From Academic Kids
Architectural practice has often used proportional systems to generate or constrain the forms considered suitable for inclusion in a building. In almost every building tradition there is a system of mathematical relations which govern the relationships between aspects of the design. These systems of proportion are often quite simple; whole number ratios or easily constructed geometric shapes (such as the Vesica Piscis or the Golden Ratio).
Generally the goal of a proportional system is to produce a sense of coherence and harmony among the elements of a building.
Among the Cistercians, Gothic, Renaissance, Egyptian, Semitic, Babylonian, Arab, Greek and Roman traditions; the harmonic proportions, human proportions, cosmological/astronomical proportions and orientations, and various aspects of sacred geometry (Vesica Piscis, the Pentagram, Golden Ratio and small whole-number ratios) were all applied as part of the practice of Architectural design.
Part of the practice of Feng Shui is a proportional system based on the double tatami mat. Feng Shui also includes within it the ideas of cosmic orientation and ordering, as do most systems of "Sacred Proportions".
The Classical Orders are largely known through the writings of Vitruvius, particularly De Archetura (The Ten Books of Architecture) and studies of classical architecture by Renaissance architects and historians. Within a classical order elements from the positioning of triglyphs to the overall height and width of the building were controlled by principles of proportionality.
Vitruvius described as the principle source of proportion among the orders the proportion of the human figure. 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, prefering 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 Section (most famously, in the Parthenon of Athens), and the Pythagorean divisions of the octave. Care should be taken in reading too much into this, however, as simple geometric transformations can quite readily produce these proportions, thus it is possible that the originators of the design may not have been aware of the particular proportions they were generating as they worked.
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.
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 Leone Battista Alberti produced proportional systems for classically-based architecture.
The work of de Chambrey, Desgodet and Perrault eventually demonstrated that classical buildings were not rigidly defined and perfectly mathematical, but were instead loose and readily adapted. (Tzonis and Lefaivre, 1986, pp 39)
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 Plastic Number
The Plastic Number is of interest primarily for its method of genesis. Its creator, Hans van der Laan performed experiments on human subjects to attempt to discover the limits of human beings ability to perceive relationships between objects. From these discovered limits he extrapolated a system of proportions (the particular set he chose are quite close to the Pythagorean divisions of the octave). The range of scales over which the Plastic Number is considered functional is limited, so it is possible to construct a set of all proportional forms within it. The Plastic Number has not been widely adopted by practicing architects.