Beautiful Repetitions
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1 Beautiful Repetitions 5-minute introduction to Iterations & Fractals Gaurish Korpal (gaurish4math.wordpress.com) National Institute of Science Education and Research, Bhubaneswar March 28, 2015 Gaurish Korpal(gaurish4math.wordpress.com) (NISER, Bhubaneswar) Beautiful Repetitions March 28, / 16
2 Is this ugly? Gaurish Korpal(gaurish4math.wordpress.com) (NISER, Bhubaneswar) Beautiful Repetitions March 28, / 16
3 Technical Terms - I Iteration OR Iterative Process Any action which is performed repeatedly will be referred to as an iteration. The specific rule used to do the iteration is the iterative function. The input value with which we start the iteration is the seed of the iteration. Gaurish Korpal(gaurish4math.wordpress.com) (NISER, Bhubaneswar) Beautiful Repetitions March 28, / 16
4 Technical Terms - I Iteration OR Iterative Process Any action which is performed repeatedly will be referred to as an iteration. The specific rule used to do the iteration is the iterative function. The input value with which we start the iteration is the seed of the iteration. Iteration Sequence OR Orbit OR Trajectory The sequence which is generated by an iterative function together with a seed is an iteration sequence. A convenient short form for this is: < x; f >, where x is seed and f is iterative function. Gaurish Korpal(gaurish4math.wordpress.com) (NISER, Bhubaneswar) Beautiful Repetitions March 28, / 16
5 Technical Terms - I Iteration OR Iterative Process Any action which is performed repeatedly will be referred to as an iteration. The specific rule used to do the iteration is the iterative function. The input value with which we start the iteration is the seed of the iteration. Iteration Sequence OR Orbit OR Trajectory The sequence which is generated by an iterative function together with a seed is an iteration sequence. A convenient short form for this is: < x; f >, where x is seed and f is iterative function. Cycle An orbit that returns to its starting value after a certain number of steps is called a cycle. The number of entries in a cycle is called its length. Gaurish Korpal(gaurish4math.wordpress.com) (NISER, Bhubaneswar) Beautiful Repetitions March 28, / 16
6 Algebraic Iterations The 3n + 1 conjecture concerns the following innocent seeming arithmetic procedure applied to integers described by iterative function, given by : f (n) = { 3n+1 2 if n is odd n 2 if n is even Gaurish Korpal(gaurish4math.wordpress.com) (NISER, Bhubaneswar) Beautiful Repetitions March 28, / 16
7 Algebraic Iterations The 3n + 1 conjecture concerns the following innocent seeming arithmetic procedure applied to integers described by iterative function, given by : Statement of Conjecture f (n) = { 3n+1 2 if n is odd n 2 if n is even Starting from any positive integer n, iterations of the function f (n) will eventually reach the number 1. Thereafter iterations will cycle, taking successive values 1, 2, 1,... Gaurish Korpal(gaurish4math.wordpress.com) (NISER, Bhubaneswar) Beautiful Repetitions March 28, / 16
8 Algebraic Iterations The 3n + 1 conjecture concerns the following innocent seeming arithmetic procedure applied to integers described by iterative function, given by : Statement of Conjecture f (n) = { 3n+1 2 if n is odd n 2 if n is even Starting from any positive integer n, iterations of the function f (n) will eventually reach the number 1. Thereafter iterations will cycle, taking successive values 1, 2, 1,... Can you Prove it? This is around 85 year old conjecture. This conjecture was independently discovered by many mathematicians so is also called Collatz conjecture, Ulam conjecture, Kakutanis problem, Thwaites conjecture, Hasses algorithm and Syracuse problem. Gaurish Korpal(gaurish4math.wordpress.com) (NISER, Bhubaneswar) Beautiful Repetitions March 28, / 16
9 Geometric Iterations Let S 0 be a square, and let its vertices be labelled in cyclic order as A,B,C,D. D C A B Gaurish Korpal(gaurish4math.wordpress.com) (NISER, Bhubaneswar) Beautiful Repetitions March 28, / 16
10 Geometric Iterations Let S 0 be a square, and let its vertices be labelled in cyclic order as A,B,C,D. D C A B Let k be a number lying between 0 and 1. Gaurish Korpal(gaurish4math.wordpress.com) (NISER, Bhubaneswar) Beautiful Repetitions March 28, / 16
11 Geometric Iterations Let points E,F,G,H, respectively, be located on the sides AB, BC, CD, DA of S 0, according to the following rule: AE AB = BF BC = CG CD = DH DA = k The segments EF, FG, GH, HE are drawn, giving another square, S 1, which we take to be the output of a function with input S 0. Gaurish Korpal(gaurish4math.wordpress.com) (NISER, Bhubaneswar) Beautiful Repetitions March 28, / 16
12 Geometric Iterations For k = 0.1 we will get following S 1 : D H G C A E F B Gaurish Korpal(gaurish4math.wordpress.com) (NISER, Bhubaneswar) Beautiful Repetitions March 28, / 16
13 Geometric Iterations For k = 0.1 we will get following S 1 : D H G C A E F B Observe that S 1 is smaller than S 0 and is wholly contained within it; it has same center as its parent square, but has been turned through a small angle. Gaurish Korpal(gaurish4math.wordpress.com) (NISER, Bhubaneswar) Beautiful Repetitions March 28, / 16
14 Geometric Iterations Now what we defined above was a process which we will iterate again and again. If we iterate the process defined above, then after 20 iterations we will get: Gaurish Korpal(gaurish4math.wordpress.com) (NISER, Bhubaneswar) Beautiful Repetitions March 28, / 16
15 Geometric Iterations Now what we defined above was a process which we will iterate again and again. If we iterate the process defined above, then after 20 iterations we will get: [Source: gldfsh.deviantart.com] Voilà! aurish Korpal(gaurish4math.wordpress.com) (NISER, Bhubaneswar) Beautiful Repetitions March 28, / 16
![We are surrounded by Fractals [Source: Hudhud Cyclone, ABP Live] The spiral is an extremely common fractal in nature, found over](/docs-images/95/126321966/images/16-0.jpg "a huge range of scales. Gaurish Korpal(gaurish4math.wordpress.")
16 We are surrounded by Fractals [Source: Hudhud Cyclone, ABP Live] The spiral is an extremely common fractal in nature, found over a huge range of scales. Gaurish Korpal(gaurish4math.wordpress.com) (NISER, Bhubaneswar) Beautiful Repetitions March 28, / 16
17 Technical Terms - II Fractal A fractal is a never ending pattern that repeats itself at different scales. A fractal is made by iterations. Eg: Cauliflower; coastline aurish Korpal(gaurish4math.wordpress.com) (NISER, Bhubaneswar) Beautiful Repetitions March 28, / 16
18 Technical Terms - II Fractal A fractal is a never ending pattern that repeats itself at different scales. A fractal is made by iterations. Eg: Cauliflower; coastline Self-Similarity The property of fractal to repeat itself at different scales is called self-similarity.this is analogous to zooming in on digital images to uncover finer, previously invisible, new structure. If this is done on fractals, however, no new detail appears; the same pattern repeats over and over. Eg: Mandelbrot set fine detail resembles the detail at low magnification. c Nadim National Ghaznavi Science Day aurish Korpal(gaurish4math.wordpress.com) (NISER, Bhubaneswar) Beautiful Repetitions March 28, / 16
19 Algebraic Fractal We can create fractals by iterating a simple equation. Since the equations must be calculated thousands of times, we need computers to study them. Gaurish Korpal(gaurish4math.wordpress.com) (NISER, Bhubaneswar) Beautiful Repetitions March 28, / 16
20 Algebraic Fractal We can create fractals by iterating a simple equation. Since the equations must be calculated thousands of times, we need computers to study them. Mendelbrot Set c Wolfgang Beyer [Wikipedia] This is a false colour image of a set, colour coded according to behaviour of elements of the set towards an iterative function. Gaurish Korpal(gaurish4math.wordpress.com) (NISER, Bhubaneswar) Beautiful Repetitions March 28, / 16
21 Algebraic Fractal We can create fractals by iterating a simple equation. Since the equations must be calculated thousands of times, we need computers to study them. Mendelbrot Set c Wolfgang Beyer [Wikipedia] This is a false colour image of a set, colour coded according to behaviour of elements of the set towards an iterative function. I can t explain the Mathematics behind this fractal since it involves iterations on set of complex numbers (C)! Gaurish Korpal(gaurish4math.wordpress.com) (NISER, Bhubaneswar) Beautiful Repetitions March 28, / 16
22 Geometric Fractal 1 Start with a solid equilateral triangle Gaurish Korpal(gaurish4math.wordpress.com) (NISER, Bhubaneswar) Beautiful Repetitions March 28, / 16
23 Geometric Fractal 1 Start with a solid equilateral triangle 2 Divide it into four equal pieces, and remove the middle piece. Gaurish Korpal(gaurish4math.wordpress.com) (NISER, Bhubaneswar) Beautiful Repetitions March 28, / 16
24 Geometric Fractal 1 Start with a solid equilateral triangle 2 Divide it into four equal pieces, and remove the middle piece. 3 Then divide each of the three remaining triangles into four equal pieces, and remove the middle piece of each one. Gaurish Korpal(gaurish4math.wordpress.com) (NISER, Bhubaneswar) Beautiful Repetitions March 28, / 16
25 Geometric Fractal If this pattern is continued forever, then the result is Sierpinski s triangle. Gaurish Korpal(gaurish4math.wordpress.com) (NISER, Bhubaneswar) Beautiful Repetitions March 28, / 16
26 Geometric Fractal If this pattern is continued forever, then the result is Sierpinski s triangle. If you made three copies of the triangle, and shrunk them down by a factor of one-half, then you could cover the original triangle. Because the figure has this self-similarity property, it is a fractal Gaurish Korpal(gaurish4math.wordpress.com) (NISER, Bhubaneswar) Beautiful Repetitions March 28, / 16
27 Fractals in Everyday life The bronchial network in human respiratory system is a fractal c E. Weibel Gaurish Korpal(gaurish4math.wordpress.com) (NISER, Bhubaneswar) Beautiful Repetitions March 28, / 16
28 Fractals in Everyday life The bronchial network in human respiratory system is a fractal c E. Weibel The artificial landscapes like Mountains, etc. which you see in cinema are actually fractals. (they just appear to be mountains) [Source: djconnel.com/vue] Gaurish Korpal(gaurish4math.wordpress.com) (NISER, Bhubaneswar) Beautiful Repetitions March 28, / 16
29 This is just start... Fractals constitute a relatively modern discovery; they date to the latter half of the 20th century and may be said to have originated in the work of the French Mathematician Benoit Mandelbrot. So, in this presentation I have shown you just tip of iceberg. Go explore yourself... Video Fractals and the art of Roughness by Benoit Mandelbrot; TED Talk Books A Gateway to Modern Mathematics : Adventures in Iterations I & II by Shailesh A Shirali; Ramanujan Mathematical Society : Little Mathematical Treasures, ISBN: & Chaos, Fractals and Self-Organisation by Arvind Kumar; National Book Trust, ISBN: Gaurish Korpal(gaurish4math.wordpress.com) (NISER, Bhubaneswar) Beautiful Repetitions March 28, / 16
30 Thank you! Curious? For any further discussion on mathematics feel free to write me at: Download This presentation can be downloaded from: gaurish4math.wordpress.com/downloads/my-notes/ Gaurish Korpal(gaurish4math.wordpress.com) (NISER, Bhubaneswar) Beautiful Repetitions March 28, / 16
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