Φ - Phi : The Golden Ration

Fibonacci Numbers

The Fibonacci numbers
 are a sequence of integers, starting with 0, 1 and continuing 1, 2, 3, 5, 8, 13, ..., each new number being the sum of the previous two. 

The nature is the living proof for the importance of this series:









If you go little deeper into the behavior of this numbers, you get to see more interesting detail.

Fibonacci Series	Ratio of 2 #s	Inverse of the Ratio	Difference of the ratios
0
1
1	1.000000000000	1.000000000000	0.000000000000
2	2.000000000000	0.500000000000	1.500000000000
3	1.500000000000	0.666666666667	0.833333333333
5	1.666666666667	0.600000000000	1.066666666667
8	1.600000000000	0.625000000000	0.975000000000
13	1.625000000000	0.615384615385	1.009615384615
21	1.615384615385	0.619047619048	0.996336996337
34	1.619047619048	0.617647058824	1.001400560224
55	1.617647058824	0.618181818182	0.999465240642
89	1.618181818182	0.617977528090	1.000204290092
144	1.617977528090	0.618055555556	0.999921972534
233	1.618055555556	0.618025751073	1.000029804483
377	1.618025751073	0.618037135279	0.999988615794
610	1.618037135279	0.618032786885	1.000004348393
987	1.618032786885	0.618034447822	0.999998339064
1597	1.618034447822	0.618033813400	1.000000634422
2584	1.618033813400	0.618034055728	0.999999757673
4181	1.618034055728	0.618033963167	1.000000092561
6765	1.618033963167	0.618033998522	0.999999964645
10946	1.618033998522	0.618033985017	1.000000013504
17711	1.618033985017	0.618033990176	0.999999994842
28657	1.618033990176	0.618033988205	1.000000001970
46368	1.618033988205	0.618033988958	0.999999999247
75025	1.618033988958	0.618033988670	1.000000000287
121393	1.618033988670	0.618033988780	0.999999999890
196418	1.618033988780	0.618033988738	1.000000000042
317811	1.618033988738	0.618033988754	0.999999999984
514229	1.618033988754	0.618033988748	1.000000000006
832040	1.618033988748	0.618033988751	0.999999999998
1346269	1.618033988751	0.618033988750	1.000000000001
2178309	1.618033988750	0.618033988750	1.000000000000
3524578	1.618033988750	0.618033988750	1.000000000000
5702887	1.618033988750	0.618033988750	1.000000000000
9227465	1.618033988750	0.618033988750	1.000000000000
14930352	1.618033988750	0.618033988750	1.000000000000
24157817	1.618033988750	0.618033988750	1.000000000000
39088169	1.618033988750	0.618033988750	1.000000000000
63245986	1.618033988750	0.618033988750	1.000000000000
102334155	1.618033988750	0.618033988750	1.000000000000
165580141	1.618033988750	0.618033988750	1.000000000000
267914296	1.618033988750	0.618033988750	1.000000000000
433494437	1.618033988750	0.618033988750	1.000000000000
701408733	1.618033988750	0.618033988750	1.000000000000
1134903170	1.618033988750	0.618033988750	1.000000000000
1836311903	1.618033988750	0.618033988750	1.000000000000
2971215073	1.618033988750	0.618033988750	1.000000000000
4807526976	1.618033988750	0.618033988750	1.000000000000

The first column defines the Fibonacci numbers starting from 0. 

The second column is the ratio between two adjacent Fibonacci numbers

The third column is the inverse of the value in the second column i.e. the inverse of the ratio.

The fourth column defines the difference between the values in second and third columns.

The highlighted number '1.618033988750' has a special behavior that the inverse of it is exactly the decimal value of itself i.e. the difference between itself and its own inverse is '1'. This is referred as the Golden Ratio.

Golden Ratio:

This is also referred as Golden Mean or Golden Section. To understand the ratio little better, refer to the below image.

Our body is the proof of this ratio:

A perfect face or a natural face or a healthy face is one which has the Golden ratio proportions.

Now the nature does not limit the application of Golden ratio to the face but also to the body.

This image depicts, at a broader level, the Golden ratios present in face, hands and in the body.

Nature is always unique, mysterious and interesting to explore as you learn and grow deeper!!