Fibonacci Sequence & the Golden Ratio

< 32.2 The Number π | Topic Index | 32.4 Fermat’s Last Theorem >

Write down a sequence of numbers which starts with 0 &amp; 1. Every subsequent number in the sequence is the sum of its two previous numbers.

Hence after 0, 1 it is 1 (0 + 1). So 0, 1, 1

The next number is 2 since 1 + 1 is 2. So, we get 0, 1, 1, 2

The next number is 3 since 1 + 2 is 3. So, 0, 1, 1, 2, 3

The next number is 5 since 2 + 3 is 5.

So the first 12 numbers in this sequence are 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89&hellip;&hellip;..

Almost any primary school student can write down this sequence. But they would be surprised that this is one of the famous sequences in math called Fibonacci sequence.

Fibonacci in Nature

In many flowers, the seeds are arranged so that numbers from the Fibonacci Sequence appear. It is something to do with the efficient packing of seeds in a flower!

Golden Ratio

The ratio of any number in the Fibonacci sequence, to its previous number tends towards a number which is called the Golden Ratio and has a symbol &phi;. It is pronounced as "Phi". The further we go in the sequence (or as the numbers become bigger &amp; bigger) the ratio becomes closer and closer to the value of &phi;. For example

5/3 = 1.6666666&hellip;

233/144 = 1.618055556&hellip;

377/233 = 1.618025751&hellip;

The actual value of &phi; is about 1.6180339887&hellip;..

Greek aestheticians said that a rectangle with sides which are in this ratio is aesthetically the most pleasing to the eye. The rectangular fa&ccedil;ade of the Parthenon in Athens seems to have this ratio. Such rectangles are also called golden rectangles.

The concept of certain rectangles being pleasing to the eye is not widely accepted.

&phi; also has a mathematical property which can be described as &phi; = 1 + 1/&phi;.

&phi; is the most irrational number (degrees of irrationality)

Fibonacci Numbers in Indian Mathematics

Hemachandra, a Jain sage and mathematician described the  Fibonacci sequence  in around 1150, about fifty years before  Fibonacci  (1202). He was considering the number of cadences of length n, and showed that these could be formed by adding a short syllable to a cadence of length n &minus; 1, or a long syllable to one of n &minus; 2. This recursion relation F(n) = F(n &minus; 1) + F(n &minus; 2) is what defines the Fibonacci sequence.

He (c. 1150 AD) studied the rhythms of Sanskrit poetry. Syllables in Sanskrit are either long or short. Long syllables have twice the length of short syllables. The question he asked is How many rhythm patterns with a given total length can be formed from short and long syllables? For example, how many patterns have the length of five short syllables (i.e. five &ldquo;beats&rdquo;)? There are eight: SSSSS, SSSL, SSLS, SLSS, LSSS, SLL, LSL, LLS As rhythm patterns, these are xxxxx, xxxx., xxx.x, xx.xx, x.xxx, xx.x., x.xx., x.x.x  [ 

Mathematically both the above enquires are same

< 32.2 The Number π | Topic Index | 32.4 Fermat’s Last Theorem >