Golden ratio: What is it and how to find it?

We delve into geometry, the golden ratio and its definition as a concept that enhances the harmony of objects, design and architecture with many examples!

✎ Autor:  Pau Segui

The golden section and its proportion

Although some planners or designers overlook the subject of the golden ratio, the reality is that throughout history it has been successfully applied in many projects, designs, buildings, photography, playing an important role in mathematics.

Those illustrious artists and craftsmen understood that the divine proportionality before the viewer is necessary to obtain a visual with harmony and composition enhancing the beauty of things.

The golden number is the mathematical key to the beauty and harmony that permeates our world.

composición aurea y número dorado Represented by the Greek letter Phi = 1.618034 in honor of the Greek sculptor Phidias. A phi number(Golden number) that possesses many interesting properties that was discovered in antiquity, not as a “unit” but as a ratio or proportion in geometric figures(Ideal of beauty).

That discovery provided a new aesthetic rule that crossed borders and remains today – for many designers and artists – as a key element in the artistic process.

What is the golden ratio and its history

The golden ratio is an irrational number found in many forms of nature, art, architecture or design and is associated with beauty and harmony.

proporción aurea arte arquitectura
Greek Parthenon – Mona Lisa – Vitruvian Man all elements with proportionality – Via Wikipedia – Juntaandalucia

Throughout history, it has been assigned many definitions and names; The golden number, the golden number or golden ratio, fi number, golden section, golden ratio, golden ratio, golden ratio, golden measure or divine proportion.

  • Art and architecture: This balance has been used in art and architecture since ancient times. The ancient Greeks considered it the perfect proportion, and used it in the construction and visual composition of many iconic buildings.

Architecture with divine proportion is not just construction, but a symphony of spaces that resonates with universal harmony.

For example, from the famed Greek Parthenon or the Temple of Ceres. Further on, with Leonardo da Vinci used the technique in the Mona Lisa (Gioconda), Michelangelo in the Sistine Chapel or the ideal proportion in the Vitruvian Man.

  • Nature: It is also found in many forms in nature, such as flowers, seashells and galactic spirals. For example, the spiral of a Nautilus shell approximates golden proportionality, and the arrangement of the petals of a sunflower flower is also based on this harmony.
  • Design and photography: golden harmony is used to create aesthetically pleasing spaces and objects, in photography the golden rule is used to compose balanced images, or in web applications, it creates more attractive and harmonious designs.

The Fibonacci Succession and the golden ratio

If we remember the history in search of the concept of divine proportion. Leonardo Pisano, also known as Fibonacci, was a famous mathematician from Italy who dedicated himself to spread the Arabic numbering system (1, 2, 3…) with a decimal base and with a null value (zero) throughout Europe in his Book of the Abacus in 1202.

But the great discovery of this mathematician was the Fibonacci Succession, which later gave rise to the golden ratio in art.

The golden ratio is the mathematical key to the beauty and harmony that pervades our world.

What is the Fibonacci Succession?…. It is a numerical series: 0, 1, 1, 1, 2, 2, 3, 3, 5, 8, 13, 21, 34, 55, etc. It is an infinite series in which the sum of two consecutive numbers always results in the next number (1+1=2; 13+21=34). The ratio that exists between each pair of consecutive numbers (that is, if we divide each number by its previous one) approaches the golden number 1.618034.

numero phi

So, with the 1.618(Golden Ratio or Golden Ratio)? How is this succession of numbers applied in art, architecture or design? Well, the composition of the so-called golden rectangle is used because it facilitates its use.

The golden rectangle and the golden spiral

The golden rectangle is a geometric figure whose ratio between length and width is the golden ratio, approximately 1.618, based on the Fibonacci sequence, and whose result is the formation of the golden spiral. To understand it better, we break down the creation step by step in a three-part drawing:

  • A.- If we transfer the above numerical sequence to a rectangle we find the following example for better understanding:

rectangulo aureo

  • B.- If we follow the division with the Fibonacci succession:

numero de oro

  • C.- By joining different vertices of the golden rectangle with a line we will get the famous Golden Spiral that is very present in nature resulting visually a “natural” proportion.

divina proporcion

This process is translated in the following summary scheme that you can obtain the PDF from HERE and that surely will be useful for many (I remind you that further on there is a calculator that we can insert values in centimeters to obtain the desired figure)

plantilla rectángulo aureo

Subsequently, the fascination has been such throughout history that an Italian mathematician and theologian Luca Pacioli published a book entitled The Divine Proportion ( 1509) in which he gave five reasons to unravel why the golden ratio is divine:

  • The fact that it is defined by three segments of a straight line, which resembles the Trinity.
  • The uniqueness of the number itself, which resembles that of God.
  • If we look at the incommensurability of number, just as God is incommensurable.
  • God gave being to the universe through the quintessence, represented at one time by a dodecahedron, and the golden number gave being to the dodecahedron.
  • Our God is omnipresent and unchanging, just as this number is.

Before this numerical sequence and its derivations, we can see this video that explores geometry, the Golden Number and much more in the field of nature:

Calculating Golden Proportionality

A useful tool to obtain the measurements in a fast and practical way is the following golden ratio calculator that will help us to find the measurements:

Golden Rectangle Calculator

calculadora de numero aureo
Reflects mathematical proportions on the above calculator – The Golden Ratio

The following video explains how to draw a golden or golden spiral step by step in a simple way:

ejemplo de sección aurea y divina proporción

Examples divine proportion

If we look around us, there are many examples of golden ratio, in architecture, nature, in the human body, in design or photography, but this time we are going to focus on some that we particularly see many times without realizing it.

1.- In flowers and plants

In nature we can find many similarities with harmonious geometries; from plants and flowers, through the typical snail shells.

divina proporción aurea en la naturaleza
Via OVACEN

2.- In web page design

In digital design there is also the golden ratio. An example of this can be found on the Twitter website. It’s that simple. Although currently we see it in reverse, it still maintains an aesthetic balance.

diseño digital proporcion aurea
Golden ratio ideas

The Golden Ratio transforms an ordinary design into a masterpiece of harmony.

3.- On a computer mouse

The design of a simple computer mouse. Many everyday objects, begin their beginnings in a design based on the golden rectangle although later they are deformed according to needs or utility objectives.

seccion aurea en diseno
Example golden number applied to object design

4.- In the construction of houses

In the case of architecture there are modern buildings and houses that are raised from a golden point of view and perspective. This is the case of the Nautilus house:

arquitectura proporción aurea
Example divine harmony applied in architecture. Via Exploded Diagram – archdaily.com

5.- Spiral staircase

Many examples of divine proportion can be found in perfectly designed architectural works or recognize the perfect measures of the human being. One example is the spiral staircase:

proporción aurea con ejemplo en arquitectura y diseño
Vía OVACEN

6.- In the ancient Greek constructions

We know that there are many buildings built by the Greeks that bear the “mark” of the Divine Proportion in their exterior visual:

ejemplos de proporciones aureas
Composition buildings and human

7.- In paintings

Painters were the first to use the golden ratio in a systematic way. Leonardo da Vinci was one of the most outstanding exponents of this use. In his most famous works, such as The Vitruvian Man, La Gioconda and The Last Supper, da Vinci used this technique to create balanced and harmonious compositions.

proporción aurea pintura
Leonardo Da Vinci – Diego Velázquez in Las Meninas – Via Wikipedia

But there is a long list of painters; from Francisco de Goya, Vincent van Gogh, Diego Velázquez, etc.

It makes an image more attractive and visually interesting and can suggest a sense of depth and movement.

8.- In sculpture

In sculpture, it is commonly used to divide the sculpture into two equal parts. Some sculptors who have used this technique in their works are: Michelangelo’s David, Auguste Rodin, Leonardo da Vinci, Jean-Antoine Houdon, etc.

proporción aurea en la escultura
Via OVACEN

9.- In photography

In the same way as in painting, framing in photography is essential. A common way is to use it to divide an image into two equal parts, so that one part is slightly larger than the other. Some examples

divina proporción en fotografía
Via OVACEN

The two-thirds rule is a simplification of the golden rule. This rule divides an image into nine equal parts, using two horizontal and two vertical lines. The intersecting lines are considered strong points in the image, and important objects are often placed at these points.

How to observe the goldenratio

tablero proporcion aurea y medidas The aesthetic harmony and beauty changes the way we see objects, for example, objects that psychologically could have obvious negative connotations such as cigarette packs or credit cards, are golden rectangles because it gives them a certain aesthetic beauty, this is called “marketing”…

To quickly find out how to get the golden ratio in an object just put it next to another, short side next to long side and draw a diagonal from the top and bottom corner of the set, if three vertices are aligned, is that the golden composition is met in design of objects. The representative example would be:

proporcion perfecta y aurea tecnologia
An example of golden measure

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A “toy” that has fascinated us for its simplicity and the way it superimposes the golden spiral on any shape is the Golden Section Finder designed by the Areaware studio. A thin card, the size of a pocket that helps to locate perfection and proportionality in everyday elements or in nature itself.

proporcion aurea
Via: areaware.com

To learn more in PDF documents

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