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A NUMBER FOR BEAUTY


Beautiful numbers:

1,2,3,5,8,13,21,34,...
  In the above sequence each number is a sum of previous two numbers{of course starting with 1}.  This sequence is called Fibonacci sequence.  

      This sequence is found everywhere in nature.  You can easily observe in a garden that the leaves arrangement on a branch or a petals arrangements in a flower follows 2,3,5,8,... pattern.  Fibonacci arrangement naturally save space and also gives good exposure of sunlight.  The bracts of a pine cone; the scales of a pine apple; the reproduction of rabbits follows Fibonacci order.  The Fibonacci numbers are therefore applicable to the growth of every living thing. 



A divine proportion:

If you divide the consecutive numbers in the sequence, that is : 2/1,3/2, 5/3, 8/5,...  The ratio will gradually zero in on 1.618...  This number is called golden ratio phi ⏀.  It has got a  divine touch.  

      The angular arcs between the petals and leaves are in the ratio 1.618... The segments of our finger bones are in this ratio.  Our facial features and body structure also follows the golden ratio.  Many animals like star fish has the golden ratio property.  

Basic property:

1. Take any two consecutive fibonacci numbers and draw lines proportional to them.
Let a=8 and b=13

 [-------------]  [----------------------]
        a                    b

 b/a= a+b/a= 1.618... =phi
This is always true in Fibonacci sequence.
2. Any whole number can be written using Fibonacci numbers.
Example : 32=21+8+3

Not just a rectangle:

Again take any two consecutive numbers in the sequence and draw a rectangle using them as length and breadth.

      In our example let us take last two numbers 34 and 21.
34/21= 1.618... golden ratio.

In our day today life we always end up with golden rectangles.  Because these rectangles always appeal to our eyes and also has utility values.  Example:  credit cards, debit cards, website layout, poster layout etc.

Golden spiral:

Take the rectangle in our previous example and take the next numbers in the sequence in the reverse order.  That is 13.  Now draw vertical line at 13 mm from right end of the rectangle.  We end up with one big square and a rectangle.

 a = 21;  b = 13

  Now take the next digit 8 and draw a horizontal line at 8 mm from the bottom in the small rectangle.  We now have two squares and one rectangle.  Repeat the process till the number 1.  All the rectangles are converted into squares as given below.
     Draw circular arcs connecting the opposite corners of squares{ starting from innermost square}.  We get a golden spiral or also called logarithmic spiral.

    Golden spirals are found in seashells, sunflowers, pine cones, spiral galaxies etc,.

     Hence the golden ratio phi= 1.618... should be respected on par with the constants e=2.71.. and pi =3.14...These numbers are mathematical constants that reveal the nature's hidden secrets.  

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