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

2. 2 The Cantor Set Lecture 3

3. GEM2505M 3 Today’s Lecture Self-Similarity Cantor Set Sierpinski Gasket Pascal’s Triangle The Chaos Game

4. GEM2505M 4 Self-Similarity Good approximate examples from daily life are cauliflower and broccoli. Roughly, self-similar means that a part looks just like the whole:

5. GEM2505M 5 Self-Similarity The meter stick! A more scientific example is: 1m = 10dm 1dm = 10cm 1cm = 10mm Looks the same regardless of the scale.

6. GEM2505M 6 Self-Similarity This is far less trivial then it seems! In Europe, e.g. the decimal system was only introduced in 1202 by Leonardo Fibonacci. The modern metric system was introduced by the French around 1795. And indeed, it’s still not used everywhere. Think USA! So how many feet is three miles and 12 inches? (Compare this to: How many meters is 512 centimeters ).

7. GEM2505M 7 Introduction Georg Cantor 1845 - 1918 The Cantor Set was first published in 1883 in an investigation into sets with exceptional properties. It is easy to construct this set! Start with a line of unit length (i.e. it goes from 0 to 1) and remove the middle third. This leaves us with two lines (from 0 to 1/3 and from 2/3 to 1). Now remove the middle third of these lines. Do this ad infinitum! But there are many fascinating properties as we shall see.

8. GEM2505M 8 Construction Geometrically 01 remove middle third Note: The open interval is removed, not the start and end points. I.e. (1/3,2/3) is removed. This is just like in an iterative map 1-31-3 0-30-3 2-32-3 3-93-9 0-90-9 6-96-9 1-91-9 2-92-9 7-97-9 8-98-9 9-99-9 3-33-3 9 - 27 0 - 27 18 - 27 3 - 27 6 - 27 21 - 27 24 - 27 27 - 27 8 - 27 1 - 27 19 - 27 2 - 27 7 - 27 20 - 21 25 - 27 26 - 27 The first few steps in the construction of the Cantor Set.

9. GEM2505M 9 Construction Self-similarity 01 remove middle third magnify 81 times Looks exactly the same! We can magnify any little line and get exactly the same graph back.

10. GEM2505M 10 So what is the Cantor Set? Definition Points?? Wouldn’t the remaining parts be very-very small lines? The answer is no! (Infinity is a strange beast). Examples of such points clearly are the end-points of the intervals we started with: 1/3,2/3,1,2/9 etc. The Cantor Set is the set of points that remains after repeating the ‘middle third’ procedure an infinite number of times.

11. GEM2505M 11 Not just end points! Ah! Then the Cantor set is just the set of all the little intervals’ end points. Wrong! It can be shown that the Cantor Set is uncountable. However, the set of end points is countable as can be understood by looking at this picture: 01 23 4567 8 Note: Countable in mathematics means that there’s a one to one correspondence to integers.

12. GEM2505M 12 What else is there? Consequently, there are more points than just end points. Rather amazing! Let’s explore this a bit further and try to find some points which are not end points. For this we need to use triadic numbers ‘0’‘1’‘2’ I.e. we’ll use the digits

13. GEM2505M 13 Triadic Numbers In our decimal system, the value of a digit is a power of 10 (we say it has base 10) This is easy for us, but one can write numbers in any base. E.g. binary for computers. Here, lets look at base 3. 1319.6 = 1 x 1000 + 3 x 100 + … + 6 x 10 -1 12.1 = 1 x 3 + 2 x 1 + 1 x 3 -1 = 5.3333333 TriadicDecimal The line on top of the last three means repeat forever.

14. GEM2505M 14 Triadic Tree Note that in decimal 0.9 = 1. In the same way, in triadic, 0.2 = 1. Thus we can visualize any number in the unit interval (where the Cantor Set lives) by this tree: 01 0 1 2 000102101112202122

15. GEM2505M 15 01 Triadic Tree Now let us superimpose the Cantor Set on this triadic number expansion. 0 1 2 000102101112202122 What we see is that the removed intervals start with a 1!

16. GEM2505M 16 Triadic Tree This allows us to make the following statement: 01 0 1 2 000102101112202122 The Cantor set C is the set of points in [0,1] for which there is a triadic expansion that does not contain the digit ‘1’. Clearly this is true for points like 2/3 or 2/9 which can be written as 0.2 and 0.02 respectively. How about 1/3 though? Isn’t it also part of the set? ?

17. GEM2505M 17 Triadic Tree Usually 1/3 would be written as 0.1. But we can also write it as 0.0222. Ergo there is an expansion without a one. Hence the rule works for this point. So, how about 0.11 (i.e. 1/3 + 1/9) which is not a part of the Cantor set? Can this be expanded as well? Yes and no! We can expand the last ‘1’ but not the first. The best we can do is: 0.10222. Always contains a one. Ergo not a part of the Cantor set.

18. GEM2505M 18 Triadic Tree So we can draw a powerful conclusion now. Points which end in ‘0’ or infinitely many ‘2’ are end points of the intervals. All other numbers are not end points. Can you think of some? ?

19. GEM2505M 19 Triadic Tree Here are a few examples of points that are not end-points: 0.202200222000222200002222200000….. 0.02202020020002220202202….. Create points through some kind of a rule: 0.202202220222202222202222202222220….. Create points through a random sequence of ‘0’s and ‘2’s: 0.000202002202202020200022202…..

20. GEM2505M 20 Triadic Tree And thus we have the proof that there are more points than just end-point in the Cantor set! It’s quite amazing that such a fundamental statement can be proven with such simple Mathematics.

21. GEM2505M 21 The Sierpinski Gasket The construction of the Sierpinski Gasket is quite similar to the construction of the Cantor Set. Take a filled triangle, pick the midpoints of the three sides and remove the triangle formed by these three points. StartStep 1Step 2

22. GEM2505M 22 The Sierpinski Gasket Eventually, it looks like this. As is the case with the Cantor set, this structure is perfectly self-similar.

23. GEM2505M 23 Pascal’s Triangle This describes the coefficients of the expansion of the polynomial (x + 1) n. Pascal 1654

24. GEM2505M 24 Pascal’s Triangle Despite it’s name, this was known well before Pascal. China 1303 Japan 1781

25. GEM2505M 25 Pascal’s Triangle When color coding the even entries as white and the uneven entries as black …. we get the Sierpinski gasket!

26. GEM2505M 26 1.Make a triangle and label the corners A,B,C 2.Choose any point inside the triangle, make a dot 3.Roll a dice, if it’s 1 or 2 move halfway to A, if it’s 3 or 4, move halfway to B and if it’s 5 or 6 move half way to C. Make a dot at the new position 4.Repeat 3 until dots cover the entire triangle! Roll your dice! A B C The Chaos Game 1,2 3,4 5,6

27. GEM2505M 27 Example A B C The Chaos Game 1,2 3,4 5,6 1 st throw: 4 A B C 1,2 3,4 5,6 A B C 1,2 3,4 5,6 A B C 1,2 3,4 5,6 2 nd throw: 2 3 rd throw: 3 4 th throw: 6

28. GEM2505M 28 Stunning! The chaos game yields the Sierpinsky gasket. http://math.bu.edu/DYSYS/applets/fractalina.html This is a nice applet to illustrate this: And then there’s a game: http://math.bu.edu/DYSYS/applets/chaos-game.htmlhttp://math.bu.edu/DYSYS/applets/chaos-game.html

29. GEM2505M 29 Can we understand this? The Chaos Game Yes! Start anywhere, e.g. in the big triangle in the center. ? Where can we be next?

30. GEM2505M 30 Possible next points The Chaos Game Hey, they are all within the next smaller white triangle! Let’s try that again.

31. GEM2505M 31 Possible next points The Chaos Game Cool, they are all within the next smaller white triangle again! Indeed, this is how it works: After every step you get into one size smaller triangle until the triangle is smaller than the dot of your drawing. Once this is the case, every dot you draw represents a triangle of the gasket. So drawing many dots is like drawing many triangles.

32. GEM2505M 32 Generation of Fractals by Iteration. Addresses of Points in Fractals Understanding the Chaos Game Key Points of the Day

33. GEM2505M 33 Do we encounter fractals everyday? Think about it! Fractal, Mountain, Skiing, Switzerland!

34. GEM2505M 34 References