Fractal Dimension of Julia Sets
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1 Fractal Dimension of Julia Sets Claude Heiland-Allen March 6, 2015
2 Fractal Dimension of Julia Sets Fractal Dimension How Long is a Coast? Box-Counting Dimension Examples
3 Fractal Dimension of Julia Sets Fractal Dimension How Long is a Coast? Box-Counting Dimension Examples Julia Sets Complex Dynamics Image Generation Examples
4 Fractal Dimension of Julia Sets Fractal Dimension How Long is a Coast? Box-Counting Dimension Examples Julia Sets Complex Dynamics Image Generation Examples Fractal Dimension of Julia Sets Concept Implementation Results
5 Fractal Dimension Fractal Dimension How Long is a Coast? Box-Counting Dimension Examples Julia Sets Complex Dynamics Image Generation Examples Fractal Dimension of Julia Sets Concept Implementation Results
6 How Long is a Coast? Fractal Dimension How Long is a Coast? Box-Counting Dimension Examples Julia Sets Complex Dynamics Image Generation Examples Fractal Dimension of Julia Sets Concept Implementation Results
7 How Long is a Coast? How long is a coast?
8 How Long is a Coast? It looks this long.
9 How Long is a Coast? But as you look closer,
10 How Long is a Coast? more details appear,
11 How Long is a Coast? giving a longer length,
12 How Long is a Coast? so when do you stop?
13 How Long is a Coast? How long is a coast?
14 How Long is a Coast? It gets longer the closer you look.
15 How Long is a Coast? The concept of length is usually meaningless for geographical curves. They can be considered superpositions of features of widely scattered characteristic sizes; as even finer features are taken into account, the total measured length increases, and there is usually no clearcut gap or crossover, between the realm of geography and details with which geography need not be concerned. B. B. Mandelbrot How long is the coast of Britain? Science: 156, 1967,
16 How Long is a Coast? A better question:
17 How Long is a Coast? A better question: How much longer does a coast get the closer you look?
18 Box-Counting Dimension Fractal Dimension How Long is a Coast? Box-Counting Dimension Examples Julia Sets Complex Dynamics Image Generation Examples Fractal Dimension of Julia Sets Concept Implementation Results
19 Box-Counting Dimension The idea:
20 Box-Counting Dimension The idea: Pick a box size r.
21 Box-Counting Dimension The idea: Pick a box size r. Cover the boundary with boxes of size r.
22 Box-Counting Dimension The idea: Pick a box size r. Cover the boundary with boxes of size r. Count how many boxes are needed N r.
23 Box-Counting Dimension The idea: Pick a box size r. Cover the boundary with boxes of size r. Count how many boxes are needed N r. See how quickly N r increases as r gets smaller.
24 Box-Counting Dimension The formal definition: dim = lim r 0 log N r log r
25 Box-Counting Dimension The formal definition: dim = lim r 0 log N r log r Converges very slowly, not practical.
26 Box-Counting Dimension A more practical definition: dim = lim r 0 log 2 N r N 2r
27 Box-Counting Dimension A more practical definition: dim = lim r 0 log 2 N r N 2r But finite computers have issues with infinite limits.
28 Box-Counting Dimension The formula I actually use: dim = 1 2 log N 2r0 2 N 8r0 r 0 = pixel size
29 Box-Counting Dimension The formula I actually use: dim = 1 2 log N 2r0 2 N 8r0 r 0 = pixel size More on the trade-offs involved later...
30 Examples Fractal Dimension How Long is a Coast? Box-Counting Dimension Examples Julia Sets Complex Dynamics Image Generation Examples Fractal Dimension of Julia Sets Concept Implementation Results
31 Example: circle circle
32 Example: circle r = 2 1 N = 4
33 Example: circle r = 2 2 N = 12
34 Example: circle r = 2 3 N = 28
35 Example: circle r = 2 4 N = 52
36 Example: circle r = 2 5 N = 116
37 Example: circle r = 2 6 N = 244
38 Example: circle r = 2 7 N = 444
39 Example: circle r = 2 8 N = 860
40 Example: circle r = 2 9 N = 1412
41 Example: circle dim
42 Example: circle dim = 1 (limit as r 0)
43 Example: Ireland Ireland
44 Example: Ireland r = 2 1 N = 7
45 Example: Ireland r = 2 2 N = 18
46 Example: Ireland r = 2 3 N = 44
47 Example: Ireland r = 2 4 N = 94
48 Example: Ireland r = 2 5 N = 217
49 Example: Ireland r = 2 6 N = 485
50 Example: Ireland r = 2 7 N = 1033
51 Example: Ireland r = 2 8 N = 2021
52 Example: Ireland r = 2 9 N = 3520
53 Example: Ireland dim
54 Example: Norway Norway
55 Example: Norway r = 2 1 N = 9
56 Example: Norway r = 2 2 N = 25
57 Example: Norway r = 2 3 N = 68
58 Example: Norway r = 2 4 N = 180
59 Example: Norway r = 2 5 N = 488
60 Example: Norway r = 2 6 N = 1310
61 Example: Norway r = 2 7 N = 3333
62 Example: Norway r = 2 8 N = 7641
63 Example: Norway r = 2 9 N = 13070
64 Example: Norway dim
65 Example: carpet carpet
66 Example: carpet r = 2 1 N = 16
67 Example: carpet r = 2 2 N = 36
68 Example: carpet r = 2 3 N = 140
69 Example: carpet r = 2 4 N = 528
70 Example: carpet r = 2 5 N = 1771
71 Example: carpet r = 2 6 N = 6418
72 Example: carpet r = 2 7 N = 23340
73 Example: carpet r = 2 8 N = 82680
74 Example: carpet r = 2 9 N =
75 Example: carpet dim
76 Example: carpet dim = log 8 log (limit as r 0)
77 Example: dust dust
78 Example: dust r = 2 1 N = 16
79 Example: dust r = 2 2 N = 36
80 Example: dust r = 2 3 N = 100
81 Example: dust r = 2 4 N = 256
82 Example: dust r = 2 5 N = 441
83 Example: dust r = 2 6 N = 1024
84 Example: dust r = 2 7 N = 2304
85 Example: dust r = 2 8 N = 4096
86 Example: dust r = 2 9 N = 4096
87 Example: dust dim
88 Example: dust dim = log 4 log (limit as r 0)
89 Julia Sets Fractal Dimension How Long is a Coast? Box-Counting Dimension Examples Julia Sets Complex Dynamics Image Generation Examples Fractal Dimension of Julia Sets Concept Implementation Results
90 Complex Dynamics Fractal Dimension How Long is a Coast? Box-Counting Dimension Examples Julia Sets Complex Dynamics Image Generation Examples Fractal Dimension of Julia Sets Concept Implementation Results
91 Complex Dynamics Consider the quadratic polynomial: f c (z) = z 2 + c
92 Complex Dynamics Consider the quadratic polynomial: Here z, c C, complex numbers. f c (z) = z 2 + c
93 Complex Dynamics The quadratic polynomial can be iterated: f n c = f c (f c (... (f c (f c (z)))...)) }{{} n times
94 Complex Dynamics The quadratic polynomial can be iterated: f n c = f c (f c (... (f c (f c (z)))...)) }{{} n times Or in more manageable notation: f 0 c (z) = z fc n+1 (z) = fc n (f c (z))
95 Complex Dynamics What is the behaviour of f n c (z) as n?
96 Complex Dynamics When c = 2 there are 2 distinct cases:
97 Complex Dynamics When c = 2 there are 2 distinct cases: f n c (z) as n
98 Complex Dynamics When c = 2 there are 2 distinct cases: f n c (z) as n none of the above
99 Complex Dynamics When c = 0 there are 3 distinct cases:
100 Complex Dynamics When c = 0 there are 3 distinct cases: f n c (z) as n
101 Complex Dynamics When c = 0 there are 3 distinct cases: f n c (z) as n f n c (z) 0 as n
102 Complex Dynamics When c = 0 there are 3 distinct cases: fc n (z) as n fc n (z) 0 as n none of the above
103 Complex Dynamics When c = 0 there are 3 distinct cases: fc n (z) as n fc n (z) 0 as n none of the above z > 1
104 Complex Dynamics When c = 0 there are 3 distinct cases: fc n (z) as n fc n (z) 0 as n none of the above z > 1 z < 1
105 Complex Dynamics When c = 0 there are 3 distinct cases: fc n (z) as n fc n (z) 0 as n none of the above z > 1 z < 1 z = 1
106 Complex Dynamics When c = 1 there are 4 distinct cases:
107 Complex Dynamics When c = 1 there are 4 distinct cases: f n c (z) as n
108 Complex Dynamics When c = 1 there are 4 distinct cases: f n c (z) as n fc 2n (z) 0 as n
109 Complex Dynamics When c = 1 there are 4 distinct cases: f n c (z) as n fc 2n (z) 0 as n (z) 1 as n f 2n c
110 Complex Dynamics When c = 1 there are 4 distinct cases: f n c (z) as n fc 2n (z) 0 as n (z) 1 as n f 2n c none of the above
111 Complex Dynamics The initial cases are Fatou components F m (f c ).
112 Complex Dynamics The initial cases are Fatou components F m (f c ). Within each Fatou component the behaviour is the same.
113 Complex Dynamics The initial cases are Fatou components F m (f c ). Within each Fatou component the behaviour is the same. Moreover, nearby points stay nearby under iteration.
114 Complex Dynamics The initial cases are Fatou components F m (f c ). Within each Fatou component the behaviour is the same. Moreover, nearby points stay nearby under iteration. (In fact, they usually get closer.)
115 Complex Dynamics The last case (none of the above) is the Julia set J(f c ).
116 Complex Dynamics The last case (none of the above) is the Julia set J(f c ). Nearby points get further apart under iteration.
117 Complex Dynamics The last case (none of the above) is the Julia set J(f c ). Nearby points get further apart under iteration. But they stay within the Julia set.
118 Complex Dynamics The last case (none of the above) is the Julia set J(f c ). Nearby points get further apart under iteration. But they stay within the Julia set. The Julia set is the boundary of the Fatou components.
119 Image Generation Fractal Dimension How Long is a Coast? Box-Counting Dimension Examples Julia Sets Complex Dynamics Image Generation Examples Fractal Dimension of Julia Sets Concept Implementation Results
120 Image Generation What do Julia sets look like?
121 Image Generation The first step is to determine the number of Fatou components.
122 Image Generation The first step is to determine the number of Fatou components. There is always one component F 1 with f n c (z).
123 Image Generation The first step is to determine the number of Fatou components. There is always one component F 1 with f n c (z). Call the number of other components p.
124 Image Generation The first step is to determine the number of Fatou components. There is always one component F 1 with f n c (z). Call the number of other components p. Then there are components F m, 0 m < p with f pn c (z) z m.
125 Image Generation The first step is to determine the number of Fatou components. There is always one component F 1 with f n c (z). Call the number of other components p. Then there are components F m, 0 m < p with f pn c (z) z m. The algorithm for determining p from c is quite involved, so I won t go into it now.
126 Image Generation The second step is to determine the attractor of F 0 : (if p = 0, skip this step)
127 Image Generation The second step is to determine the attractor of F 0 : (if p = 0, skip this step) Define z 0 = lim n f pn c (0).
128 Image Generation The second step is to determine the attractor of F 0 : (if p = 0, skip this step) Define z 0 = lim n f pn c (0). z 0 satisfies f p c (z 0 ) = z 0.
129 Image Generation The second step is to determine the attractor of F 0 : (if p = 0, skip this step) Define z 0 = lim n f pn c (0). z 0 satisfies f p c (z 0 ) = z 0. The solution can be found using Newton s method.
130 Image Generation Now we can iterate f c (z) with z set by the coordinates of the pixel within the image, to determine which Fatou component z is in:
131 Image Generation Now we can iterate f c (z) with z set by the coordinates of the pixel within the image, to determine which Fatou component z is in: We need two numbers, a large E for detecting attraction to and a small e for detecting attraction to z 0.
132 Image Generation Now we can iterate f c (z) with z set by the coordinates of the pixel within the image, to determine which Fatou component z is in: We need two numbers, a large E for detecting attraction to and a small e for detecting attraction to z 0. If f n c (z) > E, then z F 1.
133 Image Generation Now we can iterate f c (z) with z set by the coordinates of the pixel within the image, to determine which Fatou component z is in: We need two numbers, a large E for detecting attraction to and a small e for detecting attraction to z 0. If f n c (z) > E, then z F 1. If f n c (z) z 0 < e, then z F n mod p.
134 Image Generation Now we can iterate f c (z) with z set by the coordinates of the pixel within the image, to determine which Fatou component z is in: We need two numbers, a large E for detecting attraction to and a small e for detecting attraction to z 0. If f n c (z) > E, then z F 1. If f n c (z) z 0 < e, then z F n mod p. If n > N, where N is a maximum iteration count necessary for finite computers, then we give up, and don t know much about z.
135 Image Generation While iterating, keep track of the derivative:
136 Image Generation While iterating, keep track of the derivative: z f 0 c (z) = 1
137 Image Generation While iterating, keep track of the derivative: z f 0 c (z) = 1 z f c n+1 (z) = 2fc n (z) z f c n (z)
138 Image Generation While iterating, keep track of the derivative: z f 0 c (z) = 1 z f n+1 c (z) = 2f n c (z) z f n c (z) In imperative programming language pseudo-code: z := pixel coordinates d := 1 for n := 0 to N d := 2 * z * d z := z * z + c
139 Image Generation Why do we need the derivative?
140 Image Generation Why do we need the derivative? The Julia set is the boundary of the Fatou components.
141 Image Generation Why do we need the derivative? The Julia set is the boundary of the Fatou components. How to detect the boundary, if there is only one Fatou component?
142 Image Generation Why do we need the derivative? The Julia set is the boundary of the Fatou components. How to detect the boundary, if there is only one Fatou component? Even if there are more components, the boundary might be very thin in places.
143 Image Generation The derivative provides an estimate of the distance to the Julia set:
144 Image Generation The derivative provides an estimate of the distance to the Julia set: d = f c n (z) log fc n (z) z f c n (z)
145 Image Generation The derivative provides an estimate of the distance to the Julia set: d = f c n (z) log fc n (z) z f c n (z) If d is small compared to the pixel size, the Julia set passes through the pixel.
146 Image Generation The derivative provides an estimate of the distance to the Julia set: d = f c n (z) log fc n (z) z f c n (z) If d is small compared to the pixel size, the Julia set passes through the pixel. This formula is only valid for F 1 with f n c (z), so it s best to make E as large as reasonably possible.
147 Image Generation The derivative provides an estimate of the distance to the Julia set: d = f c n (z) log fc n (z) z f c n (z) If d is small compared to the pixel size, the Julia set passes through the pixel. This formula is only valid for F 1 with f n c (z), so it s best to make E as large as reasonably possible. (There are other formulae for the other cases, but I couldn t get them to work reliably.)
148 Image Generation Now we know the Fatou component F m for each pixel, and for m = 1 we also have a distance estimate d. This is enough to generate an image:
149 Image Generation Now we know the Fatou component F m for each pixel, and for m = 1 we also have a distance estimate d. This is enough to generate an image: If m = 1 and d is small, then colour the pixel black.
150 Image Generation Now we know the Fatou component F m for each pixel, and for m = 1 we also have a distance estimate d. This is enough to generate an image: If m = 1 and d is small, then colour the pixel black. If m = 1 and d is large, then colour the pixel white.
151 Image Generation Now we know the Fatou component F m for each pixel, and for m = 1 we also have a distance estimate d. This is enough to generate an image: If m = 1 and d is small, then colour the pixel black. If m = 1 and d is large, then colour the pixel white. Compare our m with the m for neighbouring pixels:
152 Image Generation Now we know the Fatou component F m for each pixel, and for m = 1 we also have a distance estimate d. This is enough to generate an image: If m = 1 and d is small, then colour the pixel black. If m = 1 and d is large, then colour the pixel white. Compare our m with the m for neighbouring pixels: If any are different, then colour the pixel black.
153 Image Generation Now we know the Fatou component F m for each pixel, and for m = 1 we also have a distance estimate d. This is enough to generate an image: If m = 1 and d is small, then colour the pixel black. If m = 1 and d is large, then colour the pixel white. Compare our m with the m for neighbouring pixels: If any are different, then colour the pixel black. If all are the same, then colour the pixel white.
154 Image Generation Now we know the Fatou component F m for each pixel, and for m = 1 we also have a distance estimate d. This is enough to generate an image: If m = 1 and d is small, then colour the pixel black. If m = 1 and d is large, then colour the pixel white. Compare our m with the m for neighbouring pixels: If any are different, then colour the pixel black. If all are the same, then colour the pixel white. This image has black pixels near the Julia set, and white pixels elsewhere.
155 Image Generation Now we know the Fatou component F m for each pixel, and for m = 1 we also have a distance estimate d. This is enough to generate an image: If m = 1 and d is small, then colour the pixel black. If m = 1 and d is large, then colour the pixel white. Compare our m with the m for neighbouring pixels: If any are different, then colour the pixel black. If all are the same, then colour the pixel white. This image has black pixels near the Julia set, and white pixels elsewhere. An extension is to colour pixels by the value of m, instead of white, as in the following examples.
156 Examples Fractal Dimension How Long is a Coast? Box-Counting Dimension Examples Julia Sets Complex Dynamics Image Generation Examples Fractal Dimension of Julia Sets Concept Implementation Results
157 Example: needle dust c = i dim
158 Example: elephant dust c = i dim
159 Example: seahorse dust c = i dim
160 Example: needle tip dendrite c = 2 + 0i dim
161 Example: 2-way hub dendrite c = i dim
162 Example: 3-way hub dendrite c = i dim
163 Example: period 1 c = i dim
164 Example: period 2 c = 1 + 0i dim
165 Example: period 3 c = i dim
166 Example: period 1 near 2 over 5 c = i dim
167 Example: period 5 near 2 over 5 c = i dim
168 Example: period 5 c = i dim
169 Example: period 3 island c = i dim
170 Example: period 3 island 1 over 3 bulb c = i dim
171 Example: period 4 island c = i dim
172 Fractal Dimension of Julia Sets Fractal Dimension How Long is a Coast? Box-Counting Dimension Examples Julia Sets Complex Dynamics Image Generation Examples Fractal Dimension of Julia Sets Concept Implementation Results
173 Concept Fractal Dimension How Long is a Coast? Box-Counting Dimension Examples Julia Sets Complex Dynamics Image Generation Examples Fractal Dimension of Julia Sets Concept Implementation Results
174 Concept The Mandelbrot set is a c-plane plot of whether J(f c ) is connected.
175 Concept The Mandelbrot set is a c-plane plot of whether J(f c ) is connected. Some visualisations of the Mandelbrot set use psychedelic colours, but the mathematical object is binary.
176 Concept The Mandelbrot set
177 Concept What would a c-plane plot of dim J(f c ) look like?
178 Concept What would a c-plane plot of dim J(f c ) look like? 0 dim J(f c ) 2, so a spectrum of colours would be necessary.
179 Implementation Fractal Dimension How Long is a Coast? Box-Counting Dimension Examples Julia Sets Complex Dynamics Image Generation Examples Fractal Dimension of Julia Sets Concept Implementation Results
180 Implementation The implementation is written in C99:
181 Implementation The implementation is written in C99: C99 supports complex numbers.
182 Implementation The implementation is written in C99: C99 supports complex numbers. Low-level and efficient.
183 Implementation The implementation is written in C99: C99 supports complex numbers. Low-level and efficient. Most libraries have C interfaces.
184 Implementation The implementation uses OpenGL for graphics:
185 Implementation The implementation uses OpenGL for graphics: OpenGL supports programmable graphics hardware (GPUs).
186 Implementation The implementation uses OpenGL for graphics: OpenGL supports programmable graphics hardware (GPUs). GPUs are very good at parallel number-crunching tasks, like rendering a Julia set.
187 Implementation The implementation uses OpenGL for graphics: OpenGL supports programmable graphics hardware (GPUs). GPUs are very good at parallel number-crunching tasks, like rendering a Julia set. Mipmap generation reduces images to progressively coarser pixel resolutions.
188 Implementation The implementation uses OpenGL for graphics: OpenGL supports programmable graphics hardware (GPUs). GPUs are very good at parallel number-crunching tasks, like rendering a Julia set. Mipmap generation reduces images to progressively coarser pixel resolutions. Occlusion queries can be used for counting pixels.
189 Implementation The implementation uses a few fragment shaders:
190 Implementation The implementation uses a few fragment shaders: to compute Fatou components and distance estimates;
191 Implementation The implementation uses a few fragment shaders: to compute Fatou components and distance estimates; to post-process the Fatou component index and distance estimate into a black and white image of the Julia set;
192 Implementation The implementation uses a few fragment shaders: to compute Fatou components and distance estimates; to post-process the Fatou component index and distance estimate into a black and white image of the Julia set; to discard pixels below a threshold.
193 Implementation Mipmap reduction averages groups of pixels:
194 Implementation Mipmap reduction averages groups of pixels:
195 Implementation Mipmap reduction averages groups of pixels: Box-counting should count if any subpixel was black.
196 Implementation Mipmap reduction averages groups of pixels: Box-counting should count if any subpixel was black. The solution is to threshold the grey level in each mipmap level.
197 Implementation Mipmap reduction averages groups of pixels: Box-counting should count if any subpixel was black. The solution is to threshold the grey level in each mipmap level. The threshold should between the lightest grey and white.
198 Implementation Box-counting is performed with occlusion queries:
199 Implementation Box-counting is performed with occlusion queries: First clear the depth buffer.
200 Implementation Box-counting is performed with occlusion queries: First clear the depth buffer. Then draw the Julia set, discarding pixels below a threshold.
201 Implementation Box-counting is performed with occlusion queries: First clear the depth buffer. Then draw the Julia set, discarding pixels below a threshold. Then draw again, but further away.
202 Implementation Box-counting is performed with occlusion queries: First clear the depth buffer. Then draw the Julia set, discarding pixels below a threshold. Then draw again, but further away. The depth test prevents pixels that were rendered the first time from being drawn, so only the previously discarded pixels pass.
203 Implementation Box-counting is performed with occlusion queries: First clear the depth buffer. Then draw the Julia set, discarding pixels below a threshold. Then draw again, but further away. The depth test prevents pixels that were rendered the first time from being drawn, so only the previously discarded pixels pass. The occlusion query counts the number of passed pixels in the second draw.
204 Implementation Performance:
205 Implementation Performance: Faster than a CPU-based implementation.
206 Implementation Performance: Faster than a CPU-based implementation. But it s still time-consuming.
207 Implementation Performance: Faster than a CPU-based implementation. But it s still time-consuming. The final image took over 5 hours to render.
208 Implementation Performance: Faster than a CPU-based implementation. But it s still time-consuming. The final image took over 5 hours to render. Watching the image appear pixel-by-pixel brings back memories of rendering fractals a couple of decades ago...
209 Results Fractal Dimension How Long is a Coast? Box-Counting Dimension Examples Julia Sets Complex Dynamics Image Generation Examples Fractal Dimension of Julia Sets Concept Implementation Results
210 Results dim
211 Results But is it accurate?
212 Results But is it accurate? No.
213 Results But is it accurate? No. But it s pretty close.
214 Results Recall the formula I actually used: dim = 1 2 log N 2r0 2 N 8r0 r 0 = pixel size
215 Results Recall the formula I actually used: dim = 1 2 log N 2r0 2 N 8r0 r 0 = pixel size This formula is based on simple linear regression of log N against log r.
216 Results Recall the formula I actually used: dim = 1 2 log N 2r0 2 N 8r0 r 0 = pixel size This formula is based on simple linear regression of log N against log r. I tried all possibilities of 0 s < t 12 for a regression range between 2 s r 0 and 2 t r 0.
217 Results t s When s = 0 and t is small, the dimension calculated is wrong because the Julia set is too inexact at the resolution of the pixel grid
218 Results t s Increasing s a little reduces this artifact of pixel resolution, but t needs to stay small or the results go bad again
219 Results t s When both s and t are large, the results are nonsense
220 Results t s The best trade-off seems to be at s = 1 and t = 3, which gives the formula I actually used
221 The End The End.
222
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