Proportion (architecture) 02

Vitruvian proportion

Vitruvius described as the principal source of proportion among the orders the proportion of the human figure. .

According to Leonardo's notes in the accompanying text, written in mirror writing, it was made as a study of the proportions of the (male) human body as described in a treatise by the Ancient Roman architect Vitruvius, who wrote that in the human body:

·         a palm is the width of four fingers or three inches

·         a foot is the width of four palms and is 36 fingers or 12 inches

·         a cubit is the width of six palms

·         a man's height is four cubits and 24 palms

·         a pace is four cubits or five feet

·         the length of a man's outspread arms is equal to his height

·         the distance from the hairline to the bottom of the chin is one-tenth of a man's height

·         the distance from the top of the head to the bottom of the chin is one-eighth of a man's height

·         the maximum width of the shoulders is a quarter of a man's height

·         the distance from the elbow to the tip of the hand is one-fifth of a man's height

·         the distance from the elbow to the armpit is one-eighth of a man's height

·         the length of the hand is one-tenth of a man's height

·         the distance from the bottom of the chin to the nose is one-third of the length of the head

·         the distance from the hairline to the eyebrows is one-third of the length of the face

·         the length of the ear is one-third of the length of the face

Leonardo is clearly illustrating Vitruvius' De architectura 3.1.3 which reads:

The navel is naturally placed in the centre of the human body, and, if in a man lying with his face upward, and his hands and feet extended, from his navel as the centre, a circle be described, it will touch his fingers and toes. It is not alone by a circle, that the human body is thus circumscribed, as may be seen by placing it within a square. For measuring from the feet to the crown of the head, and then across the arms fully extended, we find the latter measure equal to the former; so that lines at right angles to each other, enclosing the figure, will form a square.

Though he was certainly aware of the work of Pythagoras, it does not appear that he took the harmonic divisions of the octave as being relevant to the disposition of form, preferring simpler whole-number ratios to describe proportions. However, beyond the writings of Vitruvius, it seems likely that the ancient Greeks and Romans would occasionally use proportions derived from the golden ratio (most famously, in the Parthenon of Athens), and the Pythagorean divisions of the octave. These are found in the Rhynd papyrus 16. Care should be taken in reading too much into this, however, while simple geometric transformations can quite readily produce these proportions, the Egyptian were quite good at expressing arithmetic and geometric series as unit fractions. While, it is possible that the originators of the design may not have been aware of the particular proportions they were generating as they worked, it's more likely that the methods of construction using diagonals and curves would have taught them something.

The Biblical proportions of Solomons temple caught the attention of both architects and scientists, who from a very early time began incorporating them into the architecture of cathedrals and other sacred geometry.

Regarding the Pythagorean divisions of the octave mentioned above, these are a set of whole number ratios (based on core ratios of 1:2 (octave), 2:3 (fifth) and 3:4 (fourth)) which form the Pythagorean tuning. These proportions were thought to have a recognisable harmonic significance, regardless of whether they were perceived visually or auditorially, reflecting the Pythagorean idea that all things were numbers.

Renaissance orders

The Renaissance tried to extract and codify the system of proportions in the orders as used by the ancients, believing that with analysis a mathematically absolute ideal of beauty would emerge. Brunelleschi in particular studied interactions of perspective with the perception of proportion (as understood by the ancients). This focus on the perception of harmony was somewhat of a break from the Pythagorean ideal of numbers controlling all things.

Leonardo da Vinci's Vitruvian Man is an example of a Renaissance codification of the Vitruvian view of the proportions of man. Divina proportione took the idea of the golden ratio and introduced it to the Renaissance architects. Both Palladio and Alberti produced proportional systems for classically-based architecture.

Alberti's system was based on the Pythagorean divisions of the octave. It grouped the small whole-number proportions into 3 groups, short (1:1, 2:3, 3:4), medium (1:2, 4:9, 9:16) and long (1:3, 3:8, 1:4).

Palladio's system was based on similar proportions with the addition of the square root of 2 into the mix. 1:1, 1:1.414..., 3:4, 2:3, 3:5.

The work of de Chambray, Desgodetz and Perrault eventually demonstrated that classical buildings had reference to standards of proportion that came directly from the original sense of the word geometry, the measure of the earth and its division into degrees, miles, stadia, cords, rods, paces, yards, feet, hands, palms and fingers

Le modulor

Based on apparently arbitrary proportions of an "ideal man" (possibly Le Corbusier himself) combined with the golden ratio and Vitruvian Man, Le Modulor was never popularly adopted among architects, but the system's graphic of the stylised man with one upraised arm is widely recognised and powerful. The modulor is not well suited to introduce proportion and pattern into architecture (Langhein, 2005), to improve its form qualities (gestalt pragnance) and introduce shape grammar in design in building.