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First Introduction to Fractals

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Definition Snowflake Sierpinski fractals Pythagorean tree Mandelbrot set Newton fractals Natural fractals Applications
Why I...

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Definition Snowflake Sierpinski fractals Pythagorean tree Mandelbrot set Newton fractals Natural fractals Applications
Why I...

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Definition Snowflake Sierpinski fractals Pythagorean tree Mandelbrot set Newton fractals Natural fractals Applications
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First Introduction to Fractals

  1. 1. Definition Snowflake Sierpinski fractals Pythagorean tree Mandelbrot set Newton fractals Natural fractals Applications Why I Love Fractals? Dr V N Krishnachandran Why I Love Fractals
  2. 2. Definition Snowflake Sierpinski fractals Pythagorean tree Mandelbrot set Newton fractals Natural fractals Applications Why I Love Fractals? Why I Love Fractals! Dr V N Krishnachandran Vidya Academy of Science & Technology Thalakkottukara, Thrissur - 680501 Dr V N Krishnachandran Why I Love Fractals
  3. 3. Definition Snowflake Sierpinski fractals Pythagorean tree Mandelbrot set Newton fractals Natural fractals Applications Outline 1 Definition 2 van Koch Snowflake 3 Sierpinski fractals 4 Pythagorean tree 5 Mandelbrot set 6 Newton-Raphson fractals 7 Natural fractals 8 Applications Dr V N Krishnachandran Why I Love Fractals
  4. 4. Definition Snowflake Sierpinski fractals Pythagorean tree Mandelbrot set Newton fractals Natural fractals Applications Definition What is a fractal? Dr V N Krishnachandran Why I Love Fractals
  5. 5. Definition Snowflake Sierpinski fractals Pythagorean tree Mandelbrot set Newton fractals Natural fractals Applications What is a fractal? A fractal is a mathematical set or a natural phenomenon with infinite complexity and approximate self-similarity at any scale. Dr V N Krishnachandran Why I Love Fractals
  6. 6. Definition Snowflake Sierpinski fractals Pythagorean tree Mandelbrot set Newton fractals Natural fractals Applications What is a fractal? Let us see examples. Dr V N Krishnachandran Why I Love Fractals
  7. 7. Definition Snowflake Sierpinski fractals Pythagorean tree Mandelbrot set Newton fractals Natural fractals Applications What is a fractal? Romanesco Broccoli : This variant form of cauliflower is a natural fractal (see detail in next slide). Dr V N Krishnachandran Why I Love Fractals
  8. 8. Definition Snowflake Sierpinski fractals Pythagorean tree Mandelbrot set Newton fractals Natural fractals Applications What is a fractal? Romanesco Broccoli : Detail Dr V N Krishnachandran Why I Love Fractals
  9. 9. Definition Snowflake Sierpinski fractals Pythagorean tree Mandelbrot set Newton fractals Natural fractals Applications Fractal 1: van Koch snowflake Fractal 1: van Koch snowflake A curve resembling a snowflake A real snowflake Dr V N Krishnachandran Why I Love Fractals
  10. 10. Definition Snowflake Sierpinski fractals Pythagorean tree Mandelbrot set Newton fractals Natural fractals Applications Fractal 1: van Koch snowflake Construction of the curve. Dr V N Krishnachandran Why I Love Fractals
  11. 11. Definition Snowflake Sierpinski fractals Pythagorean tree Mandelbrot set Newton fractals Natural fractals Applications Fractal 1: van Koch snowflake Step 1 Dr V N Krishnachandran Why I Love Fractals
  12. 12. Definition Snowflake Sierpinski fractals Pythagorean tree Mandelbrot set Newton fractals Natural fractals Applications Fractal 1: van Koch snowflake Dr V N Krishnachandran Why I Love Fractals
  13. 13. Definition Snowflake Sierpinski fractals Pythagorean tree Mandelbrot set Newton fractals Natural fractals Applications Fractal 1: van Koch snowflake Step 2 Dr V N Krishnachandran Why I Love Fractals
  14. 14. Definition Snowflake Sierpinski fractals Pythagorean tree Mandelbrot set Newton fractals Natural fractals Applications Fractal 1: van Koch snowflake Step 3 Dr V N Krishnachandran Why I Love Fractals
  15. 15. Definition Snowflake Sierpinski fractals Pythagorean tree Mandelbrot set Newton fractals Natural fractals Applications Fractal 1: van Koch snowflake Step 4 Dr V N Krishnachandran Why I Love Fractals
  16. 16. Definition Snowflake Sierpinski fractals Pythagorean tree Mandelbrot set Newton fractals Natural fractals Applications Fractal 1: van Koch snowflake Step 5 Dr V N Krishnachandran Why I Love Fractals
  17. 17. Definition Snowflake Sierpinski fractals Pythagorean tree Mandelbrot set Newton fractals Natural fractals Applications Fractal 1: van Koch snowflake . . . and continue without stopping . . . Dr V N Krishnachandran Why I Love Fractals
  18. 18. Definition Snowflake Sierpinski fractals Pythagorean tree Mandelbrot set Newton fractals Natural fractals Applications Fractal 1: van Koch snowflake Self similarity of van Koch snowflake Dr V N Krishnachandran Why I Love Fractals
  19. 19. Definition Snowflake Sierpinski fractals Pythagorean tree Mandelbrot set Newton fractals Natural fractals Applications Fractal 1: van Koch snowflake Here is a website with animation showing the self-similarity of van Koch snowflake. https://www.tumblr.com/search/koch+snowflake Dr V N Krishnachandran Why I Love Fractals
  20. 20. Definition Snowflake Sierpinski fractals Pythagorean tree Mandelbrot set Newton fractals Natural fractals Applications Fractal 1: van Koch snowflake Step Length of Number of Perimeter Number a segment segments 1 a 3 3a 2 a/3 4 × 3 = 12 12 × (a/3) 3 a/9 4 × 12 = 48 48 × (a/9) 4 a/27 4 × 48 = 192 192 × (a/27) · · · · · · · · · · · · n a/3n−1 4 × 3n−1a 3 × (4/3)n−1a Perimeter −→ ∞ as n −→ ∞. Dr V N Krishnachandran Why I Love Fractals
  21. 21. Definition Snowflake Sierpinski fractals Pythagorean tree Mandelbrot set Newton fractals Natural fractals Applications Fractal 1: van Koch snowflake Area bounded by the snowflake = (8/5) × area of initial triangle Dr V N Krishnachandran Why I Love Fractals
  22. 22. Definition Snowflake Sierpinski fractals Pythagorean tree Mandelbrot set Newton fractals Natural fractals Applications Fractal 2: Sierpinski triangle Fractal 2 : Sierpinski triangle Dr V N Krishnachandran Why I Love Fractals
  23. 23. Definition Snowflake Sierpinski fractals Pythagorean tree Mandelbrot set Newton fractals Natural fractals Applications Fractal 2: Sierpinski triangle Construction of the fractal. Dr V N Krishnachandran Why I Love Fractals
  24. 24. Definition Snowflake Sierpinski fractals Pythagorean tree Mandelbrot set Newton fractals Natural fractals Applications Fractal 2: Sierpinski triangle . . . and continue without stopping Dr V N Krishnachandran Why I Love Fractals
  25. 25. Definition Snowflake Sierpinski fractals Pythagorean tree Mandelbrot set Newton fractals Natural fractals Applications Fractal 2: Sierpinski triangle Wikipedia page on Sierpinski triangle has an animation showing the construction of the Sierpinski triangle. Dr V N Krishnachandran Why I Love Fractals
  26. 26. Definition Snowflake Sierpinski fractals Pythagorean tree Mandelbrot set Newton fractals Natural fractals Applications Fractal 2: Sierpinski triangle Here is an animation showing the self-similarity of Sierpinski triangle. http://www.lutanho.net/fractal/sierpa.html Dr V N Krishnachandran Why I Love Fractals
  27. 27. Definition Snowflake Sierpinski fractals Pythagorean tree Mandelbrot set Newton fractals Natural fractals Applications Fractal 3: Sierpinski tetrahedron Fractal 3: Sierpinski tetrahedron Dr V N Krishnachandran Why I Love Fractals
  28. 28. Definition Snowflake Sierpinski fractals Pythagorean tree Mandelbrot set Newton fractals Natural fractals Applications Fractal 3: Sierpinski tetrahedron Construction of the fractal. Dr V N Krishnachandran Why I Love Fractals
  29. 29. Definition Snowflake Sierpinski fractals Pythagorean tree Mandelbrot set Newton fractals Natural fractals Applications Fractal 3: Sierpinski tetrahedron Steps 1, 2 Dr V N Krishnachandran Why I Love Fractals
  30. 30. Definition Snowflake Sierpinski fractals Pythagorean tree Mandelbrot set Newton fractals Natural fractals Applications Fractal 3: Sierpinski tetrahedron Steps 2, 3 Dr V N Krishnachandran Why I Love Fractals
  31. 31. Definition Snowflake Sierpinski fractals Pythagorean tree Mandelbrot set Newton fractals Natural fractals Applications Fractal 3: Sierpinski tetrahedron Step 6, and continue without stopping Dr V N Krishnachandran Why I Love Fractals
  32. 32. Definition Snowflake Sierpinski fractals Pythagorean tree Mandelbrot set Newton fractals Natural fractals Applications Fractal 4: Pythagorean tree Fractal 4: Pythagorean tree Dr V N Krishnachandran Why I Love Fractals
  33. 33. Definition Snowflake Sierpinski fractals Pythagorean tree Mandelbrot set Newton fractals Natural fractals Applications Fractal 4: Pythagorean tree Construction of the fractal. Dr V N Krishnachandran Why I Love Fractals
  34. 34. Definition Snowflake Sierpinski fractals Pythagorean tree Mandelbrot set Newton fractals Natural fractals Applications Fractal 4: Pythagorean tree Steps 1, 2 Dr V N Krishnachandran Why I Love Fractals
  35. 35. Definition Snowflake Sierpinski fractals Pythagorean tree Mandelbrot set Newton fractals Natural fractals Applications Fractal 4: Pythagorean tree Steps 3, 4 and continue without stopping to get the image shown in the next slide (with added colors). Dr V N Krishnachandran Why I Love Fractals
  36. 36. Definition Snowflake Sierpinski fractals Pythagorean tree Mandelbrot set Newton fractals Natural fractals Applications Fractal 4: Pythagorean tree Dr V N Krishnachandran Why I Love Fractals
  37. 37. Definition Snowflake Sierpinski fractals Pythagorean tree Mandelbrot set Newton fractals Natural fractals Applications Fractal 4: Pythagorean tree Modifications to Pythagorean tree Dr V N Krishnachandran Why I Love Fractals
  38. 38. Definition Snowflake Sierpinski fractals Pythagorean tree Mandelbrot set Newton fractals Natural fractals Applications Fractal 4: Pythagorean tree Dr V N Krishnachandran Why I Love Fractals
  39. 39. Definition Snowflake Sierpinski fractals Pythagorean tree Mandelbrot set Newton fractals Natural fractals Applications Fractal 4: Pythagorean tree Dr V N Krishnachandran Why I Love Fractals
  40. 40. Definition Snowflake Sierpinski fractals Pythagorean tree Mandelbrot set Newton fractals Natural fractals Applications The complex world! Dr V N Krishnachandran Why I Love Fractals
  41. 41. Definition Snowflake Sierpinski fractals Pythagorean tree Mandelbrot set Newton fractals Natural fractals Applications The Mandelbrot set Dr V N Krishnachandran Why I Love Fractals
  42. 42. Definition Snowflake Sierpinski fractals Pythagorean tree Mandelbrot set Newton fractals Natural fractals Applications Fractal 5: Mandelbrot set Dr V N Krishnachandran Why I Love Fractals
  43. 43. Definition Snowflake Sierpinski fractals Pythagorean tree Mandelbrot set Newton fractals Natural fractals Applications Fractal 5: Mandelbrot set Mandelbrot set discovered by Benoit Mandelbrot in 1979. Dr V N Krishnachandran Why I Love Fractals
  44. 44. Definition Snowflake Sierpinski fractals Pythagorean tree Mandelbrot set Newton fractals Natural fractals Applications Fractal 5: Mandelbrot set The mathematics of Mandelbrot set Dr V N Krishnachandran Why I Love Fractals
  45. 45. Definition Snowflake Sierpinski fractals Pythagorean tree Mandelbrot set Newton fractals Natural fractals Applications Fractal 5: Mandelbrot set Consider the iteration scheme: zn+1 = z2 n + c Dr V N Krishnachandran Why I Love Fractals
  46. 46. Definition Snowflake Sierpinski fractals Pythagorean tree Mandelbrot set Newton fractals Natural fractals Applications Fractal 5: Mandelbrot set 1 Choose a fixed value for c, say, c = 1 + i. 2 Mark the point represented by c in the complex plane. 3 Let z0 = 0. 4 Compute z1, z2, z3, . . . as follows: z1 = z2 0 + c z2 = z2 1 + c z3 = z2 2 + c · · · 5 If the values z1, z2, z3, . . . increases indefinitely, mark the point c with red color. Otherwise, mark the point c with blue color. 6 Repeat this by choosing different points in the complex plane. 7 The blue colored region is the Mandelbrot set. Dr V N Krishnachandran Why I Love Fractals
  47. 47. Definition Snowflake Sierpinski fractals Pythagorean tree Mandelbrot set Newton fractals Natural fractals Applications Fractal 5: Mandelbrot set We can use a computer program to generate the Mandelbrot set. The Mandelbrot set obtained by computing z1, z2, . . . pixel by pixel, starting with z0 = 0. If the values do not go to infinity after a large number of iterations the present pixel value is in the Mandelbrot set and is then colored blue. If the sequence diverge then the pixel is colored red. Sometimes, the pixel is colored according to how fast the divergence is. Dr V N Krishnachandran Why I Love Fractals
  48. 48. Definition Snowflake Sierpinski fractals Pythagorean tree Mandelbrot set Newton fractals Natural fractals Applications Fractal 5: Mandelbrot set Dr V N Krishnachandran Why I Love Fractals
  49. 49. Definition Snowflake Sierpinski fractals Pythagorean tree Mandelbrot set Newton fractals Natural fractals Applications Fractal 6 : Newton-Raphson fractals Fractal 6 : Newton-Raphson fractals Dr V N Krishnachandran Why I Love Fractals
  50. 50. Definition Snowflake Sierpinski fractals Pythagorean tree Mandelbrot set Newton fractals Natural fractals Applications Fractal 6 : Newton-Raphson fractals Dr V N Krishnachandran Why I Love Fractals
  51. 51. Definition Snowflake Sierpinski fractals Pythagorean tree Mandelbrot set Newton fractals Natural fractals Applications Fractal 6 : Newton-Raphson fractal We consider the Newton-Raphson fractal corresponding to the equation f (z) ≡ z3 − 1 = 0. The roots of this equation are 1, 1 2(−1 + i √ 3), 1 2(−1 − i √ 3). Dr V N Krishnachandran Why I Love Fractals
  52. 52. Definition Snowflake Sierpinski fractals Pythagorean tree Mandelbrot set Newton fractals Natural fractals Applications Fractal 6 : Newton-Raphson fractal Newton-Raphson method. 1 Choose an initial approximation z0 to a root. 2 Compute z1, z2, . . . by zn+1 = zn − f (zn) f (zn) = zn − z3 n − 1 3z2 n 3 The numbers z1, z2, . . . approach a root. Dr V N Krishnachandran Why I Love Fractals
  53. 53. Definition Snowflake Sierpinski fractals Pythagorean tree Mandelbrot set Newton fractals Natural fractals Applications Fractal 6 : Newton-Raphson fractal 1 Choose a fixed complex number c. Mark the point c in the complex plane. 2 Let z0 = c. 3 Compute z1, z2, . . . using the formula zn+1 = zn − z3 n − 1 3z2 n . 4 If z0, z1, z2, . . . approach 1, color the point c red. If z0, z1, z2, . . . approach 1 2 (−1 + i √ 3), color the point c green. If z0, z1, z2, . . . approach 1 2 (−1 − i √ 3), color the point c blue. 5 Repeat this by choosing various points in the complex plane. 6 The resulting figure is the Newton-Raphson fractal corres-ponding to the equation z3 − 1 = 0 (see next slide). Dr V N Krishnachandran Why I Love Fractals
  54. 54. Definition Snowflake Sierpinski fractals Pythagorean tree Mandelbrot set Newton fractals Natural fractals Applications Fractal 6 : Newton-Raphson fractals Dr V N Krishnachandran Why I Love Fractals
  55. 55. Definition Snowflake Sierpinski fractals Pythagorean tree Mandelbrot set Newton fractals Natural fractals Applications Fractal 6 : Newton-Raphson fractal The equation z4 + z3 − 1 = 0 produces the fractal shown in the next slide. Dr V N Krishnachandran Why I Love Fractals
  56. 56. Definition Snowflake Sierpinski fractals Pythagorean tree Mandelbrot set Newton fractals Natural fractals Applications Fractal 6 : Newton-Raphson fractal Dr V N Krishnachandran Why I Love Fractals
  57. 57. Definition Snowflake Sierpinski fractals Pythagorean tree Mandelbrot set Newton fractals Natural fractals Applications Software We can explore the complexity of the Mandelbrot set, the Newton-Raphson fractal and many others using the Xaos program available at: http://matek.hu/xaos/doku.php Dr V N Krishnachandran Why I Love Fractals
  58. 58. Definition Snowflake Sierpinski fractals Pythagorean tree Mandelbrot set Newton fractals Natural fractals Applications Into the natural world! Dr V N Krishnachandran Why I Love Fractals
  59. 59. Definition Snowflake Sierpinski fractals Pythagorean tree Mandelbrot set Newton fractals Natural fractals Applications Fractal 7 : Natural fractals Romanesco Broccoli : This variant form of cauliflower is the ultimate fractal vegetable (see detail in next slide). Dr V N Krishnachandran Why I Love Fractals
  60. 60. Definition Snowflake Sierpinski fractals Pythagorean tree Mandelbrot set Newton fractals Natural fractals Applications Fractal 7 : Natural fractals Romanesco Broccoli : Detail Dr V N Krishnachandran Why I Love Fractals
  61. 61. Definition Snowflake Sierpinski fractals Pythagorean tree Mandelbrot set Newton fractals Natural fractals Applications Fractal 7 : Natural fractals Oak tree, formed by a sprout branching, and then each of the branches branching again, etc. Dr V N Krishnachandran Why I Love Fractals
  62. 62. Definition Snowflake Sierpinski fractals Pythagorean tree Mandelbrot set Newton fractals Natural fractals Applications Fractal 7 : Natural fractals River network in China, formed by erosion from repeated rainfall flowing downhill for millions of years. Dr V N Krishnachandran Why I Love Fractals
  63. 63. Definition Snowflake Sierpinski fractals Pythagorean tree Mandelbrot set Newton fractals Natural fractals Applications Fractal 7 : Natural fractals Our lungs are branching fractals with a surface area of approximately 100 m2. Dr V N Krishnachandran Why I Love Fractals
  64. 64. Definition Snowflake Sierpinski fractals Pythagorean tree Mandelbrot set Newton fractals Natural fractals Applications Fractal 7 : Natural fractals The plant kingdom is full of spirals. An agave cactus forms its spiral by growing new pieces rotated by a fixed angle. Dr V N Krishnachandran Why I Love Fractals
  65. 65. Definition Snowflake Sierpinski fractals Pythagorean tree Mandelbrot set Newton fractals Natural fractals Applications What are fractals useful for? Dr V N Krishnachandran Why I Love Fractals
  66. 66. Definition Snowflake Sierpinski fractals Pythagorean tree Mandelbrot set Newton fractals Natural fractals Applications Applications of fractals Fractal antennas A fractal antenna is an antenna that uses a fractal, self-similar design to maximize the length, or increase the perimeter of material that can receive or transmit electromagnetic radiation within a given total surface area or volume. Dr V N Krishnachandran Why I Love Fractals
  67. 67. Definition Snowflake Sierpinski fractals Pythagorean tree Mandelbrot set Newton fractals Natural fractals Applications Applications of fractals Fractal compression Fractals have been used for developing algorithms for image compression. Dr V N Krishnachandran Why I Love Fractals
  68. 68. Definition Snowflake Sierpinski fractals Pythagorean tree Mandelbrot set Newton fractals Natural fractals Applications Applications of fractals Fractal compression Natural fern leaves (left) and fractal generated fern leaves (right) Dr V N Krishnachandran Why I Love Fractals
  69. 69. Definition Snowflake Sierpinski fractals Pythagorean tree Mandelbrot set Newton fractals Natural fractals Applications Applications of fractals Dr V N Krishnachandran Why I Love Fractals
  70. 70. Definition Snowflake Sierpinski fractals Pythagorean tree Mandelbrot set Newton fractals Natural fractals Applications Thanks for patient hearing. Dr V N Krishnachandran Why I Love Fractals

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