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DUST OF NUMBERS

    


     Draw a straight line. Remove the middle one-third and draw the two segments of line again. In each segment, remove the middle one-third portion again, and draw the four pieces of the line below it. Just go on repeating the process. We finally get the dust of points.

     The first line is having the dimension one. The next segments of lines have a dimension less than 1 because they do not occupy the entire line fully. This pattern has only fractional dimension and hence it is called 'fractal'. 

      Imagine, the length of the first line is marked as in the scale from 0 to 1 in fractions. When the middle portion is cut off, the ends of two segments must have the marking 0 - 1/3 and 2/3 -1. When the two segments are redrawn by applying the same rule, the marking at the ends of the four segments is 0 -1/9, 2/9-1/3, 2/3 - 7/9 and 8/9 - 1. The process can be repeated without end. We get a set of numbers called 'cantor set'. Here we learn that between 0 and 1 there are no integers, but countless fractions and real numbers. That is, even between 0 and 1, the numbers go to infinity. Dust of numbers in the mathematical universe.

      How can we draw cantor set using computers? Say'1' means line and '0' means no line. Rule 1. 1 yields 101 2. 0 yields 000 Using these two simple rules, one can draw cantor's fractal in the digital words as illustrated in the figure. A fractal always has self-similarity. The structure of a branch or a leaf is a replica of the entire tree. Similarly, any portion of this fractal is the replica of the whole pattern. Cantor set is the first and simple fractal and it is the gateway to the amazing fractal world.






                       1
                      1  0  1     
                 101  000   101
101  000   101                 101  000 101    AND ON
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