Fractals and the Chaos Game

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1 Math: Outside the box! Fractals and the Chaos Game Monday February 23, :30-4:20 IRMACS theatre, ASB Randall Pyke Senior Lecturer Department of Mathematics, SFU

2 A Game.

3 Is this a random walk? Far from it! For example, a game point will never land in this circle

45 Another chaos game

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

49 Regular (Euclidean) geometry

50 Fractal geometry....

55 Self-Similarity The whole fractal

56 Self-Similarity Pick a small copy of it

57 Self-Similarity

58 Self-Similarity Rotate it

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

60 Self-Similarity Now continue with that piece

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

62 Self-Similarity

63 Self-Similarity

65 Four self-similar pieces Infinitely many self-similar pieces

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

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

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

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

70 4 self-similar pieces

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

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

73 Fractal dust Fractal curves Fractal areas Fractal volumes

74 Fractals in Nature

81 Real Mountains

82 Fractal Mountains

83 Fractal Art

90 Technically speaking Fractals are obtained by iterating a certain kind of function beginning with an image (any image). The function is called an Iterated Function System (IFS) and is made up of (several) affine ( linear ) transformations. The IFS is a contraction on the space of all images and so (by the Contractive Mapping Theorem) has a unique fixed point which is the fractal. Each affine transformation making up the IFS determines the type of self-similarity of the fractal.

91 Blueprint: W(S), S = square

92 Blueprint: W(S), S = square 2 iterations; W(W(S))

93 Blueprint: W(S), S = square 2 iterations; W(W(S)) 3 iterations

94 2 iterations; W(W(S)) 3 iterations Blueprint: W(S), S = square 4 iterations

95 2 iterations; W(W(S)) 3 iterations Blueprint: W(S), S = square 4 iterations Fixed point F

98 How do we draw fractals?

99 Drawing fractals Here are four ways: - Iteration of an IFS ( decoding )

100 Drawing fractals Here are four ways: - Iteration of an IFS - Removing pieces

101 Drawing fractals Here are four ways: - Iteration of an IFS - Removing pieces - Adding pieces

102 Drawing fractals Here are four ways: - Iteration of an IFS - Removing pieces - Adding pieces - The Chaos Game

103 Removing pieces

104 Wikipedia; fractals

105 Removing pieces

106 Or in 3-D

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108 Now you try.

109 Start with a triangle: Now you try

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

111 Now you try Remove sub-triangle 3:

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

113 Partition each remaining triangle into 4 smaller triangles

114 Partition each remaining triangle into 4 smaller triangles

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

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123 Another type: Remove triangles 1 and 2

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133 Shapes produced with the 4 triangle partition;

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

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

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

137 Can do the same with a square

138 What is the recipe for this one?

139 Remove all number 5 squares

140 What is the recipe for this image?

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143 Remove all 4, 6, and 8 squares

144 Adding pieces

145 Finally..

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

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149 The Chaos Game

150 Playing the Chaos game to draw fractals

151 Playing the Chaos game to draw fractals

152 Playing the Chaos game to draw fractals

153 Playing the Chaos game to draw fractals

154 Playing the Chaos game to draw fractals

155 Playing the Chaos game to draw fractals

156 Playing the Chaos game to draw fractals

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158 Why the Chaos Game Works

159 Why the Chaos Game Works Addresses:

160 Why the Chaos Game Works Addresses:

161 Why the Chaos Game Works Addresses:

162 Address length 1 Address length 2

163 Address length 3

164 Address length 3

165 Address length 3

166 Address length 3

167 Address length 3

168 Address length 3

169 Address length 3

170 Address length 3

171 Address length 3

172 Address length 3

173 Address length 3

174 Address length 3

175 Address length 4

176 Address length 4

177 Address length 4

178 Address length 4

179 Address length 4

180 Address length 4

181 Address length 4

182 Address length 4

183 Address length 4

184 Address length 4

185 What is Sierpinski s Triangle?

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

187 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

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

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

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

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

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

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

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

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

196 Suppose first game number is a 2;

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

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

199 And if the next game number is a 3;

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

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

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

203 Next game number is a 2;

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

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

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

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

208 Next game number is a 1;

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

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

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

212 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.

213 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. (Addresses of points are reversed from sequence) 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.

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

215 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

216 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

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

218 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.

219 Another sequence that contains all patterns is a random sequence

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

221 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

222 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

223 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

224 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.

225 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;

226 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);

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228 More points will land in regions with a 3 in their address

229 Are equal probabilities always the best?

230 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;

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232 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%

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234 What probabilities to use? Sometimes it is very difficult to determine the best probabilities to draw the fractal. But it is easy to calculate pretty good probabilities (you want the density of points to be equal in address regions of the same address length).

235 Pseudo Fractals

236

237 Remove all 12 s from game numbers..

238 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

239 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

240 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

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

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243 Back to adjusting probabilities

244 Back to adjusting probabilities

245 Back to adjusting probabilities Equal probabilities

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249 Unequal probabilities;

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251 1 Game sequence: 2 or /2 2

252 1 22 or 33 or 23 or /4 2

253 1 21 or 24 or 31 or /4 2

254 1 21 or 24 or 31 or /8 2

255 1 211 or 214 or 241 or 244 or /8 2

256 1 211 or 214 or 241 or 244 or /8 9/16

257 1 211 or 214 or 241 or 244 or /8 9/ , 2112, 2143, 2144,...

258 1 Why so few points in this region? 3 4 2

259 1 Why so few points in this region? 3 4 Because points must first go all the way here before going back. 2

260 We can estimate the relative intensities of the lines P= P=0.1 P= P=0.1

261 Assume a line of initial game points here P= P=0.1 P= P=0.1

262 Assume a line of initial game points here P= P=0.1 P= P=0.1 Then we play the chaos game for each of these points.

263 P=0.4 1 Game sequence: 2 or 3 3 P=0.1 P=0.4 4 P = =0.2 2 P=0.1

264 P=0.4 1 Game sequence: 2 or 3 3 P=0.1 P=0.4 4 P = =0.2 This line is (approximately) 20% as dark as the initial line 2 P=0.1

265 P= or 33 or 23 or 32 3 P=0.1 P=0.4 4 P = 4 (0.1)*(0.1)= P=0.1

266 P= or 24 or 31 or 34 3 P=0.1 P=0.4 4 P = 4 (0.1)*(0.4)= P=0.1

267 Unequal probabilities (again);

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271 Assign 0 probability to 3

272 Using other sequences for the chaos game random sequences non-random, naïve sequences the best sequences

273 What is the shortest game sequence needed to draw the fractal? That is, what is the most efficient game sequence?

274 Naïve sequence for Sierpinski (all addresses of length 3) Addresses of game points ( * are redundant!);

275 Left with the 27 distinct addresses of length 3.

276 Left with the 27 distinct addresses of length 3. The best sequence;

277 So the naïve sequence is not very efficient (lots of redundancy). How efficient are random sequences? We know that, in order of decreasing redundancy; naïve sequence < random sequence < best sequence By experiment we can quantify redundancy in a sequence. The experiment is to draw the Sierpinski fractal to the resolution of the computer screen (level 11, say). The naïve sequence needs to be 3^11*(11) ~ 2x10^6 digits long, while the best sequence will be 3^11 ~ 177,000 digits long. We find that to draw the fractal to this level of detail requires a random sequence with ~ 300,000 digits. Redundancy r = length of sequence/length of best sequence; r(naïve) = 11, r(random) = 1.7

278 Playing the chaos game with non-random sequences

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

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

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

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

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

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

316 Call these 3 points A, B, and C. Let denote the address of point A.

317 Call these 3 points A, B, and C. Let denote the address of point A. Now play the game with the game numbers

318 Call these 3 points A, B, and C. Let denote the address of point A. Now play the game with the game numbers Address of point B is

319 Call these 3 points A, B, and C. Let denote the address of point A. Now play the game with the game numbers Address of point B is 1...

320 Call these 3 points A, B, and C. Let denote the address of point A. Now play the game with the game numbers Address of point B is 1... Address of point C is

321 Call these 3 points A, B, and C. Let denote the address of point A. Now play the game with the game numbers Address of point B is 1... Address of point C is 21...

322 Call these 3 points A, B, and C. Let denote the address of point A. Now play the game with the game numbers Address of point B is 1... Address of point C is Address of point A then is

323 Call these 3 points A, B, and C. Let denote the address of point A. Now play the game with the game numbers Address of point B is 1... Address of point C is Address of point A then is

324 Call these 3 points A, B, and C. Let denote the address of point A. Now play the game with the game numbers Address of point B is 1... Address of point C is Address of point A then is =...

325 So the address of point A must be And so address of B must be And the address of point C must be

326 So the address of point A must be And so address of B must be And the address of point C must be The game cycles through these points in this order.

327 Testing sequences for randomness

328 For sequences of 1,..., 9 Move 8/9 distance towards pin #k

329 Addresses

330 All digits occurring equally likely

331 What are the probabilities for this sequence?

332 More points..

333 Addresses

334 Looks like p3 is too low.

335 Testing sequences for randomness 10,000 random 10,000 digits of Pi (digits 1,..., 9)

336 This is how we define a sequence of numbers, each in [0,1] The histogram (distribution) of these numbers

337 Graphical iteration

338 Graphical iteration x1

339 Graphical iteration x1

340 Graphical iteration x1 x2

341 Graphical iteration x1 x2

342 Graphical iteration x1 x2 x3

343 Graphical iteration x1 x2 x3

344 Graphical iteration x1 x4 x2 x3

345 Graphical iteration x1 x4 x2 x3

346 Graphical iteration x1 x4 x2 x3 x5

347 Graphical iteration x1 x4 x2 x3 x5

348 Creating a sequence of digits 1,2,3 using the logistic equation

349 Creating a sequence of digits 1,2,3 using the logistic equation 1 2 3

350 Logistic sequences; 1,, 9

351 Logistic equation

352 Logistic equation

353 Logistic equation

354 Logistic equation x1

355 Logistic equation x1 x2

356 Logistic equation x1 Never a 34, but 38 ok x2

357 Addresses

358 Logistic sequences; 1,, 9

359 Order matters for randomness! This could be the histogram for a random sequence...

360 But there is still hope.. The logistic equation creates a chaotic system; the movement of the points x1, x2,... is chaotic ( effectively unpredictable ). An important property of chaos is mixing ; any small interval of [0,1] will eventually be spread throughout all of [0,1]. We use this property to create random sequences.

361 Small initial interval Mixing

362 Logistic sequences Delay = 1 Delay = 2 x2 = f(x1) x2 = f(f(x1))

363 Logistic sequences Delay = 3 Delay = 4

364 Logistic sequences Delay = 6 Random

365 Using the chaos game in bioinformatics

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369 Random Fractals (randomize the actions) Random midpoints to define triangle decomposition Remove random triangle at each iteration

370 Random fractal curves

371 Using fractals to create real life images

372 Fractal Clouds

373 Fractal Clouds Decide the self-similar pieces

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

375 now blur a bit Create a realistic cloud!

376 Creating fractal mountains..

377 Fractal Mountains

378 Fractal image compression

379 Fractal image compression Recall the IFS.

380 Fractal image compression A fractal can be reconstructed by playing the chaos game (or iteration of the IFS) One only needs the game rules (or the affine transformations of the IFS) Equations need only ~ 10 s of KB Images need ~1,000 of KB!

381 Many fractals can be made to look like real-life images. Using a real image as a guide to finding an appropriate IFS Encoding an IFS

382 and many real life images look like (many) fractals

383 Determine which small pieces of the image can be used to re-create other pieces of the image. Collect all these IFS s into a multi IFS that will re-create the entire image.

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

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389 For more information: This presentation: More info: Fractals

Hei nz-ottopeitgen. Hartmut Jürgens Dietmar Sau pe. Chaos and Fractals. New Frontiers of Science

Hei nz-ottopeitgen. Hartmut Jürgens Dietmar Sau pe. Chaos and Fractals. New Frontiers of Science Hei nz-ottopeitgen Hartmut Jürgens Dietmar Sau pe Chaos and Fractals New Frontiers of Science Preface Authors VU X I Foreword 1 Mitchell J. Feigenbaum Introduction: Causality Principle, Deterministic