Fractal Concept
( Reference: Fractal Programming in C )
 

What are Fractals ?

Mandelbrot’s classical definition:

"A fractal is a curve whose Hausdorff-Besicovitch dimension is larger than its Euclidian dimension."

Recursive techniques and iterated expressions were found that could describe curves that have fractional dimensions.

　
What are Fractals good for ?

Ignoring the artistic value, "Fractals appear to provide solutions to many previously unanswered questions at the frontiers of the physical sciences."

How are Fractals Used

Fractal curves are the natural way of representing objects and as a means of representing natural scenes. Moreover, fractals occur naturally in the expressions for mathematical phenomena as varied as the prediction of weather systems, the describing of turbulent flow of liquid, and the growth and decline of population. Fractals are also useful in dimensional transformations that can used for expressing and compressing graphical data.

Basic Considerations

Investigation of the chaotic field of fractal:

1. Intuition leads us to believe that fractal curves should have a dimension greater than their traditional geometric dimension
2. There is now sound mathematical grounding for accepting this premise
3. Fractal curves are associated with many physical and natural phenomena.
4. Fractal often possess a rare and unusual beauty. No doubt , this is partly true because fractals correspond to the way in which nature produces those shapes that we are most familiar with that basically define our ideas of "the beautiful
5. Fractals have the unusual characteristic that they can be defined totally by relatively simple mathematical equations, yet they are not periodic. Thus, the progression of the fractal curve may differ widely if we start at just slightly different points in space, so unless we can measure where we are with absolute precision we cannot be sure just what the progression of the curve will like. This is in spite of the fact that the curve is defined through all of its wanderings by very simple iterated expressions.
6. Most fractals are self-similar, so that the shape that we identify in the plot of a fractal curve repeats itself on a smaller and smaller scale as we enlarge the image further and further.

　
Fractal Dimension

Suppose that we start with an initiator that is some simple geometric figure consisting of a number of connected line segments. It may be a triangle or a square or even straight a straight line. We now define a generator. This generator is a series of line segments that is going to replace every line segment of the initiator. The generator consists of N line segments, each of length r, where re is a fraction of the line segment being replaced. The arrangement of the N line segments is such that the distance from the beginning of the generator to its end is the same as the length of the line segment being replaced. The replacement process repeats an infinite number of times, each time replacing each line segment of the previous level curve with a scaled-down replica of the generatro. It can then be shown that the Hausdorff-Besicovithc dimension of the resulting fractal curve is

Comparing this demension with the Euclidian dimension gives us some idea of the properties of fractal . For example, a D of 1.0 is imply an ordinary line, whereas a D of 2.0 means that the curve completely fills the plane.

Algorithm for calculate power

Very often, the calculation of a to power is necessary in mathematics program. However, not all programming language support the power or it provides only integer parameter. If the programming language support natural log and anti-log e, you may try this

ln y = b ln a

Now, you may write you own power function which support non-integer no. Here is an example:
 
 

Function power(a, b As Variant) power = Exp(b * Log(a)) End Function

　
 

Draw a line

To draw a line, it requires two end points of a line. The length, point and angle of line are changing for every iteration. For a line information

A point (x1,y1)

Angle (a)

Length ( r )

Then another point (x2,y2) can be calculated by the following formular

,

=> , 

Cos (a) = (x2-x1)/r -> x2=x1 +r cos (a)

Sin (a) = (y2-y1)/r -> y2=y1 +r sin (a)

It should not be difficult, right ?
 

Recursive concept

To plot a fractal graph, a "replaced segment" is imposed on every segment. A recursive method is somehow better than a "for loop" for infinite iterations and structure.

Here is a simple program for factorial function(Basic):
 
 

Function fractorial(n As Integer) If n = 1 Then fractorial = 1 Else fractorial = n * fractorial(n - 1) End If

Then
　

1! = 1

2!= 2*1!

3!=3 *2!

4! =4*3!

……

So, the concept can be applied to the fractal making on replacing segment.