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GEM2505M

Taming Chaos. GEM2505M. Frederick H. Willeboordse frederik@chaos.nus.edu.sg. The Cantor Set. Lecture 3. Today’s Lecture. Self-Similarity Cantor Set Sierpinski Gasket Pascal’s Triangle The Chaos Game. Self-Similarity. Roughly, self-similar means that a part looks just like the whole:.

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GEM2505M

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

  2. The Cantor Set Lecture 3

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

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

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

  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. Introduction The Cantor Set was first published in 1883 in an investigation into sets with exceptional properties. It is easy to construct this set! Georg Cantor But there are many fascinating properties as we shall see. 1845 - 1918 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!

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

  9. Construction Self-similarity We can magnify any little line and get exactly the same graph back. 0 1 remove middle third magnify 81 times Looks exactly the same!

  10. So what is the Cantor Set? Definition The Cantor Set is the set of points that remains after repeating the ‘middle third’ procedure an infinite number of times. 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.

  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: 0 1 Note: Countable in mathematics means that there’s a one to one correspondence to integers. 2 3 4 5 6 7 8

  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 I.e. we’ll use the digits ‘0’ ‘1’ ‘2’

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

  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: 0 2 1 00 01 02 10 11 12 20 21 22 0 1

  15. Triadic Tree Now let us superimpose the Cantor Set on this triadic number expansion. 0 2 1 00 01 02 10 11 12 20 21 22 0 1 What we see is that the removed intervals start with a 1!

  16. Triadic Tree 0 2 1 00 01 02 10 11 12 20 21 22 0 1 This allows us to make the following statement: 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. 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. 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. Triadic Tree Here are a few examples of points that are not end-points: Create points through some kind of a rule: 0.202200222000222200002222200000….. 0.202202220222202222202222202222220….. Create points through a random sequence of ‘0’s and ‘2’s: 0.02202020020002220202202….. 0.000202002202202020200022202…..

  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. 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. Start Step 1 Step 2

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

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

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

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

  26. The Chaos Game C 5,6 Roll your dice! A B 3,4 1,2 • Make a triangle and label the corners A,B,C • Choose any point inside the triangle, make a dot • 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 • Repeat 3 until dots cover the entire triangle!

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

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

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

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

  31. The Chaos Game Cool, they are all within the next smaller white triangle again! Possible next points 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. Key Points of the Day • Generation of Fractals by Iteration. • Addresses of Points in Fractals • Understanding the Chaos Game

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

  34. References

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