**A. A brief
History of the Golden Ratio**

There are many different names for the golden ratio; The Golden Mean, Phi, the Divine Section, The Golden Cut, The Golden Proportion, The Divine Proportion, and tau(t).

The Great Pyramid of Giza built around 2560 BC is one of the earliest examples of the use of the golden ratio. The length of each side of the base is 756 feet, and the height is 481 feet. So, we can find that the ratio of the vase to height is 756/481=1.5717Ö.. The Rhind Papyrus of about 1650 BC includes the solution to some problems about pyramids, but it does not mention anything about the golden ratio Phi.

Euclid (365BC - 300BC) in his "Elements" calls dividing
a line at the 0.6180399.. point dividing a line in the extreme
and mean ratio. This later gave rise to the name Golden Mean.
He used this phrase to mean the ratio of the smaller part of this
line, GB to the larger part AG (GB/AG) is the same as the ratio
of the larger part, AG, to the whole line AB (AG/AB).Then the
definition means that GB/AG = AG/AB.

(proposition 30 in book VI)

Plato, a Greek philosopher theorised about the Golden Ratio. He believed that if a line was divided into two unequal segments so that the smaller segment was related to the larger in the same way that the larger segment was related to the whole, what would result would be a special proportional relationship.

Luca Pacioli wrote a book called De Divina Proportione (The Divine Proportion) in 1509. It contains drawings made by Leonardo da Vinci of the 5 Platonic solids. Leonardo Da Vinci first called it the sectio aurea (Latin for the golden section).

Today, mathematicians also use the initial letter of the Greek
Phidias who used the golden ratio in his sculptures.

**1) Numeric definition**

Here is a 'Fibonacci series'.

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, .ÖÖÖ.

If we take the ratio of two successive numbers in this series and divide each by the number before it, we will find the following series of numbers.

1/1 = 1

2/1 = 2

3/2 = 1.5

5/3 = 1.6666...

8/5 = 1.6

13/8 = 1.625

21/13 = 1.61538...

34/21 = 1.61904...

The ratio seems to be settling down to a particular value,
which we call the golden ratio(Phi=1.618Ö..).

We can notice if we have a 1 by 1 square and add a square with side lengths equal to the length longer rectangle side, then what remains is another golden rectangle. This could go on forever. We can get bigger and bigger golden rectangles, adding off these big squares.

Step 1 Start with a square 1 by 1

Step 2 Find the longer side

Step 3 Add another square of that side to whole thing

Here is the list we can get adding the square;

1 x 1, 2 x 1, 3 x 2, 5 x 3, 8 x 5, 13 x 8, 21 x 13, 34 x 21Ö.

with each addition coming ever closer to multiplying by Phi.

**3) Algebraic and Geometric definition**

We can realize that Phi + 1 = Phi * Phi.

Start with a golden rectangle with a short side one unit long.

Since the long side of a golden rectangle equals the short side multiplied by Phi, the long side of the new rectangle is 1*Phi = Phi.

If we swing the long side to make a new golden rectangle, the short side of the new rectangle is Phi and the long side is Phi * Phi.

We also know from simple geometry that the new long side equals the sum of the two sides of the original rectangle, or Phi + 1.

Since these two expressions describe the same thing, they are
equivalent, and so Phi + 1 = Phi * Phi.