1. Computer Graphics Lecture 40 Fractals Taqdees A. Siddiqi cs602@vu. edu
Computer Graphics Lecture 40 Fractals Taqdees A. Siddiqi

2. Fractal are geometric patterns that is repeated at ever smaller scales to produce irregular shapes and surfaces that can not be represented by classical geometry.

3. Fractals are used in computer modeling of irregular patterns and structure in nature.

4. According to Webster's Dictionary a fractal is defined as being "derived from the Latin fractus meaning broken, uneven: any of various extremely irregular curves or shape that repeat themselves at any scale on which they are examined."

5. Mandelbrot, the discoverer of fractals gives two definitions:
"I coined fractal from the Latin adjective fractus. The corresponding Latin verb frangere means 'to break:' to create irregular fragments"

6. Every set with a non-integer (Hausdorff-Besicovitch) dimension (D) is a fractal. However not every fractal has an integer D. A fractal is by definition a set for which D strictly exceeds the topological dimension (D^).

7. Hausdorff-Besicovitch(Fractal Dimension)

8. To understand the second definition we need to be able to understand the fractal dimension. So first we have to look at understanding how to calculate the dimension of an object. Below we have three different objects.

9. As you can see the line is broken into 4 smaller lines
As you can see the line is broken into 4 smaller lines. Each of these lines is similar to the original line, but they are all 1/4 the scale. This is the idea of self similarity.

10. The square ahead is also broken into smaller pieces
The square ahead is also broken into smaller pieces. Each of which is 1/4th the size of the original. In this case it takes 16 of the smaller pieces to create the original.

12. As with the others the cube is also broken down into smaller cubes of 1/4 the size of the original. It takes 64 of these smaller cubes to create the the original cube.

14. By looking at this we begin to see a pattern:
4 = 4^1
16 = 4^2
64 = 4^3
This gives us the equation:
N = S^D

15. Where N is the number of small pieces that go into the larger one, S is the scale to which the smaller pieces compare to the larger one and D is the dimension.

16. Now solve for D in the previous equation; we find:
D = log N / log S
This dimension is the Hausdorff-Besicovitch dimension.

17. Koch Curve

18. Euclidean Geometry is the geometry of lines, planes, circles etc
Euclidean Geometry is the geometry of lines, planes, circles etc. It's simple and it works, and for a long time, mathematicians thought it was a reasonable representation of nature.

19. However, people soon discovered that they could draw (or at least begin to draw) certain curves and surfaces that could not be described by the classical geometry.

20. How hard can it be to draw a curve. Let us attempt to describe
How hard can it be to draw a curve? Let us attempt to describe. This is the Koch curve:

22. Draw a triangle.
If we say that each line is of length 1, then the total length of the curve is 3.

23. Original Triangles
New Triangles

24. Now take each edge in turn and add another triangle, a third of the size. So now there are 12 edges and 12 points. The length of the curve is now 4.
Repeat the process again, and again, forever.

26. length =

27. length =

28. length =
As we continue adding edges, the length of the curve increases. If we add edges forever, then length of the curve reaches infinity, but the whole curve nevertheless covers a finite area.

29. The curve is infinitely detailed
The curve is infinitely detailed. No matter how closely we zoom into the image, it always shows up more detail.

30. Self Similarity
So what do these mathematical curiosities have to do with the real world? Well, everything as it turns out. Such objects turn up all the time in the natural world.

31. Lets take a look at a common plant, the fern
Lets take a look at a common plant, the fern. A fern consists of a leaf, which is made up from many similar, but smaller leaves, each of which, in turn, is made from even smaller leaves.

32. The following figure is a standard fern
The following figure is a standard fern. We will see the overall theme of repeating leaves. Each smaller leaf looks similar to the larger leaf.

34. Looking a little closer, we can see that those small leaves are made up from even smaller leaves.

37. Of course, in reality, a fern does have a smallest leaf, though we’re sure every fern aspires to be like that one. What is interesting it that the program to generate this image is only a few lines long.

38. Fractal Geometry

39. Almost all geometric forms used for building man made objects belong to Euclidean geometry, they are comprised of lines, planes, rectangular volumes, arcs, cylinders, spheres, etc. These elements can be classified as belonging to an integer dimension, either 1, 2, or 3.

40. This concept of dimension can be described both intuitively and mathematically. Intuitively we say that a line is one dimensional because it only takes 1 number to uniquely define any point on it.

41. That one number could be the distance from the start of the line
That one number could be the distance from the start of the line. This applies equally well to the circumference of a circle, a curve, or the boundary of any object.

42. Any point “a” on one dimension curve can be represented by one number, the distance d from the start point.

43. A plane is two dimensional since in order to uniquely define any point on its surface we require two numbers. There are many ways to arrange the definition of these two numbers but we normally create an orthogonal coordinate system.

44. One of the two possible methods is to grid the surface and measure two distances along the grid lines.

45. The volume of some solid object is 3 dimensional on the same basis as above, it takes three numbers to uniquely define any point within the object.

47. Any point “a” on three dimensions can be uniquely represented by three numbers. Typically these three numbers are the coordinates of the point using an orthogonal coordinate system.

48. A more mathematical description of dimen0sion is based on how the "size" of an object behaves as the linear dimension increases. In one dimension consider a line segment.

49. If the linear dimension of the line segment is doubled then obviously the length (characteristic size) of the line has doubled. In two dimensions, if the linear dimensions of a rectangle for example is doubled then the characteristic size, the area, increases by a factor of 4.

50. In three dimensions if the linear dimension of a box are doubled then the volume increases by a factor of 8. This relationship between D, L and the resulting increase in size S can be generalised and written as

51. This is just telling us mathematically what we know from everyday experience. If we scale a two dimensional object for example then the area increases by the square of the scaling.

52. If we scale a 3-dimensional object the volume increases by the cube of the scale factor. Rearranging the above gives an expression for dimension depending on how the size changes as a function of linear scaling, namely

53. In the examples above the value of D is an integer, either 1, 2, or 3, depending on the dimension of the geometry. This relationship holds for all Euclidean shapes.

54. There are however many shapes which do not conform to the integer based idea of dimension given above in both the intuitive and mathematical descriptions.

55. That is, there are objects which appear to be curves for example but which a point on the curve cannot be uniquely described with just one number. If the earlier scaling formulation for dimension is applied the formula does not yield an integer.

56. There are shapes that lie in a plane but if they are linearly scaled by a factor L, the area does not increase by L squared but by some non integer amount. These geometries are called fractals!

57. One of the simpler fractal shapes is the von Koch snowflake
One of the simpler fractal shapes is the von Koch snowflake. The method of creating this shape is to repeatedly replace each line segment with the following 4 line segments.

59. The process starts with a single line segment and continues for ever
The process starts with a single line segment and continues for ever. The first few iterations of this procedure are shown ahead.

61. This demonstrates how a very simple generation rule for this shape can generate some unusual (fractal) properties. Unlike Euclidean shapes this object has detail at all levels.

62. If one magnifies an Euclidean shape such as the circumference of a circle it becomes a different shape, namely a striaght line. If we magnify this fractal more and more detail is uncovered, the detail is self similar or rather it is exactly self similar.

63. Note also that the "curve" on the right is not a fractal but only an approximation of one. This is no different from when one draws a circle, it is only an approximation to a perfect circle. At each iteration the length of the curve increases bu a factor of 4/3.

64. Thus the limiting curve is of infinite length and indeed the length between any two points of the curve is infinite. This curve manages to compress an infinite length into a finite area of the plane without intersecting itself!

65. Considering the intuitive notion of 1 dimensional shapes, although this object appears to be a curve with one starting point and one end point, it is not possible to uniquely specify any position along the curve with one number as we expect to be able to do with Euclidean curves which are 1 dimensional.

66. Although the method of creating this curve is straightforward, there is no algebraic formula the describes the points on the curve. Some of the major differences between fractal and Euclidean geometry are outlined in the following table.

67. Fractal
Euclidean
Modern Invention
No specific size or scale
Appropriate for geometry in nature
Described by an algorithm
Traditional
Based on a characteristic size or scale
Suits description of man made objects
Described by a usually simple formula

68. Firstly the recognition of fractal is very modern, formally been studied in the last 10 years compared to Euclidean geometry back over 2000 years. Secondly whereas Euclidean shapes normally have a few characteristic sizes or length scales fractals have so characteristic sizes.

69. Fractal shapes are self similar and independent of size or scaling
Fractal shapes are self similar and independent of size or scaling. Third, Euclidean geometry provides a good description of man made objects whereas fractals are required for a representation of naturally occurring geometries.

70. It is likely that this limitation of our traditional language of shape is responsible for the striking difference between mass produced objects and natural shapes. Finally, Euclidean geometries are defined by algebraic formulae, for example