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

2. 2 The Game of Life Lecture 5

3. GEM2505M 3 Today’s Lecture Pascal’s Triangle Revisited Cellular Automata The Game of Life

4. GEM2505M 4 Pascal’s Triangle revisited Previously, we had seen how Pascal’s triangle relates to the Sierpinski gasket. How can we build up the graph systematically? Since we color even coefficients white and the uneven ones black, a start would be to obtain an expression for all the coefficients.

5. GEM2505M 5 Pascal’s Triangle revisited Pascal’s Triangle With some puzzling, one can find that the coefficients are given by: with

6. GEM2505M 6 Pascal’s Triangle revisited Hence we can introduce a coordinate system where n = 0,1,2,.. is the row index and k = 0,1,2, …. The column index. Then the coefficient with coordinates (n,k) is and we can immediately determine whether that’s even or not That’s excellent. Unfortunately, n! becomes very big very rapidly!

7. GEM2505M 7 Pascal’s Triangle revisited The problem can be alleviated a bit by recalling the way the triangle was constructed (a new coefficient is obtained by adding the coefficients to the left and right in the row above). Using this relationship, we can compute the coefficients without the factorials. But, again even the coefficients become very large very rapidly …..

8. GEM2505M 8 Pascal’s Triangle revisited Fortunately, we do not need the actual values of the coefficients at all. We only need to know whether they are even or odd. If we examine the addition rule: We can see that even and uneven coefficients are simply generated by this set of rules.

9. GEM2505M 9 Pascal’s Triangle revisited This may look familiar to you if you like Wittgenstein or Computers (or both). It describes a logical XOR XOR

10. GEM2505M 10 Pascal’s Triangle revisited This now allows us to construct the Pascal’s triangle derived Sierpinski gasket as follows: Color a 1 black and make a 0 white (here white means ‘unused’ as well) The Pascal Triangle starts with a single 1 Note: in the Pascal Triangle, each row is shifted by half a column as with regards to the previous column to stress the symmetry. Mathematically, this is the same as a square grid.

11. GEM2505M 11 Pascal’s Triangle revisited We then obtain the colors of the next row by applying our rule table (slightly relabeled). Note: Outside of the boundary colors are considered as ‘0’ and hence white. ? What color do you think we have to paint here?

12. GEM2505M 12 Pascal’s Triangle revisited Step by step ….

13. GEM2505M 13 Pascal’s Triangle revisited Indeed, after some time, the pattern of the Sierpinski gasket emerges (albeit shifted …).

14. GEM2505M 14 Cellular Automata In fact, what we have done just now is the construction of a simple cellular automaton. In a one-dimensional cellular automaton one has a row of cells that can assume two or more colors and a set of rules that determines how the colors change from one time step to the next. code 912

15. GEM2505M 15 Cellular Automata Ergo, for our Pascal’s Triangle we had two possible colors and a set with four rules. staterules Pascal’s Triangle Other rules Of course we can make other rules as we wish. In total, there are 2 4 rules we can make this way.

16. GEM2505M 16 Cellular Automata The 16 1-neighbor rules

17. GEM2505M 17 Cellular Automata Do you recognize a pattern? ? It’s the pattern of zeros and ones when counting binary! E.g. 10 in binary is 1010

18. GEM2505M 18 Cellular Automata Of course, if one can make 1-neighbor rules, one can also make 2 neighbor rules. state rules This is called rule 90.

19. GEM2505M 19 Cellular Automata Rule 90 yields …. the Sierpinski gasket!!!

20. GEM2505M 20 Cellular Automata In total, there are 2 8 rules, some typical examples are: Chaotic – can be used as random number generator Repetitive Localized Structures Nested Structures

21. GEM2505M 21 Cellular Automata

22. GEM2505M 22 Cellular Automata

23. GEM2505M 23 Cellular Automata And of course, one does not need to restrict oneself to two colors and two neighbors …

24. GEM2505M 24 The Game of Life A good idea in 1 dimension can be a great idea in 2 dimensions. Indeed, the game of life is nothing but a 2 dimensional cellular automaton with a specific set of rules. The rules are: If there are less than 2 or more than 3 neighbors: Death, the cell becomes white If there are exactly 3 neighbors and the cell is white: Birth, the cell becomes black Other cases: no change (i.e. white remains white and black remains black). State: Consider a cell in 2D with 8 neighbors.

25. GEM2505M 25 The Game of Life Examples Death Survival DeathSurvival - - Birth -

26. GEM2505M 26 The Game of Life The dynamics of the game of life are astonishingly rich. On nice feature is that there are small blocks that return to their initial shape after a certain number of time steps. These are sometimes referred to as ‘life forms’. Indeed there are life forms that do not change shape even from one generation to the next, such forms are sometimes called ‘still lifes’.

27. GEM2505M 27 The Game of Life Examples of still lifes.

28. GEM2505M 28 The Game of Life Oscillators: Patterns that return to their original shape. Examples of oscillators Queen Bee Shuttle One of several period 4 oscillators One of several period 5 oscillators

29. GEM2505M 29 The Game of Life Spaceships: moving patterns that leave no trail. The dragon The Glider The Weekender Examples of Spaceships Another Glider

30. GEM2505M 30 The Game of Life Guns: Patterns that emit ‘spaceships’ The ‘New Gun’ Examples of guns Gosper Gun (this is the first glider gun ever discovered. Named after Bill Gosper who found it.)

31. GEM2505M 31 The Game of Life Puffers: Spaceships that leave a trail Example of a puffer Noah’s Ark The first ever puffer – discovered again by Bill Gosper

32. GEM2505M 32 Simple Rules. Amazing Dynamics! Key Points of the Day

33. GEM2505M 33 Is life a game? Think about it! Game, Rhinoceros, Africa, Origin of Humans

34. GEM2505M 34 References http://mathworld.wolfram.com/ Alan Hensel’s Life Applet