1. Fractals

2. The Koch Snowflake
First iteration
After
2 iterations

3. After 3 iterations

5. After n iterations

6. After iterations
(work with me here, people)

7. The Koch snowflake is six of these put together to form . . .
. . . well, a snowflake.

8. Notice that the perimeter of the Koch snowflake is infinite . . .
. . . but that the area it bounds is finite (indeed, it is
contained in the white square).

9. Area of Koch Snowflake
First, the area of triangle = a

10. Area of First iteration

11. Area of the Second Iteration

12. Area of the Third iteration

13. Formula for area of Koch Snowflake

15. Each of the six sides of the Koch snowflake is
self-similar: If you take a small copy of it . . .
. . . then dilate by a factor of
. . . you get four copies of the original.

16. But self-similarity is not what makes the Koch snowflake
a fractal! (Contrary to a common misconception.)
After all, many common geometric objects exhibit
self-similarity. Consider, for example, the humble
square.

17. If you take a small square . . .
. . . and dilate by a factor of
. . . then you get 4 copies of the original.
A square is self-similar, but it most certainly is not a fractal.

18. If you take a small square . . .
. . . and dilate by a factor of
. . . then you get 9 copies of the original.

19. Let k be the scale factor.
Let N be the number of copies of the original that you get.
Note that for the square, we have that:
Or in other words, we have:

20. Let’s compute
for some other shapes.

21. Line segment
Original
Dilated
k = scale factor = 2
N = number of copies of original = 2

22. Triangle
Original
Dilated
k = scale factor = 2
N = number of copies of original = 4

23. Cube
Original
Dilated
k = scale factor = 2
N = number of copies of original = 8

25. Shape Square 2 Line segment 1 Triangle Cube 3 What does
tell us about a shape?

26. That’s right:
tells us the dimension of the shape.
(Note that for this to make sense, the shape has to be
self-similar.)
So for a self-similar shape, we can take
to be the definition of its dimension.
(It turns out that this definition coincides with a much more
general definition of dimension called the fractal dimension.)

27. Now let’s recall what k and N were for one side of the
Koch snowflake:
k = scale factor = 3
N = number of copies of original = 4

28. So each side of the Koch snowflake is approximately
1.261-dimensional.
That’s what makes the Koch snowflake a fractal – the fact that
its dimension is not an integer.
Even shapes which are not self-similar can be fractals. The
most famous of these is the Mandelbrot set.

29. The Sierpinski Carpet

30. perimeter = 4(3) + 4(1)
area =
Start with a square of side length 3, with a square of side
length 1 removed from its center.

31. perimeter =
area =
Think of this shape as consisting of eight small squares, each
of side length 1.
From each small square, remove its central square.

32. Iterate.

33. perimeter =
area =
Iterate.

34. perimeter =
area =
Iterate.

35. perimeter =
area =
Iterate.

36. area =
perimeter =
Iterate.

37. The Sierpinski carpet is what’s left after
you’re finished removing everything.
In other words, it’s the intersection of all the previous sets.

38. perimeter =

39. area =
= 0

40. Weird So the Sierpinski carpet has an infinite perimeter – but
it bounds a region with an area of zero!
Weird

41. Your turn: compute the fractal dimension of the Sierpinski carpet.

42. The Sierpinski carpet has a 3-dimensional analogue called
the Menger sponge.

43. Its surface area is infinite, yet it bounds a region of zero volume.

44. The fractal dimension of the Menger sponge is: