Relationship of dot, line and plane
18. Proportion and scale
To make a visually appealing composition, it is essential to determine the proportion of the elements that make it. Whether it is the proportion of dots to lines, or of red to blue, one must analyse the position and size of the constituent elements.
18 - 01 ANTHROPOCENTRIC MEASURES:
Measure must first be understood, not in terms of arithmetic, but as dimension, and the practical, functional energy of the human body. Pure geometry can exist without considerations of size or measure, since it deals only with absolute proportion. Art and architecture must deal with relative proportion, and this presupposes a unit of measure that is very human in origin.
Until the invention of the metric system at the end of the eighteenth century, all measurements of lengths, plane surfaces, weights, volumes and time were related to human functions and capacities, to belief, concept and theory, invention and commerce. The Egyptian concept of length was the cubit, which was the length of the average forearm, from the elbow to the middle finger extended. The English foot rule is also based on a working unit of the body - the foot. Besides it being the length of the average foot, it is also the distance between the rungs of a ladder; as such it relates sensibly to the amount of energy required by arms and legs in the act of climbing. The yard is the distance from the centre of the body (tip of the nose pointing forward) to the tip of the outwardly extended arm and thumb, and was closely associated with the measuring of cloth or rope. The inch-foot-yard system is now outdated, but because of its relation to the human body and human imaging, it remains a far better visual measure than the metre. However, the metric system is remarkable in its clarity, unity and versatility, and because of its scientific basis, is extremely accurate. The value of a metre as a unit of length, and of a kilogram and a litre as units of weight, constitute the essence of the metric system.
18 - 02 MODULAR :
Scholarly inquiry and analysis have determined that all important architecture of the ancient world was modular in plan and construction. Ancient techniques of weaving and plaiting, of binding saplings to form rectangle pens, and later, the making of bricks provided a repertory of forms that, in turn, served as models for thinking. They were aids to simple arithmetic and ultimately to geometry and standardized measure; for instance, the knotted cord of the Egyptians. The brick was invented in Sumeria in about 3500 B.C. or earlier, and it could be said to have led to every variation in the art of building. The brick, because it was made by hand to a certain manageable uniform size and could be repeated indefinitely, also embodied the concept of a basic unit of measure, a 'module'. Whether it's the sacred Egyptian building or Greek temples, they have all evolved from the dimensions of some basic unit: in a Greek temple, it's the circular columns derived from the trunk of a tree, which is the basic unit for its construction. The dimensions of all the rooms in a traditional Japanese house are a multiple of a thick, replaceable straw mat, measuring 3 X 6 feet, so that multiples of the unit can be used to cover the entire floor.
In the early part of the twentieth century, architect Le Corbusier developed the Modular, a measuring tool based on the human body and on mathematics. In this synthesis of form, space and structure, the inclusion of human proportion is very significant.
The Modular Man is 183 cm (6ft) tall and with left arm raised, reaches a height of 226 cm (about 7 ½ ft). His head height, if partitioned as per Golden Section or the height of his navel, is 113cm (27 ½ ft), which, curiously enough, is half the height of the raised arm. From these and a fourth key point of the human figure, the parting of the legs (or the place where the right hand rests, 86 cm (34 inches) above the base), two series of measurements have been derived: the reaching height (blue series) and the head height (red series); each divided into diminishing proportions based on the Golden Section ratio.
18 - 03 GOLDEN SECTION:
The unconscious search for relationships that are neither so well balanced as to be dull, nor so precarious as to be irritating, seems to be the basis of the Golden Section. This kind of division is neither too difficult to grasp spontaneously, nor too easy to exhaust. Equilibrium is threatened, but a kind of dynamic tension arises that is curiously binding.
The term was given in the nineteenth century to the proportion derived form the division of a line into what Euclid called "extreme and mean ratio" - according to which "a straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the less." Most attempts to explain the origin of the Golden section trail off into legend and speculation. Some say it originated in Egypt, others trace its roots to Greece, to the work of certain geometers like Pythagoras and Eudoxus, both of whom knew Egyptian achievements well. Pythagoras believed that one could demonstrate order in the universe by expressing all relationships among the parts of things in terms of single whole numbers; he believed that such relationships do exist among the perfect intervals of tones produced by sounding a stretching string. Eudoxus is said to have carried with him at all times a stick, which he asked friends and acquaintances to divide at whatever point they sensed to be the most pleasing. Much to his satisfaction they chose more often than not, the point of the Golden Section. Fact or fiction, this tale illustrates something important in art and design: the close relationship of intuitive perceptions or felt ratios to reasoned or mathematical ratios.
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