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THE COASTLINE IS REALLY UNMEASURABLE

    

        Let AB be the coastline.  If you measure the coastline with one km scale, we get the length as 2.3km.  If we measure the same coastline with 100-meter scale, we get the length as 2.9km.  From the figure, we can understand, why the length increases as we increase the magnification factor (scale).  If we use still smaller scale like 100 cm, the length of the coastline will further increase.  (Of Course, there will be a limit) 

     The length goes on increases as we increase magnification because the coastline is not a Straight-line.  It is a intricate zig-zag line.  In mathematics, it is called "fractal".  A straight line occupies two dimensions.  But the coastline is neither a straight-line nor an area.  It is 'in-between' them.  Hence, it has dimension between 1 and 2.  More accurately, it has fractional dimension, so it is called fractals. 
    How to find the fractional dimension?  Consider the squares below. 
 


 
    First is the full square.  Then, divide the square into 'two' equal halves on both sides.  We get four self-similar pieces and the magnification factor is said to be 2.  So, 2^2=4. 
    Divide the square into 3 equal parts on both sides (magnification factor=3).  we get 9 self-similar squares.  So, 3^2 =9. 
    If we divide the square into N equal parts, we will get N^2 self-similar smaller squares.  We understand the exponent of N, that is 2 stands for the dimension of the square.  Therefore, we can write, 


Number. of self-similar pieces= (magnification factor) ^dimension 

Let N = magnification factor  
LOG (number of self similar pieces) =LOG (N ^dimensions) 
Log (N^dimension) = Log (Number of self-similar pieces) 
Dimension *Log N= Log (Number of self-similar pieces). 
Dimension= Log (Number of self-similar pieces)/Log N. 


    Now, we have a formula for dimension. Coming to our coastline:  Here the magnification factor is 10 because we use 100-meter scale instead of 1 km scale.  We get 17 self-similar (assume) pieces of 100 m  as the length of coastline.  Hence the dimension of the coastline is 
=Log (17)/Log 10 = 1.23/1 =1.23 
Finally, the fractional dimension of coastline is 1.23.


    Suppose the coastal line is straight line, if 10 is the magnification factor, we will get 10 self-similar pieces.  If 100 is the magnification, we will get 100 pieces and the dimension will be one.  If we get more or less pieces than magnification,then the object has fractional dimension. 


    Many objects in this world have fractional dimension.  Picture has 2D.  Movie may be 3D.  But real-world objects have fractional D. 
                                         --------------- 
The above broken line has dimension less than one.         

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