Fractal Geometry and Programming

Transcription

1 Çetin Kaya Koç Çetin Kaya Koç Fall / 35

2 Helge von Koch Computational Thinking Niels Fabian Helge von Koch Swedish Mathematician 25 January March : A result on Riemann Hypothesis 1904: Koch Curve Çetin Kaya Koç Fall / 35

3 Sweden Computational Thinking Kingdom of Sweden, Konungariket Sverige A Scandinavian country in Northern Europe A constitutional monarchy and parliamentary democracy Borders Norway to the west and Finland to the east, connected to Denmark in the southwest via a bridge-tunnel Third largest European Union country (after France and Spain) 450,295 square kilometers of area 58 % of Turkey 9.8m population 12 % of Turkey Çetin Kaya Koç Fall / 35

4 Why Think About Sweden? intricate coastline Çetin Kaya Koç Fall / 35

5 Map of Sweden Computational Thinking Çetin Kaya Koç Fall / 35

6 Computational Thinking Fjords of Sweden C etin Kaya Koc Fall / 35

7 Koch Curve Computational Thinking Koch was (probably) thinking about the coastline of Sweden a lot.. its intricacies, seemingly endless turns and bends He invented a curve that imitates such a coastline We now call his curve, the Koch curve Each curve has a different order The order determines the length and the scale of each curve Çetin Kaya Koç Fall / 35

8 Koch Curve Order 0 and 1 Order 0 L = 1 Order /3 1/3 1/3 L = 1 1/3 1/3 Order 1 1/3 1/3 1/3 L = 4/3 Çetin Kaya Koç Fall / 35

9 Koch Curve Order 1 and 2 1/3 1/3 Order 1 L = 4/3 1/3 1/3 1/3 1/9 1/9 1/9 1/9 1/9 1/9 1/9 1/9 1/9 1/9 1/9 1/9 Order 2 L = 16/9 1/9 1/9 1/9 1/9 Çetin Kaya Koç Fall / 35

10 Koch Curve Order 0, 1, 2, 3, and 4 Order 0 L=(4/3) 0 Order 1 L=(4/3) 1 Order 2 L=(4/3) 2 Order 3 L=(4/3) 3 Order 4 L=(4/3) 4 Çetin Kaya Koç Fall / 35

11 Order of Koch Curves Computational Thinking The length of order k Koch curve: (4/3) k k L The shortest distance between the start and the end points: 1 The longest distance: lim k (4/3)k = Niels Fabian, the kid, must have liked idle walking Like Billie Çetin Kaya Koç Fall / 35

12 The Family Circus Cartoons from 1960s Çetin Kaya Koç Fall / 35

13 Fractals Computational Thinking The Koch curve is self-similar, repeating, and has varying scales Geometric structures with such properties are called fractals A fractal is a natural phenomenon or a mathematical set that exhibits a repeating pattern that displays at every scale There is some disagreement among mathematicians about the precise definition of a fractal Çetin Kaya Koç Fall / 35

14 Fractals Computational Thinking In general, they are: infinitely self-similar, iterative, recursive, and they have a fractal dimension Fractals are not just geometric objects but also processes in time Fractal patterns with various degrees of self-similarity have been studied in images, structures and sounds Bach Cello Suite No. 3 - BWV 1009 Spider, the Structural Engineer Çetin Kaya Koç Fall / 35

15 Koch Snowflakes Computational Thinking Using the Koch lines, we can create various polygons For example, we start with an equilateral triangle, each side of which is an order 0 Koch line, and increase the line orders Or, we start with a square, each side of which is an order 0 Koch line, and increase the line orders Çetin Kaya Koç Fall / 35

16 The Dimension of the Koch Curve The length of order k Koch curve: (4/3) k What is its dimension? Since it is just a line (albeit one with many turns and bends), you might think that it is dimension is 1 Length Area Volume Dimension Point Line Square Cube What is the dimension of a Koch line? Area & Volume: Zero Length: Çetin Kaya Koç Fall / 35

17 The Dimension of a Fractal The concept of the dimension of an object may be purely a human construct A definition of the dimension for self-similar objects can be given, however, it is probably subjective A clearly desirable aspect of the definition is that it agrees with the intuitive results for simple objects A self-similar object is one that looks about the same on any scale Çetin Kaya Koç Fall / 35

18 Determining the Dimension The case of d = 1 Take a straight line and cut it in half We now have two self-similar copies of the original line, each of which can be scaled by a factor of 2 to yield the original line We call this the scaling factor (S = 2) of the smaller copies of the line If we could cut the line into any number P = N of equal pieces, the scaling factor for each would be S = N 1 1/2 1/2 We can define the dimension as log P log S = log N log N = 1 Çetin Kaya Koç Fall / 35

19 Determining the Dimension The case of d = 2 Take a square and cut it in half in each direction We now have 4 self-similar copies of the original square, each of which can be scaled by a factor of 2 to yield the original square 1 1/2 1/2 1/2 1 1/2 The number of copies is P = 4 and the scaling factor is S = 2 If we could cut the square into any number P = N 2 of equal pieces, the scaling factor for each would be S = N We obtain the dimension of the square as log P log S = log N2 log N = 2 Çetin Kaya Koç Fall / 35

20 Determining the Dimension The case of d = 3 Take a cube and cut it in half in each direction We now have 8 self-similar copies of the original cube, each of which can be scaled by a factor of 2 to yield the original cube 1 1/2 1/2 1/2 1 1/2 1 1/2 1/2 The number of copies is P = 8 and the scaling factor is S = 2 If we could cut the cube into any number P = N 3 of equal pieces, the scaling factor for each would be S = N We obtain the dimension of the cube as log P log S = log N3 log N = 3 Çetin Kaya Koç Fall / 35

21 The Dimension of the Koch Curve A Koch line breaks into 3 equal to pieces, therefore S = 3 We create 4 copies of each smaller line, making P = 4 1 1/3 1/3 1/3 1/3 We find the dimension of the Koch fractal as log P log S = log 4 log 3 = Çetin Kaya Koç Fall / 35

22 The Dimension of the Koch Curve Çetin Kaya Koç Fall / 35

23 Recursive Functions Computational Thinking Recursively computable fractals are particularly interesting: the programs are simple, short and didactic A function definition that contains a call to the function being defined is said to be recursive The general outline: One or more cases in which the function accomplishes its task by using recursive calls for smaller versions of the task One or more cases in which the function accomplishes its task without the use of any recursive calls. These cases without any recursive calls are called base cases or stopping cases Çetin Kaya Koç Fall / 35

24 Recursive Functions Computational Thinking S1 S2 S3 S4 S1 S2 S1 S2 S1 S2 S1 S2 S1 S2 S4 S4 S4 S4 S4 Çetin Kaya Koç Fall / 35

25 The Koch Line Computational Thinking Order 0 L = 1 Order /3 1/3 1/3 L = 1 1/3 1/3 Order 1 1/3 1/3 1/3 L = 4/3 Çetin Kaya Koç Fall / 35

26 The Koch Line Function def kochline(order,dist,t): if(order==0): t.forward(dist) else: kochline(order-1,dist/3,t) t.left(60) kochline(order-1,dist/3,t) t.right(120) kochline(order-1,dist/3,t) t.left(60) kochline(order-1,dist/3,t) Çetin Kaya Koç Fall / 35

27 Torn Line and Torn Square Order 0 L = 1 Order Cos(a)= 0.03/0.47 a = a a L = 1 (0.5, 0.47) Order 1 L = 4*0.47 (0, 0) (0.47, 0) (0.53, 0) (1, 0) Çetin Kaya Koç Fall / 35

28 H Tree Order 0 Computational Thinking Çetin Kaya Koç Fall / 35

29 H Tree Order 1 Computational Thinking Çetin Kaya Koç Fall / 35

30 Ternary Tree - Order 0 (x, y + d) d d (x, y) d (x d p 3/2,y d/2) (x + d p 3/2,y d/2) Çetin Kaya Koç Fall / 35

31 Ternary Tree - Order 1 d/2 d/2 d/2 d d d Çetin Kaya Koç Fall / 35

32 The Minkowski Fractal Order 0 and 1 Order 0 1/4 1/4 1/4 1/4 1/4 1/4 1/4 1/2 1/4 Order 1 1/4 1/4 Çetin Kaya Koç Fall / 35

33 The Minkowski Fractal Order 3 Çetin Kaya Koç Fall / 35

34 The Dimension of the Minkowski Fractal Order 0 1/4 1/4 1/4 1/4 1/4 1/4 1/4 1/2 1/4 Order 1 1/4 1/4 A Minkowski line breaks into 4 equal to pieces, therefore S = 4 We create 8 copies of each smaller line, making P = 8 Therefore, the dimension would be log P log S = log 8 log 4 = 1.5 Çetin Kaya Koç Fall / 35

35 Comparing Length and Dimensions Length: (4/3) k Dimension: Length: 2 k Dimension: 1.5 Çetin Kaya Koç Fall / 35