Golden Section

C O N S I D E R:
[The following is from: Harlan, Calvin. Vision & Invention, An Introduction to Art Fundamentals. Englewood Cliffs, NJ: Prentice-Hall, 1986.]

The term Golden Section was given in the nineteenth century to the proportion derived from the division of a line into what Euclid (active about 300 BC) called "extreme and mean ratio." "A straight line," he explained, "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." It is often claimed that the Golden Section is esthetically superior to all other proportions, and, if it is admitted that what pleases the eye is unity in variety, it may be said that this proportion fulfills this condition better than any other.

Most attempts to explain the origin of the Golden Section, its use in antiquity, during the Middle Ages and especially in the Renaissance, and more recently in paintings by Seurat and buildings by Le Corbusier, trail off into legend and speculation. Some say it originated in Egypt as one of several proportioning schemes used in the designing of buildings, statues, and wall paintings. Other connect it, more convincingly, with the work of certain Greek savants and geometers, notably Pythagoras and Eudoxus, both of whom probably 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 stretched string. Phythagoras' work with acoustical measurements was presumably the basis of Greek tuning and the six modes or scales, which carried over in the liturgical music of the Middle Ages. Taut strings of different lengths, when related to one another according to simple numerical ratios, produce agreeable sounds or euphony. Harmonic intervals were applied, on occasion, to achitectural design during the Renaissance. Their first imaginative use was made, however, by disciples of Pythagoras in their attempt to explain the position and velocity of the heavenly bodies. They combined astronomical and musical discoveries in the famous doctrine of "the harmony of the spheres."

Eudoxus (Third century BC) is the principal figure of another legend in which the truth of fiction is perhaps as acceptable as the truth of fact. He 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.

The American artist-mathematician Jay Hambridge set out to remove every element of doubt concerning whether the architects of the Parthenon (447-438 BC) Ictinus, Callicrates, and the sculptor Phidia made use of a very evolved understanding of Golden Section and root-f proportions. His analyses of plan and elevation (façade and all vertical elements), including every detail, do provide evidence of the most incredible sensitivity and intelligence.

A spontaneous, visual-intuitive grasp of harmonious proportions must enliven all mathematical formulations, give energy and direction to every design from the moment of conception. Then, when one is working through difficulties, intuition and theory may be able to establish a partnership that is at the heart of classicism, of whatever time or place. Other modalities spring from other relations between the extremes of passion and construction, chaos and geometry. The classical spirit in individuals and in certain eras vacillates somewhere near the center of these extremes; the romantic spirit hovers more to the left of center.

Writers on the art of Europe sometimes refer to a Northern mode or tradition and a Southern mode, pointing to native, almost instinctual elements in both that pre-date classicism and romanticism, as such. The Northern tradition is said to be allied to that very large cultural complex that extended in early times across the length and breadth of what is now Russia; that is, from China to Western Europe. The "Animal style," as it is referred to is linear, restless, and affirmative. It can also be aggressive, brooding, close to the zoological buff in what would seem to be its Celtic and Scandinavian offshoots. The southern or Mediterranean mode is characterized as "Humanistic," to the point of turning nature into "background" and décor. Its art is geometrical in spirit, leaning toward the ideal and the achitectonic, as in Greek art, or actualistic, as in Roman art. Its subject is humanity and the order of humanity, not humanity in heroic or pathetic combat with nature. One has only to compare Greek pottery of the fifth, sixth, seventh, or eight centuries BC with English medieval pottery, or the Parthenon with a Viking ship, to appreciate two divergent attitudes toward nature and reality.

We could start as presumably an ancient Greek would--with little more than a piece of string, two points, a straightedge, and a smooth surface--and move quickly from one demonstration to another of Golden Section (dynamic symmetry, golden mean) proportions. [a/b = b(a+b) - [Harris, William H., and Judith S. Levey, eds. The New Columbia Encyclopedia. New York and London: Columbia University Press, 1975.]

Golden Section of the line AB.
If we had only one line to begin with . . . . Call the line AB. Draw a perpendicular line at B. Take half of AB and mark this length on the perpendicular line as C (BC). Draw the diagonal or hypotenuse AC, and then, with point of compass (or string) at A, swing an arc upward across AC from one-half AB. D marks the intersection of AC. Then, with compass point at C, measure the length CD. With this measure and the compass point at A, mark the original line AB at E, its Golden Section. In formal terms, the lesser part EB is to the greater AE as the greater part is to the whole line. (EB/AE): (AE/AB); in other words, they will be in extreme and mean ratio.

Golden Section extension.
If it isn't the point of the Golden Section of the line AB that is wanted, but the Golden Section extension, we could draw a perpendicular (with the aid of right-angle triangle) from B the same length as AB (BC); then, with the point of the compass at half the length of AB, swing an ac from C downward across the extension of AB at D. BD, then, would be the Golden proportion of AB.

Golden Section of an area or plane.
From line we could move directly to the Golden Section of an area or plane--a matter of greater interest to designers than the Golden section of a line. Here the square becomes the key figure. We could proceed in one of two ways: By starting with our line AB and drawing a perpendicular from E, its Golden Section, the same length as AE, and, with ruler or triangle, completing a Golden Section rectangle, consisting of a square and a smaller Golden Section rectangle oriented vertically; or by starting with a square and placing the point of the compass precisely at the center of the base line describing a semicircle from left to right (or right to left) across the upper corners. Extension of the base either to the right or left of the square would give the Golden Mean Proportion to the square.

The Square in the Semicircle: The Root-5 Rectangle
The two extensions, right and left, create another rectangle that was the special delight of Greek architects, if we can believe the various diagrammatic analyses of their works. The two Golden Section rectangles overlap to the extent of the square. Viewed in another way, this is a double Golden Section rectangle capable of being read both ways: to the right or left from the central square. More importantly, it can be divided into five equal parts, each having the same proportions as the parent rectangle. Parent and children are all so-called root-5 rectangles. Division of the rectangle is achieved by drawing its diagonal from corner to corner (say, upper left to lower right), then a shorter diagonal from (in this instance) the upper-right angles. The shorter one becomes the principal diagonal of a vertical rectangle one-fifth the size of the original root-5 rectangle. More divisions still may be made within this smaller rectangle.

The versatility of the root-5 rectangle (if we may take the analyses on total faith) is seen in the several arrangements of squares, Golden Section rectangles, larger and smaller root-5 rectangles, juxtaposed and interlocking, known in antiquity and still worthy of attention . . . .

During the late twelfth and early thirteenth century, at the height of the so-called first Renaissance, there lived an Italian merchant-traveler-mathematician by the name of Leonardo Fibonacci, born in Pisa in 1175. He received his advanced education among the Muslims in Barbary, North Africa. He learned the Arabic (or decimal) system of numbering, [Decimal system n [1864] 1: a number system that uses a notation in which each number is expressed in base 10 by using one of the first nine integers or ) in each place and letting each place value be a power of 10 2: a system of measurement or currency in which the basic units increase by powers of 10. . . . Arabic numeral n [ca. 1847]: any of the number symbols 0, 1, 2, 3, 4, 5, 6, 76, 8, 9, -see Number table.[Merriam-Webster's Collegiate Dictionary, 10th Edition. Springfield, MA, USA: Merriam-Webster, Inc. 1995.] as well as algebra. On his return to Italy he published his famous book of the Abacus, which stressed the advantages of the Arabic system of numeration over the Roman and brought about the triumph of the Hindu-Arabic system in Christian Europe. Fibonacci (Or Leonardo of Pisa, as he is often referred to) proposed a series of numbers that bear a close resemblance to the rule of the Golden Section. It is also called the Fibonacci Series, the Fibonacci Sequence, and the Summation Series. Starting with the number one, each unit is formed by adding together the two preceding numbers: l, l, 2, 3, 5, 8, 13, 21, 34, 55, and so on. The higher the series goes , the more closely it resembles the Golden Section ratio. It gives us the nearest whole number approximation to mean and extreme ratios. The simple and familiar ratios (straight integers) such as l:1, 1:2, 2:3, 3:4 are relatively static, inert, dividual. The Fibonacci ratios, by comparison, are dynamic and individual. They suggest a kind of sprung energy; and no doubt this is why they crop up in the study of plant and animal growth.

Fibonacci's gleanings caused great excitement at the time they were made known, but there was a lapse of some 200 years before they would strike a vital response in artists who were also mathematicians. During the first quarter of the fifteenth century, Brunelleschi (1377-1446), the scholar and architect, his protégé, the painter Masaccio (1401-1428), and a it later, those artists-mathematicians of rare spirit, Uccello (12397-1475) and Piero della Francesca (c. 1415-1492), evolved an art infused with a geometry of proportion, elegance, and space. Part and parcel of this was the new discipline of linear perspective. We may assume that their predecessors, the architects of the great Medieval cathedrals, applied more than a rule-of-thumb geometry in their marvelous, soaring structures; but the precise nature of their methods has been more difficult to pinpoint.

During the 1490s, shortly after the completion of The Last Supper, Leonardo da Vinci met Luca Pacioli, a Minorite friar and mathematician, follower of Piero della Francesca in matters pertaining to linear perspective and of Fibonacci in matters pertaining to mathematics. The two men became close friends and collaborators. Leonardo worked with Pacioli on his book Divina Proportione. This book, published in 1503 and illustrated by Leonardo, contained many of the theories about the Golden Section, or the Divine Proportion, as Pacioli chose to call it, reasoning that, if God saw fit to use this ratio in the creation of natural forms and of man, His noblest handiwork, it must indeed by divine!

This book seemed to bring into focus elements of the humanistic movement: the scientific study of perspective and the play of light and shadow (chiaroscuro: Italian Chiaro = bright + oscuro = dark) and the study of anatomy and geometrical proportion, all of which had been emerging steadily during the preceding century and had been forecast in the art of Giotto (c. 1267-1337). It was certainly to influence the character of European art for many generations. A further exploration of the mysteries of proportion and their influence in artistic structures is to be found in drawings and paintings by Raphael, Titian, Dürer, and others, and in buildings by Andrea Palladio (1508-1580) and other architects well into the present era. By the mid-nineteenth century the original spirit had become debased or exhausted. It must be emphasized that the Golden Section rectangle was not the only proportioning device used. The 5- and 6-pointed stars, the harmonic intervals of the musical scale, and other systems were tried, with greater or lesser success.

Then, during the last quarter of the nineteenth century, there came along two artists who would signal a return to an order based on perception and reinterpretation of form and space relations. They were Cézanne and Seurat. Seurat made a careful study of the Golden Section and of the composition of Piero della Francesca. The new vision would emerge in painting, as it had done in the fifteenth century. Degas and van Gogh made important structural-expressive contributions in their use of multiple points of view, or multiple cones of vision. Some or all of this was given rational definition and direction in Cubism (1907-1912).

From Cubism it radiated immediately into abstract styles in painting and sculpture, and into architecture. Cubism reinstated the straight line, the line of rationality, and, with it, a modern geometry of form, space, and movement.

Before appearing to claim all in the name of Cubism, we must take note of the fact that at least two architects, working before and apart from Cubism, created an architecture that is not unlike Cubism in its probity and form-space articulations. These were Louis Sullivan and Frank Lloyd Wright.

Evolving directly out of Cubism is the work of another man of genius, in which we find not only a synthesis of form, space, and structure but a convergence of the old dual considerations of human proportion and harmonic proportion. Le Corbusier's Modularis an important achievement, whether it has gained or will ever gain universal acceptance. It expresses a heroic spirit, a modern/ancient dream of man in an ordered world, where mind or spirit and body complement each other.

A final explanation of the special appeal of the Golden Section ratio, or any other harmonic ratio arrived at mathematically or intuitively, has yet to be given, as far as I know. Perhaps there is no simple, easy explanation beyond the assumption that the laws of proportion are, in some strange way, inherent in visual perception. The unconscious searching for relationships that are neither so well balanced as to be dull, nor so precarious as to be irritating, are all involved to some extent. The eye quickly exhausts any area that is divided in strict symmetry. The Golden Section division is neither too difficult to grasp spontaneously nor easy to exhaust. Equilibrium is threatened, but a kind of dynamic tension arises that is curiously binding. The eye will try to coax the division back to the center of the area or, failing that, to some other "regular" position. Unequal partitioning and the back-and-forth, long-and-short rhythm of perception will join in such a way as to vitalize every square inch of the area.

The first four defining (delimiting) lines of a two-dimensional area are, in a sense, the first four lines of a design or composition. The internal tensions they inaugurate may, especially in the case of the Golden Section and root-5 rectangles, arouse optical interest without further partitioning. The energized blankness of these fields is arresting in and of itself. Yet these areas are seldom left vacant. More lines and shapes inscribed within them are capable of establishing other tension patterns that serve functional and a variety of esthetic purposes. Place even a point within a blank format, and horizontal and vertical divisions may be sensed immediately. These tensions, intersecting at the location of the intruding element, become the invisible governing lines of a composition. When developed further, they not only divide the composition into large and small two-dimensional areas, they also conjure fast and slow, up-down, side-to-side, diagonal, stepwise, and even circular movements on the surface and in depth.

Similar governing lines, spelled out three-dimensionally in proportion and scale, in form, space, and function, are seen in outstanding works of architecture. The architect, by the way he or she creates approaches, passageways, and cavities (wide and narrow, high and low, open and closed), more or less programs the movement of our eyes and bodies and influences both our sense of well-being and vital efficiency.

[Harlan, Calvin. Vision & Invention, An Introduction to Art Fundamentals. Englewood Cliffs, NJ: Prentice-Hall, 1986.]

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