# The golden ratio : 7 fascinating applications

The Golden Ratio, also known as the divine proportion, offers a fascinating foray into pure elementary and contemporary mathematics, revealing its influence in diverse fields such as art, geometry, and even the human body. This mysterious number is often associated with the golden rectangle, a rectangle whose length to width ratio equals the golden ratio, creating a visually pleasing aesthetic.

In the world of art, from painters to masters like Leonardo da Vinci, the Golden Ratio serves as a basis for creating balanced and harmonious compositions. It is also manifested in the construction of the Parthenon in Greece, where the uneven parts of the building follow the golden section to achieve geometric perfection.

The golden spiral, a figure obtained from the golden rectangle, reveals fascinating properties, such as the relationship with the Fibonacci sequence and its connection with the human body. From pure mathematics to practical applications in the geometric determination of φ, the Golden Ratio finds its place in the Euclidean plane and in the rule of thirds.

The uniqueness of its value, the previous calculations, as well as the sum and length ratios are key elements in understanding this mystical number. Common angles, lines and their intersections, everything converges towards a simple method, often linked to the Pythagorean theorem, to reveal the meaning of the Golden Ratio in various contexts.

By exploring Euclid and his discoveries about Euclid's proportion, as well as the speed of the calculations involved, we delve into the mysteries of mathematics, where choice equals geometric determination. The Greek letter phi becomes an essential key to unlocking these mysteries, while appealing to the discriminant and examining the arguments of the demonstration.

Whether in the circle with center O, the distance between the ends, or the positive solution of x-1, the Golden Ratio reveals itself as a mathematical object of infinite richness. This article aims to explore the multiple facets of the Golden Ratio, from Euclid's past to its impact in contemporary mathematics, providing results using the method of average reasons and equivalent squares.

## What is the meaning of the Golden Ratio, Phi or divine proportion?

What made a number so exciting that it persisted in the collective imagination for over two thousand years? So universal that it appears in the writings of an ancient Greek mathematician, the drawings of a 20th century architect and the plot of a best-selling thriller that became a blockbuster film?

So present that it is visible in the greatest architectural monument of the ancient world, in the paintings of the most eminent painter of the Renaissance and in the atomic structure of quasi-crystalline minerals.

The Golden Ratio, often represented by the Greek letter phi (φ), is a **special mathematical proportion** defined as the ratio a/b, where a is the larger part and b is the smaller, and this ratio is equal to (a+b)/a. This proportion is often expressed as (1 + √5) / 2.

The meaning of the Golden Ratio is vast and transcends several fields, including mathematics, art, nature, and even architecture.**Phi, Golden Ratio, Golden Proportion, Divine Proportion,** are synonymous expressions which designate this arithmetic ratio. This number is neither a measurement nor a dimension, it is a ratio of two homogeneous quantities.

### How to calculate the Golden ratio?

The Golden Number comes from a ratio which has only two letters a and b such that

:a / b = (a + b) / aThis

is the principle of economy.

With this ratio, we obtain an equation of the second degree.

Indeed, by posing a / b = Phi, we obtain: φ² = φ + 1

Equation which has a positive root:

A **golden rectangle** is a rectangle in the ratio length to width is equal to the number Phi.

## What is the value of the Golden Number?

Phi ( Φ = 1.618033988749895... ), is simply an irrational number like pi ( p = 3.14159265358979... ), but with many unusual mathematical properties. Unlike pi, which is a transcendental number, phi is the solution of a quadratic equation.

The ratio, or proportion, determined by Phi (1.618...) was known to the Greeks as the "division of a line in the extreme and middle ratio" and to Renaissance artists as the "Divine Proportion", ratio and golden number.

### Why is the Golden ratio called Divine proportion?

When the parts of a whole have **harmonious relationships** with each other and with the whole, we can speak of beauty, of harmony. Nature has invented such proportions in her creation and man, by intuition, has recognized them.

This beauty, this harmony is generated by the ratio between length and width of a rectangle or between height and depth of a nerve ... it happens that this ratio, so often present, and the Golden Number called Divine Proportion or Golden Section.

It is thanks to this number that the artists of the 5th century B.C. triggered the emotion of the spectators: this same emotion, this same **extraordinary impression of purity, for the eye and for the spirit**, is felt in front of an abbey, a cathedral, a simple village church built in the Middle Ages.

## Where does the Golden ratio come from ?

### The Golden ratio in Antiquity

Building has always been man's great ambition and the discovery of the Golden Number probably dates back to ancient times.

The Golden Number, in fact, is found more than 20 times in the 5-pointed star, and the division by 5 could be suggested by the great variety of flowers with 5 petals, the five branches of a starfish, the pentagonal structure of the sea urchin and its 5 teeth... Note, also, that man has 5 fingers on each limb and 5 senses which are all **pentagonal representations.****The number 5 gives rise to a relationship with Phi.**

If the monuments prior to the Hellenic civilization, in particular those of ancient Egypt (the pyramid of Cheops is 47 centuries old) reveal the empirical use of the Golden Number, the written texts dealing with its properties or rather with the geometrical figures which are linked to it will appear only with the Greeks.

Euclid, Pythagoras and many others gave the notion of the Golden Number mathematical rigor.

Three centuries before our era, Euclid with his "Elements", brings in this field an important contribution in the form of a geometrical demonstration: construction of an isosceles triangle having each of the angles at the base double of the remaining angle (i.e., in this famous isosceles triangle each of the angles at the base is worth 72 degrees and the angle of the top is equal to 36 degrees)

Note, also, that 72 degrees is the fifth of 360 degrees and 36 degrees the 10th of 360 degrees: angles that we find in the regular pentagon.

Euclid deduced from this triangle the construction of the regular pentagon, noting that the diagonals of this pentagon intersect in the middle and extreme right and that the ratio of the diagonal to the side is equal to Phi.

He ends his "Elements" with the inscription in a sphere of the five Platonic regular bodies: tetrahedron, octahedron, hexahedron (cube), icosahedron and dodecahedron.

Meditating on the Golden Section, Vitruvius came to the following conclusion: "*As the members of the body correspond to each other, so must the parts of the building*".

### The Golden ratio in the Middle Ages

In the 11th and 12th centuries, a time when faith was written in stone, 80 cathedrals were built in France, 500 large churches in monasteries and important towns and tens of thousands of parish churches. This is the era of Romanesque art: **geometric shapes**, sound and light vibrations are combined to reproduce the model of the universe that celebrates the greatness of the creator.

At the end of the Romanesque period, the search for the sky and the light is increasing; the cathedrals, at first squat, develop higher and higher vaults, giving birth to the gothic art.

Gothic art was born in France in the middle of the 12th century (the first Gothic vault was built in 1944 in the cathedral of Saint-Denis). This gothic art is characterized essentially by the use of vaults on ribbed crossings, of the broken arch instead of the round arch and of the buttress to support the main vault

The cathedral is a book of stone that exalts faith, harmony and beauty.

### The Golden ratio in the Renaissance

In the 15th century, it is known that the "Elements" of Euclid were taken up by Luca Pacioli, a Franciscan monk and professor of sacred theology, who knew perfectly well, through Euclid and Pythagoras, the division of a segment of a straight line into the mean and the extreme reason.

His book deals with architecture, the proportions of the human body and the letters of the alphabet

It is at the time of Pacioli, that is to say during the Renaissance, that the great artists, like his friend Leonardo da Vinci, adopted the Divine Proportion as a **canon of Beauty, of Harmony**. We will deal with this subject a little further on.

### The Golden ratio in the 20th century

This Divine Proportion was, in the 20th century, designated by the letter Phi in reference to Phidias, the greatest and most famous of artists who was in the 5th century BC both painter, goldsmith, architect. He built the Parthenon in Athens based on the harmony and beauty of the golden rectangle.

In 1931, Matila Ghyka wrote a very important book on the Golden Number. Paul Valery, Le Corbusier, Cartier Bresson, and many others, emphasize the major role of this number in art. Le Corbusier, in particular, built his famous "Modulor" which he declared to be "a working tool, a precise tool" or "a keyboard, a piano, a tuned piano."

Even today, the pyramid of the Louvre or the Geode in Paris testify of its presence and other astonishing domains like soccer or cosmetic surgery find in the Golden Number the best balance of the forms of the face or the body.

## Where can we find the Golden ratio ? 7 applications

### 1- The Golden ratio and beauty, in the human body

Since ancient times it has been observed that it governs the architectural balance of the human body. The navel divides the body according to the Golden Number, it must correspond to the ratio between the total height of the body and the height of the navel above the ground.

Thus, when we consider the Venus de Milo in profile, giving the level of the soles of the feet, we note that the ratio between the height of the statue and the distance that separates the soles of the feet from the navel is equal to Phi.

It is still Phi that must regulate the harmonious relationship between the height and width of a human head. This last example brings us to the Vitruvian canon of proportion, drawn by Leonardo da Vinci, dating from the 15th century, and still present today, even in advertising.

### Vitruvian Man and the Golden ratio

A drawing made around 1490, Proporzioni del Corpo Umano Secondo Vitruvio (The proportions of the human body according to Vitruvius), is among the most famous works of Leonardo da Vinci.

As its name suggests, it is based on the **ideal human proportions** as imagined by the Roman architect and military engineer Vitruvius. In Book III of his treatise De architectura, Vitruvius mentions the human figure as the primary source of proportion in architecture, with the ideal body being eight heads high:

*"The center of the body is naturally at the navel. Let a man, indeed, lie on his back with his hands and feet extended, if one of the branches of a compass is resting on the navel, the other, describing a circular line, will touch the fingers of the feet and hands.**And just as a circle can be represented with the body thus extended, so a square can be found in it: for if we take the measurement that lies between the end of the feet and the top of the head, and relate it to those of the open arms, we will see that the width responds to the height as in a square made with a square."*

Vitruvius measured the entire human body in whole fractions of the height of a man.

The Vitruvian man also presents some dimensions suggesting a relationship of the Golden Number. Between the top of the forehead and the soles of the feet, the following organs are placed at points of the Golden Number

:- The navel (most often associated with a Golden Number of the total height);

- The nipples;

- The clavicles.

As for the distance between the elbow and the fingertips, the base of the hand begins at the point of the golden section.

### 2- The Golden ratio and the Fibonacci sequence in nature

In 1202 AD, Leonardo Fibonacci wrote in his book *"Liber Abaci"* a **simple numerical sequence** that is the basis of an incredible mathematical relationship behind phi. This sequence was known as early as the 6th century AD by Indian mathematicians, but it was Fibonacci who introduced it to the West after his travels through the Mediterranean world and North Africa. He is also known as Leonardo Bonacci, because his name is derived in Italian from words meaning "son of (the) Bonacci".

Beginning with 0 and 1, each new number in the sequence is simply the sum of the two previous ones.

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377 . . .

This sequence is shown in the right margin of a page of the *Liber Abaci*, where a copy of the book is held by the Biblioteca Nazionale di Firenze.

The relationship between the Fibonacci sequence and the golden ratio is as follows: the ratio of each successive pair of numbers in the sequence approaches Phi (1.618. . .) , because 5 divided by 3 is 1.666. . ., and 8 divided by 5 is 1.60. However, this relationship was not discovered until about 1600, when Johannes Kepler and others began to talk about it.

The ratios of the successive numbers of the Fibonacci sequence converge quickly to Phi. After the 40th number of the sequence, the ratio is accurate to 15 decimal places.

In nature, there is a very large number of flowers with 5 petals regularly distributed.

In the heart of the sunflower, two networks of spirals winding each in a direction are mixed.

These spirals called "parastiches" have a particularity: their numbers are equal to two consecutive terms of the Fibonacci sequence equal to 21 and 34, 34 and 55 or 55 and 89.

## 3 - The Golden ratio in architecture: what is it used for?

#### The Golden ratio and pyramids

For thousands of years, the architects of** sacred buildings**, such as the megalithic circles, the pyramids of Egypt and the Greek temples, have endeavored to give their works particular dimensions.

Pyramids are whole numbers, which can be combined geometrically and whose numerical value is important, even symbolic. **Temples, dedicated to God or to Gods, were built to establish a bridge between man and his deities.**

**The geometry of the** Gothic **cathedrals** and the Greek temples are not the same, but the intent was the same. Certain rules of harmony, such as those found in the Gothic cathedrals, responded to **dimensions inspired by biblical sources**.

Proportionality was used with the precise aim of **bringing God closer to man**. Cathedrals, such as those in Milan, Chartres or St. Paul's Cathedral in London, were built according to geometric data and dimensions with meaningful numbers.

Ancient megalithic buildings, such as Stonehenge, show how the **geometry of the sky** and the sacred units of measurement were applied to the construction by man of some of his most impressive temples.

The pyramid of Cheops, known as the "Great Pyramid", built about 4700 years ago, is the golden polyhedron, within which there are many triangles containing Phi, including the ratio of the height of the triangular face to half the side of the square base which is equal to the root of Phi.

In the Romanesque and Gothic numbers, Phi is omnipresent.

#### Le Corbusier and the Golden ratio

Let's be clear, I am not at all a fan of the works of Charles-Edouard Jeanneret, known as Le Corbusier, born in Switzerland in 1887. But he made his mark on his time.

In his fifties, Le Corbusier developed a system of proportions based on the golden ratio and the human body,** the Modulor**.

This system, which attempted to unite the metric and Anglo-Saxon systems, was intended as a universal standard of measurement for engineers, architects and designers to use in creating forms that were both beautiful and practical. He represented this "harmonious range of measurements" with the abstract form of a man, 1.83m tall, with his raised arm bent and aligned with the top of his head, conveniently positioned at the location of the golden ratio between his navel and the top of the raised arm.

The Australian professor of architecture Michael J. Ostwald describes it as follows:*"For Le Corbusier, the industry needed a proportional measuring system that could reconcile the needs of the human body with the inherent beauty of the gold section. If such a system could be devised, capable of simultaneously making the gold section proportional to the height of a man, it would form an ideal basis for universal standardization.*

In his attempt to use the mathematical proportions of the human body to improve both the appearance and role of architecture, Le Corbusier followed in the footsteps of Vitruvius, Leonardo da Vinci, Pacioli and the Renaissance masters who used the study of mathematics and nature to **give their masterpieces a divine quality**.

After formulating his new system in the mid-1940s, Le Corbusier applied it to several buildings, including:

- The United Nations headquarters in New York (completed in 1952);

- Several modernist building complexes in Europe, beginning with the Cité Radieuse in Marseille (completed in 1953);

- The Sainte-Marie-de-la-Tourette convent near Lyon (completed in 1961)

Today the modulor has entered the design process of almost all architects, the measures have become automatic as a universal rule that we use without even realizing it on a smaller scale of course. **These measures have become essential for the development of a quality space.**

### 4- The Golden ratio in art

"*Without mathematics there is no art"* Luca Pacioli

"*Where the mind does not work with the hand there is no art"* Leonardo da Vinci.

A French artist of the 13th century represented the god of genesis, the great architect, with a compass, the universe as in other illuminations of that time, we note that the artist has unwittingly introduced into his icon, ratios or angles related to the Golden Number.

Many other examples can be provided to show the presence of the Golden Number in all fields. This presence can be natural, unconscious or cleverly calculated.

Other harmonious proportions exist in nature and in human production but the Divine Proportion is certainly the one that has most, and since the highest antiquity, incited man to the search for beauty in harmony with himself and with nature.**Everything is harmony and beauty with the Golden Number.**

Probably one of the best illustrations of the use of the golden ratio is **Leonardo da Vinci's Last Supper**, painted between 1494 and 1498. Various plans and architectural elements show very precise relationships of the golden ratio.

For example, by examining the space between the table top and the ceiling, we can see that the top of Jesus' head is at its central point; the top of the windows is placed at the golden ratio. The width of the coats of arms is the golden number of the width of the circular arches; the bands of the central coat of arms are placed at the points of the golden number and its width

According to some, even the positions of the disciples placed around the table respect golden proportions in relation to Jesus.

The examples are numerous.

We could also mention Michelangelo's paintings in the Sistine Chapel in the Vatican. In fact, analysis of the Sistine Chapel has revealed more than two dozen instances of golden ratio dimensions in important elements of the composition (with many golden rectangles).

Probably the most striking example is the point where Adam's finger is touched by **God's finger in the iconic Creation of Adam**, a point placed at the golden ratio of its horizontal and vertical dimensions.

Michelangelo took up this theme of figures touching the point of the golden ratio in other paintings in the Sistine Chapel.

### 5- The Golden ratio in logo and product design

Now let's look at a much more contemporary aspect of the golden ratio.

In addition to its use in painting, architecture and graphics, the golden ratio is also **present in the design of many objects**. For example, many stringed instruments show golden ratio proportions, such as the famous Stradivarius violins. Renowned for the quality of their wood, construction and sound, these highly sought after violins are sold at auction for millions of dollars.

In other cases, the golden ratio adds style and aesthetic appeal. Companies investing millions in the design of their brand and logo, as these must capture the hearts and minds of as many potential customers as possible in an instant.

Here are some examples.

**Google** caught the attention of the design world in 2015 when it announced a major change to its logo, fonts and various other brand symbols and icons, but it has cleverly kept and improved its use of phi to determine the size and spacing of letters. For example, if you look closely, it's clear that the ratio of the height of the uppercase G and uppercase L to the height of the other lowercase letters (except for the small tail of the G) is equal to phi.

The ratio of the width of the upper case G to the width of the lower case G is also a golden ratio, as is the position of the search box in relation to the top of the logo and the bottom of the "search" button on the Google home page, which is, remember, the most visited website in the world.

Google is certainly not the first to use the golden ratio for its brand. Measure the three ovals that make up the **Toyota** logo and you'll see that the width of the small narrow oval in the center is bounded by two sections of gold the width of the larger oval. The inner edge of the upper middle oval is positioned at the gold section of the total logo height.

### 6- The Golden ratio and stock market analysis

Let's look at a more than incredible application since it concerns the analysis of stock markets.

Stock price variations largely reflect human opinions, evaluations and expectations. A study by the mathematical psychologist Vladimir Lefebvre has shown that humans present positive and negative evaluations of their opinions in a ratio that approaches phi, with 61.8% positive and 38.2% negative.

The phi (1.618), golden ratio and Fibonacci series numbers (0, 1, 1, 2, 3, 5, 8, ...) have been successfully used to **analyze and predict stock market movements**, called "retracements".

Forbes ASAP featured an article on scientist Stephen Wolfram's work in cellular automata (underlying rules that determine a seemingly random phenomenon) stating that "This shell may hold the secret to stock market behavior, computers that think and the future of science."

A research firm has shown that markets are perfectly structured, explaining that humans, as part of nature, create perfect geometric relationships in their behavior, much like a spider weaves a geometrically perfect web without conscious awareness of its incredible feat.

This research company applies logarithmic spirals found in shells with dynamic 3D relationships to link one market movement to others.

The golden ratio, or phi, appears frequently enough in the timing of price highs and lows and resistance points that adding this tool to technical market analysis can help identify Fibonacci "retracements," the major turning points in price movements.

### 7- The Golden ratio and symbols

The **symbols of sacred geometry** do not escape this Divine Proportion, golden proportion or the Golden Number which is the **key to harmony in the world around us**.

We can see that civilizations, religions or political movements, have borrowed their symbols from the collective heritage of humanity, then disappeared in the sands of time. But the symbols remain and do not belong to anyone. They are always there.

A harmonious symbol, balanced and respecting the Divine Proportion, elected for a long time by a group of humans and with whom we establish a conscious relationship, meets the conditions to be a successful symbol.**The world of symbols is the world of life.** Life works with symbols and manifests itself through them, every object is a symbol that contains life. To penetrate life, you have to work with the symbols and, conversely, to discover the symbols and understand everything that contains, you have to live the real life. The symbols are seeds that you can plant.

The **mandala**, the **Flower of Life**, etc., shows us what we must strive for, a perfect balance of all worlds. Our symbols are powerful tools for awakening and healing. Look at them with your child's eyes, with wonder, without putting any concept, prejudice on the virtue, the energy they carry. You will then obtain wonderful results.

Why are we so drawn to sacred geometry symbols like the Flower of Life, or vibrational Mandalas?

Simply because **symbols connect us to the living**, they reconnect us with our soul, with the being we really are (and therefore not the being we were made to believe we were). All vibratory symbols speak to our soul unconsciously.

They attract us so much because they are an invitation to walk towards ourselves, to walk towards beauty and splendor, to finally become one with them.**Symbols have a magical power: it's a power of transformation.**

They are an aid to concentration and encourage us to become more faithful to who we are inside. They are not only tools of awakening but also tools of spiritual evolution, if we know how to use them.

Listen to your heart and choose the one or two that speak to you the most.

And this is a wonderful opportunity for you because we offer **the largest selection of vibrational symbols on the market**. So listen to your feelings.

## To conclude

I hope you have enjoyed this journey of discovery of the golden ratio.

This divine, golden number is exceptional because of its mathematical properties and its frequent appearances in geometry, nature, art or architecture.

And what's even more amazing is that it's still being used today. Why?

You may be surprised to discover that the golden ratio has been right in front of your eyes the whole time, gently nudging you to buy a product or use a service.

According to Darrin Crescebzi, former director of innovation design at Interbrand New York, considered by Fast Company magazine to be one of the "most creative people in business:*"The more visually inclined - painters, architects, designers, traditionally enthusiastic observers and documentarians of both nature and the human condition, whom we can thank for much of what we know about the world - have for centuries incorporated this relationship into their work because of its inherently appealing balance between symmetry and asymmetry."*

We arrive at the end of this article. I hope you enjoyed it, feel free to comment, share and subscribe to our newsletter to be informed of future releases.

And if you want to go further in the discovery of symbols, welcome to this space / store dedicated to sacred geometry.

## Sources:

Geometry of the Golden Number at Chalagam Editions

Sacred Geometry at Vega Editions

www.goldennumber.net

Original article written on 20/06/2021; updated on 15/07/2022