# Golden Ratio

## Golden Ratio

Limit of the ratio of two consecutive terms in the Fibonacci sequence.

• An approximate value of this number is 1.618 033 989.
• The golden ratio is sometimes called the divine proportion.
• This number is generally represented by the Greek letter φ (phi</em).

### Example

The golden ratio is the algebraic number that is the real positive root of this equation:

$${x^2}\space – x\space – 1 = 0$$.

This root is:

$$x_{1}$$ = $$\dfrac{1 + \sqrt{5}}{2}$$

Its approximate value is:

$$1.618 033 989$$.

### Historical Notes

The golden ratio was studied by artists in Antiquity, especially the sculptor Phidias, hence the choice of the Greek letter phi to refer to this number. This ratio can be found everywhere in the universe, from the arrangement of branches around the trunk of a tree to how some seashells grow to the specific shapes of some fruits.

For example, if we have line segment AB on which a point C is chosen so that $$\dfrac{m\left(\overline{AB}\right)}{m\left(\overline{CB}\right)}=\dfrac{m\left(\overline{CB}\right)}{m\left(\overline{AC}\right)}$$, this proportion expresses the divine proportion or the golden ratio: 1.618 033 989 …

This proportion is found in the dimensions of the Parthenon’s facade high up on the Acropolis in Athens, in the Fibonacci sequence, in geometric shapes like a regular pentagonal star, etc.