The Golden ratio is a special ratio. Phi 
(lowercase) is used to refer to this ratio, and the value is 
, which is approximately 1.618.
The uppercase phi (
) is used for the reciprocal of the golden ratio, which is 1/.
Phi can be expressed using a line segment. We make it such that the ratio of B to A+B is equal to the ratio of A to B.
Many designs - used in buildings, sculptures and paintings use the Golden Ratio for their dimensions. Architects and artists tend to use them often as they are considered very pleasing to the eye.
For example, the Parthenon uses the golden ratio for its construction.
Another example is the Mona Lisa painting.
The golden ratio also determines how attractive a person is.
For example, the American pop singer and actress Jessica Simpson is attractive because the proportion of her face fits geometrically on the human face mask which conforms to some aspect of the Golden Ratio.
Golden Rectangle
This rectangle is a special rectangle where the ratio of the length to the width is the Golden Ratio, which is 
.
When a square is cut off from the golden rectangle, the new rectangle is still similar to the original rectangle.
Below is an illustration:
and so on…
Golden Triangle
The golden triangle is a special isosceles triangle. The top angle is 36° while the bottom two angles are 72° each. We then bisect one of these base angles. The resulting blue triangle:
is also a golden triangle! Thus, we can keep bisecting the base angle to get a set of whirling triangles:
From this, we can draw a logarithmic spiral similar to the Fibonacci spiral:
Pentagram
A pentagram is a star-shaped figure which is made out of the five diagonals on a pentagon:
As can be seen, a pentagram has five sides. From three of these sides, we can get a couple of different golden triangles:
From the figure below:
we can also conclude that two non-consecutive sides of a pentagram divide each other in the Golden Ratio.
Fibonacci number
Fibonacci number
The Fibonacci numbers are part of a sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21 and so on. As can be seen, each number is the sum of the two numbers before it, after the first two numbers. A general equation can therefore be formed:
The Fibonacci sequence is named after Leonardo of Pisa, an Italian mathematician also known as Fibonacci.
Origins
Fibonacci first thought of the sequence as a solution to a problem he posed in a book he wrote, Liber Abaci. The problem was to find out the number of rabbits produced in a rabbit population which starts from a single newly-born pair. It is assumed that each pair produces another pair every month, they each become productive from the second month onwards and the rabbits never die.
Relation to the Golden Ratio
There are many different ways in which the Fibonacci sequence is related to the Golden Ratio. Firstly, the further you go to the right of the sequence, the more the ratio of one term to the one before it estimates the Golden Ratio. The table below shows the first few numbers and their ratios:
| First number | Second number | Ratio |
| 0 | 1 | - |
| 1 | 1 | 1.0000 |
| 1 | 2 | 2.0000 |
| 2 | 3 | 1.5000 |
| 3 | 5 | 1.6667 |
| 5 | 8 | 1.6000 |
| 8 | 13 | 1.6250 |
| 13 | 21 | 1.6154 |
| 21 | 34 | 1.6190 |
| 34 | 55 | 1.6176 |
| 55 | 89 | 1.6182 |
| 89 | 144 | 1.6180 |
To describe this graphically, we can use squares. Start with a 1 by 1 square. Then, add another square of the same size. Subsequently, add squares whose sides are equal to the longest side of the existing rectangle. After we do this four times, we get a rectangle similar to the following:
The rectangle gets increasingly closer to the golden rectangle, where the ratio of the width to the height is the Golden Ratio. If we draw a quarter circle in each of the squares, we get a Fibonacci spiral as such:
