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The Golden Ratio Divine Numbers By Luca Pacioli

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So below you can see very beautiful painting of Friar Luca Pacioli. Luca Pacioli was a religious man but he was also a Mathematician . And because of his standing as mathematician he became the teacher and good friend of Leonardo Da Vinci. Who is the most famous of all the renaissance men .

Luca Pacioli 

Luca Pacioli is known as the Father of Accounting and Bookkeeping. But our interest in him is because he was the author of a book called the "De Divina Proportione"  or "On the Divine proportion". The Divine proportion is a number . It's a number that we call now the Golden Ratio. Luca Pacioli thought this number was divine, was godly number. We are going to see that this number has a very close relationship with Fibonacci Sequence.

What Is GOLDEN RATIO ??

We can understand what the Golden Ratio is by starting with a line segment and dividing the line segment into two segments. Lets Say one of length x and one of length y . Assuming that length x is larger than the length y.


The line segment to understand golden Ratio. Where length x is larger than the length y. And Ofcourse both are positive numbers.
Line Segment 

Ofcourse both x and y are positive because they are the length of a line. So what is the Golden Ratio ? We are going to call the Golden Ratio capital Phi ( Φ ) , the Greek later phi. It is the ratio of the larger segment x to the smaller segment y . Where this lines segment satisfy particular equality , that the larger segment over the smaller x\y is same as the total length x+y divided larger segment x. 


Phi ( Φ )

So let's do some mathematics on above Equation. We have x\y as our phi. On other side let's x\x=1 and y\x ( y divide by x is just reciprocal of phi ) so we have ,

                       ϕ= 1  +  1\ϕ 

Multiple both side by  ϕ   and then bring everything to the left side of the equation.  So what we get ?

                  ϕ^2 - ϕ - 1 = 0              .....(1) 

So the Golden Ratio satisfies a very simple equation called as quadratic equation . We can solve above equation by quadratic formula to solve for phi ϕ  .  

Solving by using quadratic formula ,

                 ax^2 + bx + c = 0

Comparing equation 1 and general equation we get,

  a = 1 , b = -1 , c= 1 

   We know that,

       ϕ = (-b+-√b^2-4ac) / 2a  

Substitute the value and solving, 

    Phi =(1 + √5 ) / 2 = 1.6180339… = Φ   

There are two roots but phi is the ratio of two length so it can't be negative so we neglect the other one.

So that's the Golden Ratio.

Its an irrational number but we can write it, approximate it 1.618 something . We will see later on why this Golden Ratio is divine.

Also we will see  what is connection between Fibonacci Sequence and Golden Ratio. 

Let us define another number which is called Golden Ratio Conjugate. We will denote it by φ  small phi .

This we define as  ( √5 - 1 ) / 2 

What is the difference between Golden Ratio and Golden Ratio Conjugate?

Well you can see golden Ratio Conjugate is 1 less than the Golden Ratio. If we subtract 1 from capital phi ϕ  the Golden Ratio just going to be 0.618  so it's the fractional part of the Golden Ratio. 

  Let's see we know that 

             ϕ= 1  +  1\ϕ ........ * 

               φ = ϕ - 1          .......2 

Substitute  Capital Phi value (*)  in  2 equation 

               φ = 1  +  1\ϕ -1 

               φ =  1\ϕ.  

So, Golden Ratio's Conjugate is Reciprocal of Golden Ratio. 

Because of this property Golden Ratio is special in specific way and its appearance in nature.

Thank you for Learning. 



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