The mind of architecture: November 2015

What is : Proportion ?

"The design of a temple depends on symmetry, the principles of which must be most carefully observed by the architect. They are due to proportion, in Greek ἁναλογἱα. Proportion is a correspondence among the measures of the members of an entire work, and of the whole to a certain part selected as standard. From this result the principles of symmetry. Without symmetry and proportion there can be no principles in the design of any temple; that is, if there is no precise relation between its members, as in the case of those of a well shaped man."

What is : Proportion ?

[Pythagorus and Plato on proportion] :
Pythagoras was the first to discover the diatonic scale in music, which led to the development of the system of proportions, having harmonious relationships such as the perfect fifth (3:2), minor seventh (16:9) and so on, this method is called the Pythagorean tuning which is a tuning of the syntonic temperament in which the generator is the ratio 3:2 (i.e., the untempered perfect fifth), which is 702 cents wide.
Pythagoras and Plato saw this as an evidence for an underlying harmony of nature, and a technique for achieving similar qualities in art.

[Vitruvius on proportion ]:
Chapter 1 : Vitruvius- The Ten Books of Architecture
ON SYMMETRY: IN TEMPLES AND IN THE HUMAN BODY

1. The design of a temple depends on symmetry, the principles of which must be most carefully observed by the architect. They are due to proportion, in Greek ἁναλογἱα. Proportion is a correspondence among the measures of the members of an entire work, and of the whole to a certain part selected as standard. From this result the principles of symmetry. Without symmetry and proportion there can be no principles in the design of any temple; that is, if there is no precise relation between its members, as in the case of those of a well shaped man.

2. For the human body is so designed by nature that the face, from the chin to the top of the forehead and the lowest roots of the hair, is a tenth part of the whole height; the open hand from the wrist to the tip of the middle finger is just the same; the head from the chin to the crown is an eighth, and with the neck and shoulder from the top of the breast to the lowest roots of the hair is a sixth; from the middle of the breast to the summit of the crown is a fourth. If we take the height of the face itself, the distance from the bottom of the chin to the under side of the nostrils is one third of it; the nose from the under side of the nostrils to a line between the eyebrows is the same; from there to the lowest roots of the hair is also a third, comprising the forehead. The length of the foot is one sixth of the height of the body; of the forearm, one fourth; and the breadth of the breast is also one fourth. The other members, too, have their own symmetrical proportions, and it was by employing them that the famous painters and sculptors of antiquity attained to great and endless renown.

3. Similarly, in the members of a temple there ought to be the greatest harmony in the symmetrical relations of the different parts to the general magnitude of the whole. Then again, in the human body the central point is naturally the navel. For if a man be placed flat on his back, with his hands and feet extended, and a pair of compasses centered at his navel, the fingers and toes of his two hands and feet will touch the circumference of a circle described therefrom. And just as the human body yields a circular outline, so too a square figure may be found from it. For if we measure the distance from the soles of the feet to the top of the head, and then apply that measure to the outstretched arms, the breadth will be found to be the same as the height, as in the case of plane surfaces which are perfectly square.

4. Therefore, since nature has designed the human body so that its members are duly proportioned to the frame as a whole, it appears that the ancients had good reason for their rule, that in perfect buildings the different members must be in exact symmetrical relations to the whole general scheme. Hence, while transmitting to us the proper arrangements for buildings of all kinds, they were particularly careful to do so in the case of temples of the gods, buildings in which merits and faults usually last forever.

5. Further, it was from the members of the body that they derived the fundamental ideas of the measures which are obviously necessary in all works, as the finger, palm, foot, and cubit. These they apportioned so as to form the "perfect number," called in Greek τἑλειον, and as the perfect number the ancients fixed upon ten. For it is from the number of the fingers of the hand that the palm is found, and the foot from the palm. Again, while ten is naturally perfect, as being made up by the fingers of the two palms, Plato also held that this number was perfect because ten is composed of the individual units, called by the Greeks μονἁδες. But as soon as eleven or twelve is reached, the numbers, being excessive, cannot be perfect until they come to ten for the second time; for the component parts of that number are the individual units.

6. The mathematicians, however, maintaining a different view, have said that the perfect number is six, because this number is composed of integral parts which are suited numerically to their method of reckoning: thus, one is one sixth; two is one third; three is one half; four is two thirds, or δἱμοιρος as they call it; five is five sixths, called πεντἁμοιρος; and six is the perfect number. As the number goes on growing larger, the addition of a unit above six is the ἑφεκτος; eight, formed by the addition of a third part of six, is the integer and a third, called ἑπἱτριτος; the addition of one half makes nine, the integer and a half, termed ἡμιὁλιος; the addition of two thirds, making the number ten, is the integer and two thirds, which they call ἑπιδἱμοιρος; in the number eleven, where five are added, we have the five sixths, called ἑπἱπεμπτος; finally, twelve, being composed of the two simple integers, is called διπλἁσιος.

7. And further, as the foot is one sixth of a man's height, the height of the body as expressed in number of feet being limited to six, they held that this was the perfect number, and observed that the cubit consisted of six palms or of twenty-four fingers. This principle seems to have been followed by the states of Greece. As the cubit consisted of six palms, they made the drachma, which they used as their unit, consist in the same way of six bronze coins, like our asses, which they call obols; and, to correspond to the fingers, divided the drachma into twenty-four quarter-obols, which some call dichalca others trichalca.

8. But our countrymen at first fixed upon the ancient number and made ten bronze pieces go to the denarius, and this is the origin of the name which is applied to the denarius to this day. And the fourth part of it, consisting of two asses and half of a third, they called "sesterce." But later, observing that six and ten were both of them perfect numbers, they combined the two, and thus made the most perfect number, sixteen. They found their authority for this in the foot. For if we take two palms from the cubit, there remains the foot of four palms; but the palm contains four fingers. Hence the foot contains sixteen fingers, and the denarius the same number of bronze asses.

9. Therefore, if it is agreed that number was found out from the human fingers, and that there is a symmetrical correspondence between the members separately and the entire form of the body, in accordance with a certain part selected as standard, we can have nothing but respect for those who, in constructing temples of the immortal gods, have so arranged the members of the works that both the separate parts and the whole design may harmonize in their proportions and symmetry.

Leonardo Da Vinci - Vitruvian man

[Alberti on proportion] :

"We shall therefore borrow all our Rules for the Finishing our Proportions, from the Musicians, who are the greatest Masters of this Sort of Numbers, and from those Things wherein Nature shows herself most excellent and compleat." Leon Battista Alberti (1407-1472)

Music with Arithmetic, Geometry and Astronomy, made up the Quadrivium, the four ways, or liberal arts, advocated in the Middle Ages as essential for the education of the human being, (together with their outward expression in Grammar, Rhetoric and Logic; the Trivium). Although the educated person would often have learnt to master a musical instrument, it was the mathematical and proportional aspect of music which was held to be of most relevance. Actually playing music, and even composing, because in the creative moment instinctive faculties seemed to have the upper hand, were, at least up to the end of the Middle Ages, seen as inferior to the purity of theory alone.

In Chapter VI, Alberti develops the relationship between the proportions of numbers and the measuring of areas.
Methodically, he lists three types of area;
short, middle, and long.

Alberti's own summary:
Short: 1:1, 2:3, 3:4
Middling: 2:4. 4:9, 9:16
Long:1:3, 3:8, 1:4
"By the help of these Mediocrates the Architects have discovered many excellent Things, as well as with Relation to the whole Structure, as to its several Parts; which we have not Time here to particularize. But the most common Use they have made of these Mediocrates, has been however for their Elevations

[Andrea Palladio on proportion]:

Andrea Palladio (1508-80) was an Italian architect, one of the most influential architects of our time. Palladio was born November 8, 1508, in Padua, and trained as a stonemason. Palladio moved to Vicenza in his early twenties. Originally named Andrea di Pietro della Gondola, he was named Palladio (after Pallade the goddess of wisdom daughter of Zeus) by the Italian poet and patron Giangiorgio Trissino, who oversaw Palladio's architectural studies. Trissino took him to Rome, where Palladio studied and measured Roman architectural ruins; he also studied the treatises of Vitruvius, one of the most important of the Roman architects.
In and near Vicenza he designed many residences (Villas) and public buildings (Palazzi). He also planned several churches in Venice, San Francesco della Vigna, San Giorgio Maggiore, and Il Redentore. One of his last work was the Teatro Olimpico in Vicenza, completed after his death by architect Vincenzo Scamozzi. Palladio's own use of classical motifs came through his direct, extensive study of Roman architecture.

He freely recombined elements of Roman buildings as suggested by his own building sites and by contemporary needs. At the same time he shared the Renaissance concern for harmonious proportion, and his facades have a noteworthy simplicity, austerity and repose.

Palladio was the first architect to develop a systematic organization of the rooms in a house. He was also the first to apply to houses the pedimented porticos of Roman temples-formal porches defined by a shallow triangular gable (Timpano) supported by a row of columns. Both these features are exemplified in the Villa Almerico "The Rotonda".

Palladio's buildings were highly functional. Palladio was the author of an important scientific treatise on architecture, I Quattro Libri dell'Architettura (The Four Books of Architecture), which was widely translated and influenced many later architects. Its precise rules and formulas were widely utilized, especially in England, and were basic to the Palladian style, adopted by Inigo Jones, Christopher Wren, and other English architects, which preceded and influenced the neoclassical architecture of the Georgian Style.

Palladio married Allegradonna, daugther of Marcantonio, and had five children, Leonida, Marcantonio, Orazio, Silla and Zenobia. He died on August 19, 1580 in Vicenza, or probably at Maser (Treviso), while attending construction of the Tempietto of Villa Barbaro.

"Palladio Rotonda Plan" by Andrea Palladio (1508-1580)

When Andrea Palladio, (1508-1580), in The Four Books of Architecture, published in 1570, suggested seven sets of the most beautiful and harmonious proportions to be used in the construction of rooms he chose measurements which reflect musical consonances. He suggests;
- The Circle
-The Square;
-The Rectangle in ratios [3 : 4, 1 :2, 2: 3 , 3: 5]

A perfect example of the ratios Andrea Paladio preferred in his designs :
The Path From Villa Cornaro to Drayton Hall, by Carl I. Gable.

Wordpress Blog - The Path From Villa Cornaro to Drayton Hall, by Carl I. Gable

[Le Corbusier on proportion [The Modulor]]:

Le Corbusier developed the Modulor in the long tradition of Vitruvius, Leonardo da Vinci's Vitruvian Man, the work of Leon Battista Alberti, and other attempts to discover mathematical proportions in the human body and then to use that knowledge to improve both the appearance and function of architecture. The system is based on human measurements, the double unit, the Fibonacci numbers, and the golden ratio. Le Corbusier described it as a "range of harmonious measurements to suit the human scale, universally applicable to architecture and to mechanical things".

With the Modulor, Le Corbusier sought to introduce a scale of visual measures that would unite two virtually incompatible systems: the Anglo Saxon foot and inch and the French metric system.Whilst he was intrigued by ancient civilisations who used measuring systems linked to the human body: elbow (cubit), finger (digit), thumb (inch) etc., he was troubled by the meter as a measure that was a forty-millionth part of the meridian of the earth.

In 1943, in response to the French National Organisation for Standardisation's (AFNOR) requirement for standardising all the objects involved in the construction process, Le Corbusier asked an apprentice to consider a scale based upon a man with his arm raised to 2.20 m in height. The result, in August 1943 was the first graphical representation of the derivation of the scale. This was refined after a visit to the Dean of the Faculty of Sciences in Sorbonne on 7 February 1945 which resulted in the inclusion of a golden section into the representation.