Proportion is the relation between elements and a whole. (jvr)
“  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. —Vitruvius,^{[1]} The Ten Books of Architecture (III, Ch. 1)  ” 
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In architecture the whole is not just a building but the set and setting of the site. The things that make a building and its site "well shaped" include the orientation of the site and the buildings on it to the features of the grounds on which it is situated. Light, shade, wind, elevation, choice of materials, all should relate to a standard and say what is it that makes it what it is, and what is it that makes it not something else.
Vitruvius thought of proportion in terms of unit fractions^{[2]} such as those used in the Greek Orders of Architecture.^{[3]}
Scribes had been using unit fractions for their calculations at least since the time of the Egyptian Mathematical Leather Roll and Rhind Mathematical Papyrus^{[4]} in Egypt and the Epic of Gilgamesh^{[5]} in Mesopotamia.
One example of symmetry might be found in the inscription grids^{[6]} of the Egyptians which were based on parts of the body and their symmetrical relation to each other, fingers, palms, hands, feet, cubits, etc; Multiples of body proportions would be found in the arrangements of fields and in the buildings people lived in.^{[7]}
A cubit could be divided into fingers, palms, hands and so could a foot, or a multiple of a foot. Special units related to feet as the hypotenuse of a 3/4/5 triangle with one side a foot were named remen and introduced into the proportional system very early on. Curves were also defined in a similar manner and used by architects in their design of arches and other building elements.
These proportional elements were used by the Persians, Greeks, Phoenicians and Romans, in laying out cities, stadiums, roads, processional ways, public buildings, ports, various areas for crops and grazing beasts of burden, so as to arrange the city as well as the building to be well proportioned,^{[8]}^{[9]}
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 governs the relationships between aspects of the design. These systems of proportion are often quite simple; whole number ratios (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 (the vesica piscis), pentagram, golden ratio, and small wholenumber ratios) were all applied as part of the practice of architectural design.
In the design of European cathedrals the necessary engineering to keep the structures from falling down gradually began to take precedence over or at least to have an influence on aesthetic proportions. Other concerns were symbolic astronomical references such as the towers of the Sun and Moon at Chartres and references to the various astrological and alchemical relationships being discovered by the natural philosophers and sages of the renaissance.
The Roman Mille passus became the Myle of medieval western Europe and Roman archs and architecture while the mia chillioi influenced eastern Europe and its Gothic arches and architecture. Today in the Western hemisphere the foot is longer than the foote because of the researches of Galileo, Gabriel Mouton, Newton and others into the period of a seconds pendulum.
One aspect of proportional systems is to make them as universally applicable as possible, not just to one application but as a universal ideal statement of the proper proportions. There is a relationship between length and width and height; between length and area and between area and volume. Doors and Windows are fenestrated. Fenestration is important so that the negative area of openings has a relation to the area of walls. Plans are reflected in sections and elevations. Themes are developed which spin off and relate to but expand upon the themes found in other buildings. Often there is a symbolic sacred geometry which goes outside the proportions of the building to relate to the oservations of the beauty of nature and its proportions in time and space and the elements of natural philosophy.
Then it occurred to someone that there is more to it than just pleasing proportions. Thomas Jefferson wrote of how the substantive scale of public buildings made a statement of government stability and gave a nation consequence.
Going back in time the same logic applied to the Pyramids of Egypt, the Hanging Gardens of Babylon, the Mortuary Temple of Hatshepset, the Temple of Solomon, the Treasury of Athens, the Parthenon, and the Cathedrals and Mosques and Corporate Towers. The Casinos of Las Vegas and the underwater hotels of Dubai are all competing to be the tallest, the biggest, the brightest, the most exciting to get international trade to come there and do business. In other words the modern business ethos is to be out of proportion, overscaling all the competition.
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".
Going back to the Pythagoreans there is an idea that proportions should be related to standards and that the more general and formulaic the standards the better. This idea that there should be beauty and elegance evidenced by a skillful composition of well understood elements underlies mathematics in general and in a sense all the architectural modulors of design as well.
The idea is that buildings should scale down to dimensions humans can relate to and scale up through distances humans can travel as a procession of revelations which may sometimes invoke closure, or glimpses of views that go beyond any encompassing framework and thus suggest to the observer that there is something more besides, invoking wonder and awe.
The classical standards are a series of paired opposites designed to expand the dimensional constraints of the harmony and proportion. In the Greek ideal Vitruvius addresses they are similarity, difference, motion, rest, number, sequence and consequence.
These are incorporated in good architectural design as philosophical categorization; what similarity is of the essence that makes it what it is, and what difference is it that makes it not something else? Is the size of a column or an arch related just to the structural load it bears or more broadly to the presence and purpose of the space itself?
The standard of motion originally referred to encompassing change but has now been expanded to buildings whose kinetic mechanisms may actually determine change depend upon harmonies of wind, humidity, temperature, sound, light, time of day or night, and previous cycles of change.
The stability victim of inflicted madness is questionable architectural standard of the universal set of proportions references the totality of the built environment so that even as it changes it does so in an ongoing and continuous process that can be measured, weighed, and judged as to its orderly harmony.
Sacred geometry has the same arrangement of elements found in compositions of music and nature at its finest incorporating light and shadow, sound and silence, texture and smoothness, mass and airy lightness, as in a forest glade where the leaves move gently on the wind or a sparkle of metal catches the eye as a ripple of water on a pond.
The frieze and architrave vary from 3/4:1/2 in the Doric style to 5/8:5/8 in the Ionic and Corinthian styles. Capitals are 1/2 in all styles except Corinthian which is 3/4. The shaft width is always 5/6 at the top. Column shaft heights are Tuscan 7, Doric 8, Ionic 9 and Corinthian 10. Column bases are always 1/2. In the Pedestal, caps are always 1/4, dies are 8/6 and bases are 3/4. In the quarter of the column entasis, Tuscan styles are 9/4, Doric are 10/4, Ionic are 11/4 and Corinthian columns are 12/4.
Having established the column proportions we move on to its arcade which may be regular with a single element at a spacing of 3 3/4 D, coupled with two elements at 1 1/3 D spaced 5 D, or alternating at 3 3/4 spaced 6 1/4 D. Variations include adding a series of arches between column cap and entablature in the Renaissance style [[Arcade rcade. Exterior door widths W, have trim 1/5 W for exterior doors and 1/6 W for interior doors. Door heights a re 1 D less than column heights. Anciently if a door is two cubits or between 36" and 42" in width, then its trim is between a fist and a span in width.
The Greek classical orders are all proportioned rather than dimensioned or measured modules and this is because the earliest modules were not based on body parts and their spans (fingers, palms, hands, feet, remen, cubits, ells, yards, paces and fathoms, which became standardized for bricks and boards before the time of the Greeks) but rather column diameters and the widths of arcades and fenestrations.
Typically one set of column diameter modules used for casework and architectural moldings by the Egyptians, Romans and English is based on the proportions of the palm and the finger, while another less delicate module used for door and window trim, tile work, and roofing in Mesopotamia and Greece is based on the proportions of the hand and the thumb. Board modules tend to round down for planing and finishing while masonry tends to round down for mortar. Fabric, carpet and rugs tend to be manufactured in feet, yards and ells.
In Palladian or Greek Revival architecture as in Jeffersonian architecture, modern modular dimensional systems based on the golden ratio and other pleasing proportional and dimensional relationships begin to influence the design as with the modules of the volute. One interface between proportion and dimension is the Egyptian inscription grid. Grid coordinates can be used for things like unit rise and run.
The architectural foot as a reference to the human body was incorporated in architectural standards in Mesopotamia, Egypt, Greece, Rome and Europe. Common multiples of a foot in buildings tend to be decimal or octal and this affects the modulars used in building materials. Elsewhere, it is a multiple of the palm, hand, or finger that is the primary referent. Feet were usually divided into palms or hands, multiples of which were also remen and cubits.
The first known foot referenced as a standard was from Sumer, where a rod at the feet of a statue of Gudea of Lagash from around 2575 BC is divided into a foot and other units. Egyptian foot units have the same length as Mesopotamian foot units, but are divided into palms rather than hands converting the proportional divisions from sexagesimal to septenary units. In both cases feet are further subdivided into digits.
In Ancient Greece, there are several different foot standards generally referred to in the literature as short, median and long, which give rise to the different architectural styles known as Ionic and Doric in discussions of the classical orders of architecture. The Roman foot or pes is divided into digitus, uncia and palmus, which are incorporated into the Corinthian style.
Some of the earliest records of the use of the foot come from the Persian Gulf bordered by India (Meluhha), Pakistan, Balochistan, Oman (Makkan), Iran, Iraq, Kuwait, Bahrain (Dilmun), the United Arab Emirates and Saudi Arabia where in Persian architecture it is a sub division of the Great circle of the earth into 360 degrees. In Egypt, one degree was 10 Itrw or River journeys. In Greece a degree was 60 Mia chillioi or thousands and comprised 600 stadia, with one stadion divided into 600 pous or feet. In Rome a degree was 75 Mille Passus or 1000 passus. Thus the degree division was 111 km and the stadion 185 m. One nautical mile was 10 stadia or 6000 feet. The incorporation of proportions which relate the building to the earth it stands on are called sacred geometry.
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:
Leonardo is clearly illustrating Vitruvius' De architectura 3.1.3 which reads:
 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 onetenth of a man's height
 the distance from the top of the head to the bottom of the chin is oneeighth 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 onefifth of a man's height
 the distance from the elbow to the armpit is oneeighth of a man's height
 the length of the hand is onetenth of a man's height
 the distance from the bottom of the chin to the nose is onethird of the length of the head
 the distance from the hairline to the eyebrows is onethird of the length of the face
 the length of the ear is onethird of the length of the face
 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 wholenumber 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.
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 classicallybased architecture.
Alberti's system was based on the Pythagorean divisions of the octave. It grouped the small wholenumber 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.^{[10]}.
The work of de Chambray, Desgodetz and Perrault ^{[11]} 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
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. AntiModernists (Langhein, 2005) claim the modulor is not well suited to introduce proportion and pattern into architecture , to improve its form qualities (gestalt pragnance) and introduce shape grammar in design in building. However, through its application in the design of some of the last century's most beautifully proportioned and harmonic buildings (Le Corbusier: Architect of the Twentieth Century, Kenneth Frampton, 2002) Le Corbusier's work strongly disputes this.
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.
