Skip to main content

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
-----------------------------------------------------------------

Comments

Post a Comment

Popular posts from this blog

THE EARTH, A SUPER ORGANISM

     JOIN MY COURSE: "Become a programmer in a day with python"       A man called 'love lock' (what a name) proposed a theory called Gaia theory, named after Greek Goddess.      It says, "Earth is a self-regulating organism like a human being.  The organic life in it interacts with in-organic matter and maintains atmosphere, temperature and environment".  Hence the earth is still suitable for the life to thrive.      Imagine, in a particular place, there are lot of flowers.  Some flowers are white and some are darkly coloured.  We know, white reflects light and heat while dark absorbs the same.  White flowers can thrive in hot climate.  But dark flowers requires cold climate.  The absorption and reflection balances and the environment reaches average, warm temperature at which both the flowers can co-exist.  This is the essence of "Gaia" theory.      On our earth, ...
3 comments
Read more

THE PARABOLA

          A jet of water shooting from a hose pipe will follow a parabolic path.  What is the so special about parabola.    Y= x^2 Draw a graph for the above equation.  It will result in a parabola.  This parabola is also called unit parabola.  Any equation involving square will yield a parabola. Example:  Y = 2x^2 +3x+3 (also called quadratic equation)    X= 2 and -2, both  satisfies the equation 4 = X^2.  Parabolic equations always have two solutions.     Any motion taking place freely under gravity follows parabolic path. Examples:   An object dropped from a moving train,   A bomb dropped from flying plane,  A ball kicked upwards.      If a beam of light rays fall on the parabolic shaped mirror, they will be reflected and brought to focus on a point.  This fact is made use of in Dish Antenna, Telescope mirrors, etc.   ...
Post a Comment
Read more

DISORDER IS THE "ORDER OF THE DAY"

         Imagine a balloon full of air.  The air molecules are moving randomly inside the balloon.  Let us pierce the balloon with a pin.  The air rushes out.  Why should not the air molecules stay inside the balloon safely and ignore the little hole?  That is not the way the world works.  The molecules always "want to occupy as many states as possible".  Hence the air goes out in the open to occupy more volume.   The things always goes into disorder (entropy) and the disorder increases with time.  The above statement is what we call "second law of thermodynamics".      Consider a cup of coffee on the table. Suppose the heat from entire room flows to your cup of coffee, the coffee will boil and the rest of the room will freeze.  Freezing means bringing things to order and arrangement.  It violates the second law.  Hence it will never happen.  Hence heat must flow from high ...
Post a Comment
Read more