1. The golden section in architecture Alessandro Piscopo VD

2. What is the golden section ? The golden section is a geometrical ratio based on a specific ratio. In the formula, indicating with a the greater length and with b the shorter length, the following relation holds: (a+b): a = a: b = b (a-b) This ratio is approximately 1.6180 and is expressed by the formula: The star-pentagon is the geometric shape that represents more than any other, the golden section, so it was chosen as a symbol of the Pythagorean school.

3. History of the golden section in architecture The story of the golden section is very old, but as Mario Livio says: "I think it highly improbable that the Babylonians and Egyptians knew the golden ratio and its properties, the honor of this discovery should be left to the Greek mathematicians. "This proportion represents the standard for perfection, grace and harmony in architecture and in nature. The shell of the Nautilus The ancient architects, in their works, used two basic concepts: * Symmetry (the agreement of the measures) by the repetition of certain proportions privileged; Eurytmia * (harmony) between the lengths, areas and volumes of the building, both in its entirety that in its individual parts. The technique used was that of regulatory tracks. It began with a simple shape, the square, to find, with simple projections and overturning all the main lines of the building, in plan and elevations.

4. Cheops Pyramid It was originally composed for almost two and a half million blocks of stone. The average weight of each block is about two and a half tons. Its sides are perfectly aligned in a north-south and east-west direction (the alignment error is only 3 '6 "). The ground plane is perfectly horizontal: the south east angle is just twelve millimeters higher than the north-west. In the Pyramid, the proportions between the dimensions are not random, as well as responding to the aesthetic rules, recalling some of the most important constants of mathematics.

5. Cheops Pyramid The historic Herodotus (485-425 BC) mentions in his Histories Book II, a special measure dictated by the Egyptian priests, who wanted the area of each triangular face equal to that of a square having each side the height of the pyramid, measured from the apex of the monument up to the ground. Originally, Cheops Pyramid measured 230 m. about, 147 m for the base side. Of height, with inclined faces of about 51° 50’’. Based on these details it is easy to do some simple calculation: a = 230/2m = 115 m., B = 147 m e c = 186.64 m. At this point we can test the relationship that seems close to the value of the golden section.

6. The golden section in classical Greece The ancient temples evoke a sense of balance, harmony and perfection, that enchants us with the rhythm of their proportions. And the result of an organic aesthetic concept that inspired every art of the Hellenic population. The aim was always to give the buildings the idea of balance and perfection, to achieve the universal harmony understood as "unification of the multiplicity mixed and commissioning of discordant concordance, as a perfect balance between the opposition of the principles”. The Greek architects and artists made great use of golden rectangles. If you cut a square from a golden rectangle, the rectangle that remains is a golden rectangle. These golden rectangles were used to draw the floor plan and the facade of the temple: for example, the Parthenon, the Acropolis of Athens..

7. Parthenon

8. Phidias was the artist called by Pericles to build the Parthenon. It contains many golden rectangles and the same golden proportions are found in the statues in it. Significant example is the Korai dell'Eritteo. The Parthenon, was a temple dedicated to the goddess Athena, protector of the city, and was built around the 440/430 BC. The plan of the Parthenon is a rectangle with sides having a 1:5 ratio, the length of the root is 5 times the width. The ratio between the diameter of the column and the wheelbase is 4:9 as that between length and width of the entire building and between height and width of the facade. Building were also applied some optical corrections to counter the tendency of the eye to curve straight lines. The columns of the facade are inclined inwards and the corner columns are larger.

9. Parthenon Identification of the Parthenon with golden rectangles on the front. Identification of the golden section in the geometry of the facade of the Parthenon. Identification of reports on the golden facade of the Parthenon.

10. Pantheon The Pantheon survived in good conditions thanks to its successful transformation from a pagan temple into a Christian church. The magnificence of the size of the Pantheon and the harmony of its architectural proportions have always caused special attention. The genius of construction expressed in the great hemispherical dome of the Pantheon is unique in its kind.

11. Pantheon The harmonic Directive base consists of a square divided according to the relationship φ, and a circle inscribed in it. After the square and the pentagon here appears to us the importance of the circle, which often provided the key to finding the secrets of ancient buildings and their units. Ghyka said: "The distance between the inscribed circle and the outer one is given by small φ found on the lower side of the square, everything else is adjusted according to reports." The map of Pantheon.

12. The architecture in the Middle Ages: The Cathedral of Notre Dame The Cathedral of Paris shows the surprising presence of geometry marked by the golden section, on the front and the side. According to the measures taken on the spot, the horizontal plane of the cathedral is measured, in its inside width, in m 36, while its length, also internal,is m 108, which corresponds to three squares of 36 m from the side. The width of the main facade is 42 meters. Analysis also reveals that the transverse shear is inscribed in a square, and that the main divisions are determined from the angle of 63 ° and 26 ‘ on the famous triangle whose base is equal to the height and that will meet throughout the construction.

13. The architecture in the Middle Ages: The Cathedral of Notre Dame If we take as a radius half the width of the main facade to draw the inner circle and the resulting square, it is seen that this second square enters three times in total height, just as in the horizontal plane, measuring 42 meters, and thus correspond exactly the height and length. We note that the upper side of this second square gives us the balustrade of the church, and another half of the same square, gives the parapet of the towers that remained at this height, and despite this, because of the general harmonic system, gives a sense of perfect balance. If φ is divided according to the side of the second square, we find the center of the circular central window, and also the three doors are in each part according to the same proportionate relationship. Finally, looking at the front side, we still find two squares, and even the famous triangle whose base is equal to the height.

14. Milan’s Duomo At the end of the fourteenth century, various experts were summoned to Milan to vote on the proportions to give the new cathedral and the opinions were divided into two streams: the French claimed that the circles were used, the Germans preferred the triangles. The cathedral was built on the principles “ad quadratum", the plant being formed by two squares and their subdivisions. The deliberation of the committee was for the adoption of a relative height of the equilateral triangle. Milan’s Duomo.

15. Other monuments in which trace elements related to the golden section Investigations made with strict reproductions and diagrams showed that the golden section is the rule that dominates the connection of all parts of many of his buildings. the Malatesta temple in Rimini. Taj Mahal, located in Agra, northern India, is a mausoleum built in 1632 by the emperor Mughal Shah Jahan in memory of his wife Arjumand Banu Begum. It is believed that the architect was Ustad Ahmad Lahauri. Its structure is modulated according to the golden section The CN Tower in Toronto, the tallest standing structure in the world, follows the golden section. The ratio between the total height of 553.33 meters and height of the observation deck at 342 meters,it is 1.618.

16. Le Corbusier and Castel del Monte Le Corbusier practiced plotted regulator tracks since its first architectural work, and many schemes are based on the division of a golden rectangle,applied for example in the famous Villa de Monzie / Stain Garches (France), 1927. The design of the portal of Castel del Monte, an example of Gothic architecture in Apulia, built by Frederick II of Swabia in 1240, was created by the Pentagon and its decomposition was accorded to the number 1.618. Villa de Monzie/Stein a Garches by Le Corbusier (1927). Villa Stein (x=1,25 m; y=1,00 m) The construction has some points that coincide with the vertices of a pentagon. It requires more elements that concur to a particular size as: the distance of the two columns, the angle of the gable, the height of the vertex of the tympanum; with these conditions is only possible to draw a pentagon and may think that this has been wanted. The relationship between the elements is always 1.6. This proportion gives harmony to the structure so that inside there is also a certain ease.

17. Conclusion In conclusion, it can be said that the construction of the golden ratio is not particular difficult. You may think that this choice to work in a particular manufacturing process is to be found in ancient mythology who considered this number a proportion that could be considered divine and perfect harmony. Or: "The golden section is a wonderful demonstration of the fact that man and nature used the same tools to create forms, to get to the beauty." (S. Groenman,Utrecht, 1969).