Golden Ratio

Golden Ratio, Golden Mean, Golden Section, or Divine Proportion refers to the ratio between two quantities such that the ratio of their sum to the larger of the two quantities is approximately equal to 1.618. It is denoted by the symbol ‘ϕ’ (phi), an irrational number because it never terminates and never repeats.

If the two quantities a and b are in the Golden ratio, they can be mathematically represented as

a:b = [(a+b): a] or ${\dfrac{a}{b}=\dfrac{a+b}{a}}$. For quantities a and b, a > b > 0.

The golden ratio is thus a proportional concept that describes the relative lengths of two line segments. It is important because it is found in various fields such as arts, architecture, human faces, and designs.

Who Discovered the Golden Ratio

Although the discovery of the golden ratio is a mystery, it was first thought to be mentioned around 300 BCE in Euclid’s Elements. In 1509, Luca Pacioli used the term ‘Golden Ratio’ in his book ‘The Divine Proportion.’

Now, let us expand the relation between the quantities in the golden ratio to get its value.

Formula

Let a line segment AC be divided into two parts, AB and BC, representing two quantities, a and b. AB = a is the larger part, and BC = b is the smaller part.

Now, if a and b are represented in the form of the golden ratio, then the formula is mathematically written as

${\dfrac{AB}{BC}=\dfrac{AB+BC}{AB}}$ 

=> ${\dfrac{a}{b}=\dfrac{a+b}{a}}$

The value of ϕ goes on as 1.61803398874989484820… like other typical irrational numbers with no specific pattern. The ratio also equals ${2\times \sin 54^{\circ }}$.

Derivation

Now, let us see how we obtained the above value:

${\dfrac{a}{b}=\dfrac{a+b}{a}}$

Now, splitting the right-hand side, we get

=> ${\dfrac{a}{b}=1+\dfrac{b}{a}}$ 

=> ${\phi =1+\dfrac{1}{\phi }}$

Using this formula, we can determine the value of the golden ratio by substituting the value of ϕ as follows:

=> ${\phi =1+\dfrac{1}{1.618}}$

= 1 + 0.618047…

= 1.618047…, which is the golden ratio.

The value of the ratio will be more accurate if we include more digits after the decimal places when substituting the value of ϕ.

We can obtain the value of the golden ratio mainly in 2 different ways:

By Hit and Trial Method

In this method, we will consider a value of the golden ratio (say 2) and follow the given steps until we get the value of ϕ closer to 1.618.

1. Finding the multiplicative inverse of the guessed value (=${\dfrac{1}{Value}}$), we get 0.5
2. Adding 1 to the value of step 1, we get 1.5
3. Using that value and repeating the same steps, we get the value closer to ϕ as follows.

Value	${\dfrac{1}{Value}}$	${\dfrac{1}{Value}+1}$
2 (Starting Value)	${\dfrac{1}{2}}$ = 0.5	0.5 + 1 = 1.5
1.5	${\dfrac{1}{1.5}}$ = 0.666..	0.666… + 1 = 1.666…
1.666…	${\dfrac{1}{1.666\ldots }}$ = 0.6	0.6 + 1 = 1.6
1.6	${\dfrac{1}{1.6}}$ = 0.625	0.625 + 1 = 1.625
1.625	${\dfrac{1}{1.625}}$ = 0.6153…	0.6154… + 1 = 1.6153…
1.6153…	…	…

If we proceed further, the final value gets even closer to the value of the golden ratio ϕ.

However, the hit-and-trial method needs more time and labor; thus, the value of ϕ is more commonly calculated using the quadratic formula.

By Quadratic Formula

As we know, the golden ratio formula is

${\phi =1+\dfrac{1}{\phi }}$

Now, multiplying both sides by ϕ, we get

${\phi ^{2}=\left( 1+\dfrac{1}{\phi }\right) \phi}$

${\phi ^{2}=\phi +1}$

On rearranging, we get,

${\phi ^{2}-\phi -1=0}$, which is a quadratic equation.

Thus, by using the quadratic formula, we get,

${\phi =\dfrac{-b\pm \sqrt{b^{2}-4ac}}{2a}}$

Here, a = 1, b = -1, c = -1

Thus, ${\phi =\dfrac{1\pm \sqrt{\left( 1+4\right) }}{2}}$

As the golden ratio is obtained from two positive quantities, the value of ϕ should always be positive.

Thus, ${\phi =\dfrac{1+\sqrt{\left( 1+4\right) }}{2}}$

=> ${\phi =\dfrac{1+\sqrt{5}}{2}}$ = 1.618033…

Patterns in Golden Ratio with Exponents 

Interestingly, this relation further gives us a pattern related to the value of ϕ.

As we know, the value of ϕ = 1.618 can be obtained using the formula ${\phi =1+\dfrac{1}{\phi }}$

Let us start by multiplying ϕ on both sides

${\phi ^{2}=\left( 1+\dfrac{1}{\phi }\right) \phi}$

${\phi ^{2}=\phi +1}$

Again multiplying both sides by ${\phi}$, we get

${\phi ^{3}=\phi ^{2}+\phi}$, and it goes on.

The following table shows the mentioned pattern

${\phi =1+\phi ^{-1}}$ = 1.618…

${\phi ^{2}=\phi +1}$ = 2.618…

${\phi ^{3}=\phi ^{2}+\phi}$ = 4.236…

${\phi ^{4}=\phi ^{3}+\phi ^{2}}$ = 6.854…

${\phi ^{5}=\phi ^{4}+\phi ^{3}}$ = 6.854…

Here, we observe that each power of the golden ratio is the sum of the two powers before it.

Now, let us see how the golden ratio forms using the Fibonacci number sequence, where each term is found by adding the two preceding numbers.

Relation to the Fibonacci Sequence

As we increase the value of the two consecutive Fibonacci numbers, the ratio gets closer to the golden ratio. Thus, this approximation is very close to ϕ for the pair of larger numbers. 

In the following table, let us find the value of ϕ starting from the Fibonacci number ‘2’.

Fn	Fn+1	Golden Ratio (ϕ) = Fn / Fn+1
2	3	1.5
3	5	1.66666…
…	…	…
144	233	1.6180555
…	…	…

Thus, we observe that ϕ is related to the Fibonacci sequence. 

Similarly, ϕ is also related to geometry. It relates to a notable geometric shape, the golden rectangle.

Golden Ratio and Golden Rectangle

In geometry, a rectangle formed by adding or removing the existing squares within a rectangle gives a golden rectangle. It has its sides in the golden ratio.

Let a rectangle ABCD be divided into five squares as shown. If we remove the squares AHFG and HBCE, we get a golden rectangle GFED.   

Here is another way to draw the golden rectangle.

We can also construct a golden rectangle by following the steps below.

1. A square GYXD of 1 unit is drawn.
2. A point P is marked in the midway of any side (say DX).
3. Point P is then joined to the vertex Y by a line segment PY.
4. Now, using the Pythagoras Theorem, the length of the drawn line segment (say PY) is calculated.
5. Then, using the line segment PY as the radius and the point P as the center, an arc GY is drawn along the sides of the square GYXD.
6. The line segment XE is then joined to the intersection point E of the arc GYE and the extended side DX. Finally, the rectangle DEFG (having the golden ratio) is drawn using DE.

Thus, the golden rectangle DEFG has dimensions in the golden ratio, ϕ

As we know, ϕ can be obtained from the ratio of two successive Fibonacci numbers; the golden ratio forms a spiral pattern. This spiral follows a constant angle close to ϕ and is thus known as the Golden Spiral. 

Sometimes, circles are drawn within squares instead of the spiral. Those circles are known as the Golden Circles, and the ratio of one circle to its adjacent one is found to be 1:1.618.

The gif below shows how the Golden Ratio works:

Apart from spirals and circles, ϕ is also found in other geometric shapes, such as triangles and pentagrams.summarizing all about the golden ratio

Golden Ratio in Kepler’s Triangle

As we expand the formula of ϕ and form the quadratic equation of the golden ratio, we get 

${\phi ^{2}-\phi -1=0}$

=> ${\phi ^{2}=\phi +1}$

Also, from the Pythagoras Theorem, we can write,

${c^{2}=a^{2}+b^{2}}$

Let us now consider a right-angled triangle ABC, where the length of the hypotenuse is AC, and the legs are AB and BC.

Now, if the sides of the triangle are AC = c = ϕ, AB = a = 1, and BC = b = ${\sqrt{\phi }}$, then using the Pythagoras Theorem, we can form the quadratic equation of ϕ.

The ratio of these sides is found to be in the ${1:\sqrt{\phi }:\phi}$.

It inspired Johannes Kepler to create the following triangle with Pythagoras and ${\phi}$ together.

Golden Ratio in Pentagram

The Pentagram (or Pentangle), a holy symbol, looks like a 5-pointed star. A regular Pentagram has the golden ratio in it.

Let ABCDEFGHIJ be a pentagram, where the length of AE = a, AD = b, AB = c, and BD = d.

Here, the ratio of AE to AD, AD to AB, and AB to BD gives the value of ϕ.

Real-Life Examples

• In Nature: The golden ratio is found in flowers, shells, weather, and galaxies. It also exists on the human face. A visually balanced face has a length-to-width ratio of approximately 1.618, the golden ratio. This ratio can also be seen in other parts of the human body.
• In Art and Architecture: It is used in many arts, designs, and architecture. One of the famous paintings, Leonardo Da Vinci’s Mona Lisa, was painted according to the golden ratio.
• In Logo and Design:  It is also found in many web designs and logo designs. It helps us to sketch out the proportions and shapes. Many famous logos like Twitter, Apple, and Pepsi follow this ratio.

Solved Examples