golden ratio problem solving examples

Golden Ratio

The golden ratio, which is often referred to as the golden mean, divine proportion, or golden section, is a special attribute, denoted by the symbol ϕ, and is approximately equal to 1.618. The study of many special formations can be done using special sequences like the Fibonacci sequence and attributes like the golden ratio.

This ratio is found in various arts, architecture, and designs. Many admirable pieces of architecture like The Great Pyramid of Egypt, Parthenon, have either been partially or completely designed to reflect the golden ratio in their structure. Great artists like Leonardo Da Vinci used the golden ratio in a few of his masterpieces and it was known as the "Divine Proportion" in the 1500s. Let us learn more about the golden ratio in this lesson.

What is the Golden Ratio?

The golden ratio, which is also referred to as the golden mean, divine proportion, or golden section, exists between two quantities if their ratio is equal to the ratio of their sum to the larger quantity between the two. With reference to this definition, if we divide a line into two parts, the parts will be in the golden ratio if:

The ratio of the length of the longer part, say "a" to the length of the shorter part, say "b" is equal to the ratio of their sum " (a + b)" to the longer length.

Refer to the following diagram for a better understanding of the above concept:

It is denoted using the Greek letter ϕ, pronounced as "phi". The approximate value of ϕ is equal to 1.61803398875... It finds application in geometry, art, architecture, and other areas. Thus, the following equation establishes the relationship for the calculation of golden ratio: ϕ = a/b = (a + b)/a = 1.61803398875... where a and b are the dimensions of two quantities and a is the larger among the two.

Golden Ratio Definition

When a line is divided into two parts, the long part that is divided by the short part is equal to the whole length divided by the long part is defined as the golden ratio. Mentioned below are the golden ratio in architecture and art examples.

There are many applications of the golden ratio in the field of architecture. Many architectural wonders like the Great Mosque of Kairouan have been built to reflect the golden ratio in their structure. Artists like Leonardo Da Vinci, Raphael, Sandro Botticelli, and Georges Seurat used this as an attribute in their artworks.

Golden Ratio Formula

The Golden ratio formula can be used to calculate the value of the golden ratio. The golden ratio equation is derived to find the general formula to calculate golden ratio.

Golden Ratio Equation

From the definition of the golden ratio,

a/b = (a + b)/a = ϕ

From this equation, we get two equations:

a/b = ϕ → (1)

(a + b)/a = ϕ → (2)

From equation (1),

Substitute this in equation (2),

(bϕ + b)/bϕ = ϕ

b( ϕ + 1)/bϕ = ϕ

(ϕ + 1)/ϕ = ϕ

1 + 1/ϕ = ϕ

How to Calculate the Golden Ratio?

The value of the golden ratio can be calculated using different methods. Let us start with a basic one.

Hit and trial method

We will guess an arbitrary value of the constant, then follow these steps to calculate a closer value in each iteration.

• Calculate the multiplicative inverse of the value you guessed, i.e., 1/value. This value will be our first term.
• Calculate another term by adding 1 to the multiplicative inverse of that value.
• Both the terms obtained in the above steps should be equal. If not, we will repeat the process till we get an approximately equal value for both terms.
• For the second iteration, we will use the assumed value equal to the term 2 obtained in step 2, and so on.

For example,

Since ϕ = 1 + 1/ϕ, it must be greater than 1. Let us start with value 1.5 as our first guess.

• Term 1 = Multiplicative inverse of 1.5 = 1/1.5 = 0.6666...
• Term 2 = Multiplicative inverse of 1.5 + 1 = 0.6666.. + 1 = 1.6666...

Since both the terms are not equal, we will repeat this process again using the assumed value equal to term 2 .

The following table gives the data of calculations for all the assumed values until we get the desired equal terms:

The more iterations you follow, the closer the approximate value will be to the accurate one. The other methods provide a more efficient way to calculate the accurate value.

Another method to calculate the value of the golden ratio is by solving the golden ratio equation.

ϕ = 1 + 1/ϕ

Multiplying both sides by ϕ,

ϕ 2 = ϕ + 1

On rearranging, we get,

ϕ 2 - ϕ -1 = 0

The above equation is a quadratic equation and can be solved using quadratic formula:

ϕ = \(\frac{-b \pm \sqrt{ b^2 - 4ac}}{2a}\)

Substituting the values of a = 1, b = -1 and c = -1, we get,

ϕ = \(\frac{1 \pm \sqrt{( 1 + 4 )}}{2}\)

The solution can be simplified to a positive value giving:

ϕ = 1/2 + √5/2

Note that we are not considering the negative value, as \(\phi\) is the ratio of lengths and it cannot be negative.

Therefore, ϕ = 1/2 + √5/2

What is Golden Rectangle?

In geometry, a golden rectangle is defined as a rectangle whose side lengths are in the golden ratio. The golden rectangle exhibits a very special form of self-similarity. All rectangles that are created by adding or removing a square are golden rectangles as well.

Constructing a Golden Rectangle

We can construct a golden rectangle using the following steps:

• Step 1: First, we will draw a square of 1 unit. On one of its sides, draw a point midway. Now, we will draw a line from this point to a corner of the other side.

• Step 2: Using this line as a radius and the point drawn midway as the center, draw an arc running along the square's side. The length of this arc can be calculated using Pythagoras Theorem : √(1/2) 2 + (1) 2 = √5/2 units.
• Step 3: Use the intersection of this arc and the square's side to draw a rectangle as shown in the figure below:

This is a golden rectangle because its dimensions are in the golden ratio. i.e., ϕ = (√5/2 + 1/2)/1 = 1.61803

What is the Fibonacci Sequence?

The Fibonacci sequence is a special series of numbers in which every term (starting from the third term) is the sum of its previous two terms. The following steps can be used to find the Fibonacci sequence:

• We start by taking 0 and 1 as the first two terms.
• The third term 1, is thus calculated by adding 0 and 1.
• Similarly, the next term = 1 + 2 = 3, and so on.

Fibonacci sequence is thus given as, 0, 1, 1, 2, 3, 5, 8, 13, 21,.. and so on. Fibonacci sequence and golden ratio have a special relationship between them. As we start calculating the ratios of two successive terms in a Fibonacci series, the value of every later ratio gets closer to the accurate value of ϕ.

The following table shows the values of ratios approaching closer approximation to the value of ϕ. The following table shows the values of ratios approaching closer approximation to the value of ϕ.

☛Related Topics 

Given below is the list of topics that are closely connected to the golden ratio. These topics will also give you a glimpse of how such concepts are covered in Cuemath.

• Fibonacci Numbers
• Quadratic Equations
• Ratio, Proportion, Percentages Formulas

Golden Ratio Examples

Example 1: Calculate the value of the golden ratio ϕ using quadratic equations.

Note that we are not considering the negative value, as ϕ is the ratio of lengths and it cannot be negative.

Example 2: What are the different applications of the golden ratio in our day-to-day lives?

The golden ratio is a mathematical ratio, commonly found in nature, and when used in a design, it fosters natural-looking compositions that are pleasing to the eye. There are many applications of the golden ratio in the field of architecture. For example, the Great Pyramid of Egypt and the Great Mosque of Kairouan are a few of the architectural wonders in which the concept of the golden ratio has been used. Artists like Leonardo Da Vinci, Raphael, Sandro Botticelli, and Georges Seurat used this ratio as an attribute in their artworks. It can be used to study the structure of many objects in our daily lives that resemble a certain pattern

Example 3: The 14 th term in the sequence is 377. Find the next term.

We know that 15 th term = 14 th term × the golden ratio.

\(F_{15}\) = 377 × 1.618034

≈ 609.99 = 610

Therefore, the 15 th term in the Fibonacci sequence is 610.

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Practice Questions on Golden Ratio

Faqs on golden ratio, what is the golden ratio in simple words.

The golden ratio is a mathematical ratio that exists between two quantities if their ratio is equal to the ratio of their sum to the larger quantity among the two. In other words, when a line is divided into two parts and the longer part 'a' divided by the smaller part 'b', is equal to the sum of (a + b) divided by 'a', this means the line is reflecting the golden ratio, which is equal to 1.618.

What do you Mean by Golden Rectangle?

Why is the golden ratio beautiful.

The golden ratio is a ratio, which, when used in various fields to design objects, makes the objects aesthetically appealing and pleasing to look at. Therefore, the golden ratio is referred to as a beautiful attribute. It can be noticed in various patterns of nature, like the spiral arrangement of flowers and leaves. There are many applications of the golden ratio in the field of architecture. Many architectural wonders have been built to reflect the golden ratio in their structure, like, the Great Pyramid of Egypt and the Great Mosque of Kairouan.

Why is the Golden Ratio Important?

The golden ratio is a mathematical ratio which is commonly found in nature and is used in various fields. It is used in our day-to-day lives, art, and architecture. Objects designed to reflect the golden ratio in their structure and design are more pleasing and give an aesthetic feel to the eyes. It can be noticed in the spiral arrangement of flowers and leaves.

Where is the Golden Ratio Used in Real Life?

There are many uses of the golden ratio in the field of art and architecture. Many architectural wonders have been built to reflect the golden ratio in their structure. Artists like Leo Da Vinci, Raphael, Sandro Botticelli, and Georges Seurat used this as an attribute in their artworks. It can be used to study the structure of many objects in our daily lives that resemble a certain pattern.

Who Discovered the Golden Ratio?

Ancient Greek mathematicians were the first ones to mention the golden ratio in their work. The 5th-century BC mathematician Hippacus and Euclid contributed a lot of their research work on this subject.

What is Golden Ratio Formula?

The golden ratio formula can be used to calculate the value of the golden ratio. The formula to calculate the golden ratio is given as,

where ϕ denotes the golden ratio.

Golden Ratio

The golden ratio (symbol is the Greek letter "phi" shown at left) is a special number approximately equal to 1.618

It appears many times in geometry, art, architecture and other areas.

The Idea Behind It

Have a try yourself (use the slider):

This rectangle has been made using the Golden Ratio, Looks like a typical frame for a painting, doesn't it?

Some artists and architects believe the Golden Ratio makes the most pleasing and beautiful shape.

Do you think it is the "most pleasing rectangle"?

Maybe you do or don't, that is up to you!

Many buildings and artworks have the Golden Ratio in them, such as the Parthenon in Greece, but it is not really known if it was designed that way.

The Actual Value

The Golden Ratio is equal to:

1.61803398874989484820... (etc.)

The digits just keep on going, with no pattern. In fact the Golden Ratio is known to be an Irrational Number , and I will tell you more about it later.

We saw above that the Golden Ratio has this property:

a b = a + b a

We can split the right-hand fraction then do substitutions like this:

a b = a a + b a ↓      ↓      ↓ φ =  1 + 1 φ

So the Golden Ratio can be defined in terms of itself!

Let us test it using just a few digits of accuracy:

With more digits we would be more accurate.

Powers (Exponents)

Let's try multiplying by φ :

φ = 1 + 1 φ ↓     ↓     ↓ φ 2 = φ + 1

That ended up nice and simple. Let's multiply again!

φ 2 = φ + 1 ↓     ↓     ↓ φ 3 = φ 2 + φ

The pattern continues! Here is a short list:

Note how each power is the two powers before it added together! The same idea behind the Fibonacci Sequence (see below).

Calculating It

You can use that formula to try and calculate φ yourself.

First guess its value, then do this calculation again and again:

• A) divide 1 by your value (=1/value)
• C) now use that value and start again at A

With a calculator, just keep pressing "1/x", "+", "1", "=", around and around.

I started with 2 and got this:

It gets closer and closer to φ the more we go.

But there are better ways to calculate it to thousands of decimal places quite quickly.

Here is one way to draw a rectangle with the Golden Ratio:

• Draw a square of size "1"
• Place a dot half way along one side
• Draw a line from that point to an opposite corner
• Now turn that line so that it runs along the square's side
• Then you can extend the square to be a rectangle with the Golden Ratio!

(Where did √5 2 come from? See footnote*)

A Quick Way to Calculate

That rectangle above shows us a simple formula for the Golden Ratio.

When the short side is 1 , the long side is 1 2 + √5 2 , so:

φ = 1 2 + √5 2

The square root of 5 is approximately 2.236068, so the Golden Ratio is approximately 0.5 + 2.236068/2 = 1.618034. This is an easy way to calculate it when you need it.

Interesting fact : the Golden Ratio is also equal to 2 × sin(54°) , get your calculator and check!

Fibonacci Sequence

There is a special relationship between the Golden Ratio and the Fibonacci Sequence :

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ...

(The next number is found by adding up the two numbers before it.)

And here is a surprise: when we take any two successive (one after the other) Fibonacci Numbers, their ratio is very close to the Golden Ratio .

In fact, the bigger the pair of Fibonacci Numbers, the closer the approximation.

Let us try a few:

We don't have to start with 2 and 3 , here I randomly chose 192 and 16 (and got the sequence 192, 16,208,224,432,656, 1088, 1744, 2832, 4576, 7408, 11984, 19392, 31376, ... ):

The Most Irrational

I believe the Golden Ratio is the most irrational number . Here is why ...

So, it neatly slips in between simple fractions.

Note: many other irrational numbers are close to rational numbers, such as Pi = 3.14159265... is pretty close to 22/7 = 3.1428571...)

No, not witchcraft! The pentagram is more famous as a magical or holy symbol. And it has the Golden Ratio in it:

• a/b = 1.618...
• b/c = 1.618...
• c/d = 1.618...

Read more at Pentagram .

Other Names

The Golden Ratio is also sometimes called the golden section , golden mean , golden number , divine proportion , divine section and golden proportion .

Footnotes for the Keen

* where did √5/2 come from.

With the help of Pythagoras :

c 2 = a 2 + b 2

c 2 = ( 1 2 ) 2 + 1 2

c 2 = 1 4 + 1

c = √( 5 4 )

Solving using the Quadratic Formula

We can find the value of φ this way:

Which is a Quadratic Equation and we can use the Quadratic Formula:

φ = −b ± √(b 2 − 4ac) 2a

Using a=1 , b=−1 and c=−1 we get:

φ = 1 ± √(1+ 4) 2

And the positive solution simplifies to:

Kepler Triangle

That inspired a man called Johannes Kepler to create this triangle:

It is really cool because:

• it has Pythagoras and φ together
• the ratio of the sides is 1 : √φ : φ , making a Geometric Sequence .

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What is the golden ratio:

Method-1: the recursive method, method-2: the quadratic formula, golden ratio definition:, the golden ratio and the fibonacci numbers, golden ratio in geometry, golden ratio in nature, golden ration in architecture and design, golden ratio in history:, practice questions, golden ratio – explanation and examples.

Two quantities $a$ and $b$ with $a > b$ are said to be in golden ratio if $\dfrac{ a + b}{a} = \dfrac{a}{b}$

The ratio $\frac{a}{b}$ is also denoted by the Greek letter $\Phi$ and we can show that it is equal to $\frac{1 + \sqrt{5}}{2} \approx 1.618$. Note that the golden ratio is an irrational number, i.e., the numbers of the decimal point continue forever without any repeating pattern, and we use $1.618$ as an approximation only. Some other names for the golden ratio are golden mean, golden section, and divine proportion.

Golden ration can easily be understood using the example of a stick that we break into two unequal parts $a$ and $b$, where $a>b$, as shown in the figure below

Now there are many ways in which we can break the stick into two parts; however, if we break it in a particular manner, i.e., the ratio of the long part ($a$) and the short part ($b$) is also equal to the ratio of the total length ($a + b$) and the long part ($a$), then $a$ and $b$ are said to be in the golden ratio. The figure below shows an example of when the two parts of a stick are in the golden ratio and when they are not.

Calculating the golden ratio:

We stated above that the golden ratio is exactly equal to $\frac{1 + \sqrt{5}}{2}$. Where does this number come from? We will describe two methods to find the value $\Phi$. First, we start with the definition that $a$ and $b$ are in golden ratio if

$\frac{a}{b} = \frac{a + b}{a} = 1 + \frac{b}{a}$

Let $\Phi = \frac{a}{b}$ then $\frac{b}{a} = \frac{1}{\Phi}$, so the above equation becomes

$\Phi = 1 + \frac{1}{\Phi}$.

We assume any value for the $\Phi$, lets say we assume $\Phi=1.2$. Now, we put this value in the above formula, i.e., $\Phi = 1 + \frac{1}{\Phi}$ and get a new value of $\Phi$ as follows:

$\Phi = 1 + \frac{1}{1.2} = 1.8333$.

Now, we put this new value again in the formula for the golden ratio to get another value, i.e.,

$\Phi = 1 + \frac{1}{1.8.3333} = 1.54545$.

If we keep on repeating this process, we get closer and closer to the actual value of $\Phi$. As we show in the table below

Using the fact that $\Phi = 1 + \frac{1}{\Phi}$ and multiplying by $\Phi$ on both sides, we get a quadratic equation.

$\Phi^2 = \Phi + 1$.

This can also be rearranged as

$\Phi^2 – \Phi – 1 = 0$.

By using the quadratic formula for the equation $\alpha x^2 + \beta x + c = 0$, and noting that $x=\Phi$, $\alpha=1$, $\beta=-1$ and $c=-1$, we get

$\Phi = \frac{1 \pm \sqrt{1- 4 \times 1 \times -1}}{2} = \frac{1  \pm \sqrt{5}}{2}$.

The quadratic equation always has two solutions, in this case, one solution, i.e., $\frac{1  + \sqrt{5}}{2}$ is positive and the second solution, i.e., $\frac{1  – \sqrt{5}}{2}$ is negative. Since we assume $\Phi$ to be a ratio of two positive quantities, so the value of $\Phi$ is equal to $\frac{1  + \sqrt{5}}{2}$, which is approximately equal to 1.618.

Using the above discussion, we can define the golden ratio simply as:

The golden ratio $\Phi$ is the solution to the equation $\Phi^2 = 1  + \Phi$.

Golden ratio examples:

There are many interesting mathematical and natural phenomenon where we can observe the golden ratio. We describe some of these below

The Fibonacci numbers are a famous concept in number theory. The first Fibonacci number is 0, and the second is 1. After that, each new Fibonacci number is created by adding the previous two numbers. For example, we can write the third Fibonacci number by adding the first and the second Fibonacci number, i.e., 0 + 1 = 1. Likewise, we can write the fourth Fibonacci number by adding the second and third Fibonacci numbers, i.e., 1+1 = 2, etc. The sequence of Fibonacci numbers is called a Fibonacci sequence and is shown below:

$0, \,\, 1, \,\, 1, \,\, 2,\,\, 3,\,\, 5,\,\, 8,\,\, 13,\,\, 21,\,\, 34, \cdots$

If we start dividing subsequent Fibonacci numbers, the results approach closer and closer to the golden ratio as shown in the table below:

Pentagon and pentagram

The golden ratio makes numerous appearances in a regular pentagon and its associated pentagram. We draw a regular pentagon in the figure below.

If we connect the vertices of the pentagon, we get a star-shaped geometrical figure inside, which is called a pentagram, shown below

Many lines obey the golden ratio in the above figure. For example,

$\frac{DE}{EF}$ is in golden ratio

$\frac{EF}{FG}$ is in golden ratio

$\frac{EG}{EF}$ is in golden ratio

$\frac{BE}{AE}$ is in golden ratio,

$\frac{CF}{GF}$ is in golden ratio,

to name a few.

The golden spiral

Let us take a rectangle with one side equal to 1 and the other side equal to $\Phi$. The ratio of the large side to the small side is equal to $\frac{\Phi}{1}$. We show the rectangle in the figure below.

Now let’s say we divide the rectangle into a square of all sides equal to 1 and a smaller rectangle with one side equal to 1 and the other equal to $\Phi-1$. Now the ratio of the large side to the smaller one is $\frac{1}{\Phi-1}$. The new rectangle is drawn in blue in the figure below

From the definition of the golden ratio, we note that

$\Phi^2 -\Phi -1 = 0$, we can rewrite it as

$\Phi(\Phi -1) = 1$, or

$\frac{\Phi}{1} = \frac{1}{\Phi -1}$

Hence, the new rectangle in blue has the same ratio of the large side to the small side as the original one. These rectangles are called golden rectangles. If we keep on repeating this process, we get smaller and smaller golden rectangles, as shown below.

If we connect the points that divide the rectangles into squares, we get a spiral called the golden spiral, as shown below.

The Kepler triangle

The famous astronomer Johannes Kepler was fascinated by both the Pythagoras theorem and the golden ratio, so he decided to combine both in the form of Kepler’s triangle. Note that the equation for the golden ratio is

It is similar in format to the Pythagoras formula for the right-angled triangle, i.e.,

$\textrm{Hypotenuse}^2 = \textrm{Base}^2 + \textrm{Perpendicular}^2$,

If we draw a right-angled triangle with hypotenuse equal to $\Phi$, base equal to $\sqrt{\Phi}$ and perpendicular equal to 1, it will be a right-angled triangle. Such a triangle is called the Kelper triangle, and we show it below:

There are many natural phenomena where the golden ratio appears rather unexpectedly. Most readily observable is the spiraling structure and Fibonacci sequence found in various trees and flowers. For instance, in many cases, the leaves on the stem of a plant grow in a spiraling, helical pattern, and if we count the number of turns and number of leaves, we usually get a Fibonacci number. We can see this pattern in Elm, Cherry almond, etc. However, we must remember that many plants and flowers do not follow this pattern. Hence, any claim that the golden ratio is some fundamental building block of nature is not exactly valid.

It is also claimed that the ideal or perfect human face follows the golden ratio. But, again, this is highly subjective, and there is no uniform consensus on what constitutes an ideal human face. Also, all types of ratios can be found in any given human face.

In the human body, the ratio of the height of the naval to the total height is also close to the golden ratio. However, again we must remember that many ratios between 1 and 2 can be found in the human body, and if we enumerate them all, some are bound to be close to the golden ratio while others would be quite off.

Finally, the spiraling structure of the arms of the galaxy and the nautilus shell is also quoted as examples of the golden ratio in nature. These structures are indeed similar to the golden spiral mentioned above; however, they do not strictly follow the mathematics of the golden spiral.

How much of the golden ratio is actually present in nature and how much we force in on nature is subjective and controversial. We leave this matter to the personal preference of the reader.

Many people believe that the golden ratio is aesthetically pleasing, and artistic designs should follow the golden ratio. It is also argued that the golden ratio has appeared many times over the centuries in the design of famous buildings and art masterpieces.

For example , We can find the golden ratio many times in the famous Parthenon columns. Similarly, it is argued that the pyramids of Giza also contain the golden ratio as the basis of their design.

Some other examples are the Taj Mahal and Notre Damn etc. However, it should be remembered that We cannot achieve the perfect golden ratio as it is an irrational number. Since we are good at finding patterns, it may be the case that we are forcing the golden ratio on these architectures, and the original designers did not intend it.

However, some modern architectures, such as the United Nations secretariat buildings, have actually been designed using a system based on golden ratios.

Similarly, it is thought that Leonardo Di Vinci relied heavily on the use of the golden ratio in his works such as Mona Lisa and the Vitruvian Man. Whether the golden ratio is indeed aesthetic and it should be included in the design of architecture and art is a subjective matter and we leave this matter to the artistic sense of the reader.

If you are indeed interested in using the golden ratio in your works, some simple tips would be to use fonts, such as the heading font and the body text, such that they follow a golden ratio. Or divide your canvas or screen for any painting/pictures/documents so that the golden ratio is maintained.

Once you have used the golden ratio in your work, you will be in a better position to decide the aesthetic value of the golden ratio.

We have discussed the relation of the Fibonacci sequence and the golden ratio earlier. We can find the Fibonacci sequence in the works of Indian mathematicians as old as the second or third century BC. It was later taken up by Arab mathematicians such as Abu Kamil. From the Arabs, it was transmitted to Leonardo Fibonacci, whose famous book Liber abaci introduced it to the western world.

We have already mentioned some ancient structures such as the pyramids of Giza and the Parthenon that are believed to have applied the golden ratio in their designs. We also find mentions of the golden ratio in the works of Plato. Elements is an ancient and famous book on geometry by the Greek mathematician Euclid. We find some of the first mentions of the golden ratio as “extreme and mean ratio” in Elements.

The golden ratio gained more popularity during the Renaissance. Luca Pacioli, in the year 1509, published a book on the golden ratio called divine proportion. Leonardo Da Vinci did the illustrations of this book. Renaissance artists used the concept of the golden ratio in their works owing to its aesthetic appeal.

The famous astronomer Johannes Kepler also discusses the golden ratio in his writings, and we have also described the Kepler triangle above.

The term “Golden ratio” is believed to be coined by Martin Ohm in 1815 in his book “The Pure Elementary Mathematics.”

The Greek letter Phi (i.e., $\Phi$), which we have also used in this article to denote the golden ratio, was first used in 1914 by the American Mathematician Mark Barr. Note that Greek $\Phi$ is equivalent to the alphabet “F,” the first letter of Fibonacci.

More recently, Le Corbusier, the lead architect of the UN secretariat, created a design system based on the golden ratio of the UN secretariat building. In his bestseller book ” The Da Vinci Code, “the fiction writer Dan Brown popularized the myths and legends around the golden ratio in his bestseller book “The Da Vinci Code.”

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Last modified on March 28th, 2024

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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.

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 ${2\times \sin 54^{\circ }}$.

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.

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

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

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

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

The following table shows the mentioned pattern

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’.

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.

• A square GYXD of 1 unit is drawn.
• A point P is marked in the midway of any side (say DX).
• Point P is then joined to the vertex Y by a line segment PY.
• Now, using the Pythagoras Theorem, the length of the drawn line segment (say PY) is calculated.
• 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.
• 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.

Apart from spirals and circles, ϕ is also found in other geometric shapes, such as triangles and pentagrams.

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

Calculate the value of the golden ratio ϕ using the quadratic formula.

As we know, ${\phi =1+\dfrac{1}{\phi }}$ Multiplying both sides by ${\phi}$, ${\phi ^{2}=\left( 1+\dfrac{1}{\phi }\right) \phi}$ ${\phi ^{2}=\phi +1}$ On rearranging, we get, ${\phi ^{2}-\phi -1=0}$, which is a quadratic equation. Thus, by using the quadratic formula: ${\phi =\dfrac{-b\pm \sqrt{b^{2}-4ac}}{2a}}$ Here, a = 1, b = -1, c = -1 Thus, ${\phi =\dfrac{1\pm \sqrt{\left( 1+4\right) }}{2}}$ As the ratio is for two positive quantities, the value of the Golden ratio should be the positive value. Thus, ${\phi =\dfrac{1+\sqrt{\left( 1+4\right) }}{2}}$ Or, ${\phi =\dfrac{1+\sqrt{5}}{2}}$ = 1.618033…

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The Golden Ratio

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and the human body

This exercise is divided into 3 parts:

A. The golden ratio

Distance from the ground to your belly button

Distance from your belly button to the top of your head

Distance from the ground to your knees

Distances A, B and C

Length of your hand

Distance from your wrist to your elbow

Now calculate the following ratios:

Distance from the ground to your belly button / Distance from your belly button to the top of your head

Distance from the ground to your belly button / Distance from the ground to your knees

Distance C / Distance B

Distance B / Distance A

Distance from your wrist to your elbow / Length of your hand

Write all your results on the following table:

Can you see anything special about these ratios?

B. The fibonacci sequence

Now look at the following sequence of numbers:

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

The following number is the sum of the previous two. This is Fibonacci's sequence.

Now do the following ratios on a calculator and give answers in non-fraction numbers:

As you go on and on dividing a number in the sequence by the previous number you get closer and closer to the number you discovered in the first part of the exercise, phi = $\phi$ = 1.6180339887498948482.

C. The golden rectangle

We can also draw a rectangle with the fibonacci number's ratio. From this rectangle we can then derive interesting shapes.

First colour in two 1x1 squares on a piece of squared paper:

Then draw a 2x2 square on top of this one:

Then draw a 3x3 square to the right of these:

Then draw a 5x5 square under these:

Then draw a 8x8 square to the left of these:

Then draw a 13x13 square on top of these:

We could go on like this forever, making bigger and bigger rectangles in which the ratio of length/ width gets closer and closer to the Fibonacci number.

Then place the compass tip on the bottom left corner of the 2x2 square and draw an arc like this:

Then place the compass tip on the left hand, top corner of the 3x3 square and do the same:

Do the same for the other three squares to obtain:

This shape is widely found in nature, can you find any other examples?

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7.2: The Golden Ratio and Fibonacci Sequence

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In this section, we will discuss a very special number called the Golden Ratio. It is an irrational number, slightly bigger than 1.6, and it has (somewhat surprisingly) had huge significance in the world of science, art and music. It was also discovered that this number has an amazing connection with what is called the Fibonacci Sequence, originally studied in the context of biology centuries ago. This unexpected link among algebra, biology, and the arts suggests the mathematical unity of the world and is sometimes discussed in philosophy as well.

Golden Ratio

With one number \(a\) and another smaller number \(b\), the ratio of the two numbers is found by dividing them. Their ratio is \(a/b\). Another ratio is found by adding the two numbers together \(a+b\) and dividing this by the larger number \(a\). The new ratio is \((a+b)/a\). If these two ratios are equal to the same number, then that number is called the Golden Ratio. The Greek letter \(\varphi\) (phi) is usually used to denote the Golden Ratio.

For example, if \(b = 1\) and \(a / b=\varphi\), then \(a=\varphi\). The second ratio \((a+b)/a\) is then \((\varphi+1) / \varphi\). Because these two ratios are equal, this is true:

\[\varphi=\dfrac{\varphi+1}{\varphi}\nonumber \]

(This equation has two solutions, but only the positive solution is referred to as the Golden Ratio \(\varphi\)).

One way to write this number is

\[\varphi=\dfrac{1+\sqrt{5}}{2} \nonumber \]

\(\sqrt{5}\) is the positive number which, when multiplied by itself, makes \(5: \sqrt{5} \times \sqrt{5}=5\).

The Golden Ratio is an irrational number. If a person tries to write the decimal representation of it, it will never stop and never make a pattern, but it will start this way: 1.6180339887... An interesting thing about this number is that you can subtract 1 from it or divide 1 by it, and the result will be the same.

\[\varphi-1=1.6180339887 \ldots-1=0.6180339887 \nonumber \]

\[1 / \varphi=\frac{1}{1.6180339887}=0.6180339887 \nonumber \]

Golden rectangle

If the length of a rectangle divided by its width is equal to the Golden Ratio, then the rectangle is called a "golden rectangle.” If a square is cut off from one end of a golden rectangle, then the other end is a new golden rectangle. In the picture, the big rectangle (blue and pink together) is a golden rectangle because \(a / b=\varphi\). The blue part (B) is a square. The pink part by itself (A) is another golden rectangle because \(b /(a - b)=\varphi\).

Assume that \(\varphi=\dfrac{a}{b}\) , and \(\varphi\) is the positive solution to \(\varphi^{2}-\varphi-1=0\). Then , \(\dfrac{a^{2}}{b^{2}}-\dfrac{a}{b}-\dfrac{b}{b}=0\). Multiply by \(b^{2}, a^{2}-a b-b^{2}=0\). So, \(a^{2}-a b=b^{2}\). Thus, \(a(a-b)=b^{2}\). We then get \(\dfrac{a}{b}=\dfrac{b}{a-b}\). Both sides are \(\varphi\) .

Fibonacci Sequence

The Fibonacci sequence is a list of numbers. Start with 1, 1, and then you can find the next number in the list by adding the last two numbers together. The resulting (infinite) sequence is called the Fibonacci Sequence. Since we start with 1, 1, the next number is 1+1=2. We now have 1, 1, 2. The next number is 1+2=3. We now have 1, 1, 2, 3. The next number is 2+3=5. The next one is 3+5=8, and so on. Each of these numbers is called a Fibonacci number. Originally, Fibonacci (Leonardo of Pisa, who lived some 800 years ago) came up with this sequence to study rabbit populations! He probably had no idea what would happen when you divide each Fibonacci number by the previous one, as seen below.

Here is a very surprising fact:

The ratio of two consecutive Fibonacci numbers approaches the Golden Ratio.

It turns out that Fibonacci numbers show up quite often in nature. Some examples are the pattern of leaves on a stem, the parts of a pineapple, the flowering of artichoke, the uncurling of a fern and the arrangement of a pine cone. The Fibonacci numbers are also found in the family tree of honeybees.

Meanwhile, many artists and music researchers have studied artistic works in which the Golden Ratio plays an integral role. These include the works of Michelangelo, Da Vinci, and Mozart. Interested readers can find many resources and videos online. Perhaps it is not surprising that numbers like 3, 5, 8, and 13 are rather important in music theory; just take a quick look at the piano keys!

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Saburo Matsumoto CC-BY-4.0