GROUP: Camilla Dacomo, Alessandro Marchesin (Treviso, Italy); Grigoras Andreea,

Logofatu Iulia, Preda Cristina (Bucuresti, Romania);

Italian Group:

# THE GOLDEN RATIO

The **golden ratio**, also know as the *divine proportion*, the *golden mean* or the *golden section*, is an irrational number, approximately equal to 1.618, which appears many times in geometry, art and architecture.

It is represented by the Greek letter phi (φ). This letter was proposed to represent the golden ratio by the mathematician Mark Barr, since it was the first letter in the name of the sculptor Phidias (490 – 430 BC), who made the Parthenon, who is said to have employed it.

The idea behind the golden ratio is that, once you have divided a line in two parts, the longer part divided by the smaller one is equal to the whole length of the line divided by the longer part.

The Golden Ratio in Architecture

Some architects believe the golden ratio makes the most pleasing and beautiful shape, and this is why it can be found in several buildings.

- The Parthenon

In 447 BC, architects Ictinos and Callicrates begin the construction of this temple with the supervision of Phidias, who is also responsible for the decoration of it.

This greek temple on the Athenian Acropolis was dedicated to the goddess Athena, considered the virgin patron of the city by its people.

The Parthenon's facade as well as elements of its facade and elsewhere can be circumscribed by golden rectangles.

Many other buildings and ancient constructions were built following this pleasant proportion.

Here there are a few examples:

- The Pyramid of Cheops

The oldest and largest of the three pyramids in the Giza Necropolis was built somewhere about 2560–2540 BC. The Ahmes papyrus of Egypt gives an account of this pyramid with proportions according to a "sacred ratio".

- Notre Dame Cathedral

This cathedral is a Latin Rite Catholic cathedral located in Chartres, about 80 kilometres southwest of Paris. Its construction ended in 1345. In the main facade (the west one), there are easily recognizable golden proportions.

- Taj Mahal

Probably one of the most famous buildings in the world, this white marble mausoleum, located in Agra, India, was built by Mughal emperor Shah Jahan in memory of his third wife.

The Golden Ratio in Art

Also, many artists inserted the golden ratio in their works of art. Let's find out some of them.

- Leonardo Da Vinci (1452 - 1519)

This excellent Italian Renaissance polymath (painter, sculptor, architect, musician, scientist, mathematician, engineer, inventor, anatomist, geologist, cartographer, botanist, writer) studied the human body involving in the ratios of the lengths of various body parts. After calling this ratio "divine proportion", he featured in many of his paintings.

→ An Old Man |
→ The Vitruvian Man |
→ The Mona Lisa |

Here the ratios are clearly visible. | We can draw many lines of the rectangles into this figure. | There are many more golden proportions in this painting. |

- Michelangelo (1475 - 1564)

Also Michelangelo di Lodovico Buonarroti Simoni, commonly know as Michelangelo (sculptor, painter, architect, poet, engineer), was an Italian Renaissance man who exemplified the Renaissance humanist ideal.

Michelangelo has beautifully hidden the golden proportion even in his most famous work of art: the frescos of the Sistine Chapel, in Rome.

This is a little fragment of the giant frescos that covers the entire ceiling of the chapel, which tells the Creation. In this particular picture, we can admire the central moment of all the Bible: God's fingers touching Adam's ones, to transmit the breath of life.

Maybe, now, you are wondering where the golden ratio is. Let's take a closer look.

All the lines that starts from God's phalanx converge in the same point, and each distance between these lines is a power of phi (= 1.618).

Camilla Dacomo

**Adolf Zeising**, whose main interests were mathematics and philosophy, found the golden ratio expressed in the arrangement of branches along the stems of plants and of veins in leaves. He extended his research to the skeletons of animals and the branchings of their veins and nerves, to the proportions of chemical compounds and the geometry of crystals, even to the use of proportion in artistic endeavors. In these phenomena he saw the golden ratio operating as a universal law.

Zeising wrote in 1854:

*The Golden Ratio is a universal law in which is contained the ground-principle of all formative striving for beauty and completeness in the realms of both nature and art, and which permeates, as a paramount spiritual ideal, all structures, forms and proportions, whether cosmic or individual, organic or inorganic, acoustic or optical; which finds its fullest realization, however, in the human form.*

Here there are some examples of Golden Ratio, and Golden Spiral in…

**…nature**

**…space**

**…human being** (the more the golden ratio is respected in human body, the more he looks like beautiful)

and here there some videos i found really interesting about Golden Ration and Fibonacci Sequence

Alessandro Marchesin