G1 Introduction Grade 6 Math Circles November 28/29, 2017 Math Jeopardy Centre for Education in Mathematics and Computing This lessons covers all of the material