Skip to main content

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.         

Comments

Post a Comment

Popular posts from this blog

LISSAJOUS FIGURES

  Definition:  "When a particle is subjected to two sine wave motion or two oscillatory motion at right angles, the particle describes lissajous figures".      We know sine wave motion and circular motion is basically same.  Hence we draw two circles A and B perpendicular to each other.  The circle B rotates twice faster than circle A.  That is, frequency of circle B is two times than that of A.        A particle at the intersection of two circles is subjected to two sine wave motion   A and B at 90 degree simultaneously.  The particle will describe figures depending on the frequency and phase of A and B .  In our case, the ratio of frequency is  1:2 and the two waves are in phase.        To draw lissajous figures :  A moving point in both the circles are chosen.   Here we should remember; during the time taken by the circle A to complete one rotation, circle B completes two.  Hence the points are marked on the circles according to their speed.  Then straight lines
Post a Comment
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.      Inverted parabola shape is used in the construction of buildings and bridges.  Because the shape is able to bear more weight.      A plane
Post a Comment
Read more

CASINO'S GAME

           Let us find out how the casino survives with mathematics.      Say, your friend invite you for a game of dice.  You must bet (wager) 2 dollars.  If you roll 'six' you will get back 8 dollars.  The game will go on for 30 rounds.  All sounds good.      The probability of rolling 'six' is 1/6.  Since the game will be played for 30 times, the 'expected win' is 30*1/6 = 5.  That is, you are expected to win 5 rounds out of 30.  Hence your gain will be 5 * 8 =40 dollars.  ok.  This also implies that you will loose 25 rounds.  Hence your loss will be 25*2 =50 dollars.  Your net gain will be gain-less = 40-50 = -10 dollars. For 30 rounds, the loss is -10 dollars, Hence, for one round =-10/30 = -1/3 dollars.  There will be a loss of -1/3 or 0.33 dollars per round.  It is not a fair game.     Let us make a simple formula to calculate  'Pay out per round\. The probability for a win = p The pay-out in case of win = V No. of rounds = n The expect
4 comments
Read more