This document discusses different types of fractals including the Koch snowflake, Sierpinski triangle, Pythagorean tree, and Mandelbrot set. It provides definitions of fractals and outlines their key properties like self-similarity and infinite complexity. Construction methods are described for different fractals along with examples and references to online animations showing their self-similar patterns. Applications of fractals are also briefly mentioned.

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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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