1. 1 GEM2505M Frederick H. Willeboordse frederik@chaos.nus.edu.sg Taming Chaos

2. 2 The Mandelbrot Set Lecture 4

3. GEM2505M 3 Today’s Lecture Fractal Dimension Complex Numbers Prisoners and Escapees Julia Set Mandelbrot Set

4. GEM2505M 4 Fractal Dimension Line:1 – Dimensional Plane:2 - Dimensional Usually when we talk about dimension we think of lines or surfaces. But what does this actually mean? It is (to a certain degree) related to self similarity! Previously, we saw that the meter stick is perfectly self-similar. Consequently, one can rescale it easily.

5. GEM2505M 5 Fractal Dimension Magnify by a factor 10. For example: To turn this around, when magnifying by a factor 10, the new stick contains 10 times the number of original sticks. 1m = 10dm 1dm = 10cm I.e. Magnify 1dm by a factor 10 to obtain 1m which contains 10 pieces of 1dm. 1m = 10 dm

6. GEM2505M 6 Fractal Dimension If we magnify the small red square by a factor 10. What we see is that the new big square does not contain 10 times as many small squares but 10 2 times as many squares! That’s quite obvious of course. Now what happens if we do the same trick with a square? 1 sq. m = 100 sq. dm

7. GEM2505M 7 Fractal Dimension a:number of pieces S:scaling factor (i.e. magnification) D:dimension This leads us to an important observation: In other words, the dimensions is: Defined in this way, D is called the self-similarity dimension.

8. GEM2505M 8 Fractal Dimension Can we apply this to the Cantor Set? Yes! If we take a chunk and make it 3 times bigger, how many copies of the original do we have? 2!!! So for the Cantor Set we have:

9. GEM2505M 9 Complex Numbers Complex sounds like this is very complex but in fact it’s not. In order to talk about the Mandelbrot set we need to know what complex numbers are. We all know that it’s quite easy to solve an equation like this one: Seeing that, it’s not particularly far-fetched to wonder what the solution is of:

10. GEM2505M 10 Complex Numbers Ah, easy enough! Oops! There’s no root for -1 (even if we try very hard indeed). We’re stuck and it would really be useful to be able to solve such equations. The solution is simple, if there’s no number in existence which is the root of -1, we can just introduce one. Clever solutions need not be complicated! ergo:

11. GEM2505M 11 Complex Numbers All right, we can define Twisting history a bit we can say: Since we imagined this solution let’s call it ‘imaginary’. Then the solution to or is just:

12. GEM2505M 12 Complex Numbers Excellent, well if we can do the trick once, we can do it twice! How about the solution to: That must be: Stuck again! Now we don’t know what that is… well perhaps we can think of some equation whose square is or Indeed:

13. GEM2505M 13 Complex Numbers This brings us quite closely to the general notation of a complex number: with a and b real numbers (like 0.231, 1.949, 2.000) Consequently, a complex number can be drawn as a point in a plane where the x- axis is the ‘real’ part a, and the y-axis the ‘imaginary’ part b. b a

14. GEM2505M 14 Complex Numbers Then the modulus is: And the conjugate is: If we have: conjugate

15. GEM2505M 15 Complex Numbers Addition:

16. GEM2505M 16 Complex Numbers Subtraction:

17. GEM2505M 17 Complex Numbers Multiplication:

18. GEM2505M 18 Complex Numbers Multiplication as rotation:

19. GEM2505M 19 Prisoners and Escapees Now let us consider the map: Prisoners What happens if we start with r = 0.8 and  = 10 O :

20. GEM2505M 20 Prisoners and Escapees Escapees Next, let’s consider what happens if we start with r = 1.5 and  = 50 O :

21. GEM2505M 21 Prisoners and Escapees Guards? And lastly let’s set r = 1.0 and  = 10 O :

22. GEM2505M 22 Prisoners and Escapees Graphically Prisoner Escapee Boundary

23. GEM2505M 23 Julia Set The Julia Set is the boundary between the escapees and prisoners of a complex iterative map. Definition In the case ofthis means the unit circle Often the inside of the Julia Set is filled in and the escaping points are colored according to how long it takes for the point to become larger than a certain value.

24. GEM2505M 24 Julia Set The situation changes drastically when a constant is added to the iterative map. Adding a constant Connected Julia Sets Disconnected Julia Sets

25. GEM2505M 25 Julia Set Some famous Julia sets of the complex quadratic map Dendrite Fractal c at the boundary of the Mandelbrot Set (for this picture, c = i) Rabit Fractal

26. GEM2505M 26 Another Chaos Game 1.Make a drawing of the complex plane 2.Pick any point and call it w choose a complex constant c 3.Roll a dice, if it is 1,2 or 3 move to otherwise move to. 4.Repeat 3 for a while!

27. GEM2505M 27 Another Chaos Game Result Dendrite Fractal A Julia Set!

28. GEM2505M 28 Another Chaos Game Explanation Dendrite Fractal The Julia Set is the boundary between escapees and prisoners. Hence all points not exactly on top of it move away from it. If we go backwards in the iteration, we will get closer to it. Forward iterate Backward iterates

29. GEM2505M 29 Mandelbrot Set There are two types of Julia Sets, connected Julia Sets and disconnected Julia Sets. The Mandelbrot Set is defined as the set of parameters c that lead to a connected Julia Set. Alternatively: M = {c  C | c  c 2 +c  … remains bounded } I.e. parameters c for which the orbit of z 0 is bounded.

30. GEM2505M 30 Mandelbrot Set The difference Julia Set Initial conditions z 0 Mandelbrot Set Parameters c Find boundary between prisoners and escapees. Find connected Julia Sets.

31. GEM2505M 31 Mandelbrot Set Escape? How can we know that an orbit escapes to infinity? Answer: if |z| > r(c) = max(|c|,2), an orbit will escape. Proof : Take a number such that |z| > r(c) is true. Iterating once we obtain: |z 2 + c| Applying the inequality we get: |z 2 + c – c|  |z 2 +c| + |c| |z 2 |  Ergo: |z 2 +c|  |z 2 | - |c|  |z| 2 - |c|  |z| 2 - |z|  (|z|-1)|z|  (1+  )|z| So when iterating z it grows and thus eventually escapes.

32. GEM2505M 32 Mandelbrot Set How can we make the pictures? A screen is an array of pixels: Re Mininum Re Maximum Im Mininum Im Maximum For each of the pixels, calculate whether it escapes and if so how many steps it takes to reach e.g. 2. Color the pixel according to table above. Note: Color assignment is of course arbitrary. Prisoners: black Escapees: use this

33. GEM2505M 33 Mandelbrot Set The Heart Intersects real axis from -0.75 – 0.25. Associated with a Julia set that has a period 1 attractor. Julia set c = 0 + 0i

34. GEM2505M 34 Mandelbrot Set The Buds Associated with Julia sets that have higher period attractors. Period 2 P.3 Period 4Period 5 c = -0.134 – 0.742i c = -1 + 0i

35. GEM2505M 35 Mandelbrot Set Some Features The Mandelbrot Set is connected The boundary is a fractal and infinitely long The dimension is 2 It is quasi-self-similar And some more pictures It’s better to use the applet though

36. GEM2505M 36 Fractal Dimension. Julia Set Mandelbrot Set Key Points of the Day

37. GEM2505M 37 What is our dimension? Think about it! Julia, Romeo, Singing, Dire Straits

38. GEM2505M 38 References Dave Short’s course on complex numbers http://mathworld.wolfram.com/ Peter Alfeld’s Mandelbrot Applet