This study could be done with pencil alone, with pen and pencil, with colored crayon and pencil with pencils of contrasting colors, or with felt-tip markers.

VERTICAL LINES
We continue or exploration of line into what would seem a more technical role: as definer of intervals of space. In the first of these studies, several vertical lines of identical size and station placed at equal distances apart, like a picket fence, create a single sequence. The same lines placed at irregular intervals will give up some of their significance as lines to the spatial dynamics that arise between them. Unequal pressures will result in what can be read as forward and backward pulsations in depth, from interval to interval--what the noted artist-teacher Hans Hofmann called "push and pull." This will become even more pronounced if the lines, though remaining the same length, are stationed at higher and lower positions, that is to say, staggered. Making some lines thicker and others thinner, some longer, some shorter, will draw attention to them, once again, as lines. The outcome will be an even richer pattern of perceptual forces, involving contrast of size of spatial intervals, contrast of length and thickness of lines, and thrusts forward and backward of both lines and intervals of space.

GRADIENTS
We must digress again to consider psychological qualities ordinarily taken for granted, even by artists. European art from the later years of the Middle Ages to well into the last century depended greatly on the artist's ability to achieve depth by various orderings of line, shape, value, color, and texture on a flat surface, wall, board or canvas. No wonder the elaborate picture frame became so important in time: it served to isolate the painting and establish it as a kind of window upon ever-deepening landscape space. But there is nothing in any of these elements to suggest that either deep or shallow space will be the result of their use. However, the moment they are put down with great skill and knowledge of optics, or even thrown down--two or three lines or shapes, two or more colors or values (strong and weak, warm and cool, dark and light), rough and smooth textures--the viewer will have some undeniable experience of advancing and receding parts.

Arnheim, accepting a suggestion made by James J. Gibson in his book The Perception of the Visual World (1950), says that "three-dimensional space is created by perceptual gradients." He describes these as the "gradual increase or decrease of some perceptual quality in space or time. For example, oblique parallelograms contain a gradient of location, in that the slanted figure lies at an evenly changing distance from the normal axes of the horizontal and vertical." Even a single oblique line contains a gradient of location or distance in relation to the implied horizontal and vertical axes (repeated in the sides and top and bottom of a square or rectangular format). A series of lines or shapes, wherein the elements diminish in height or width, contains a gradient of size. If the intervals of space between these elements grow narrower or wider, they too contain a gradient of size. The other gradients are those of value (light-dark gradient) and texture (smooth-rough gradient); still another pertains to sharpness/dullness or firm/fuzzy qualities.

Gradients . . . . are not dependent on anything seen in nature or the world; they pertain entirely to illusionary space, whether used to depict anything or not. The flat surface is forced to yield so that the eye may achieve order in depth.

HORIZONTALS AND VERTICALS
The structural complementary of the vertical is, of course, the horizontal. A child of about three comes up with horizontals and verticals to make so-called Greek crosses, rectangles, standing figures with outstretched arms, and, a year or two later, houses, doors, and windows. We use them as partners to describe the square, the rectangle, the cross, the post and lintel, and other peculiarly human constructs. Nature creates horizontal and vertical reticulations in the "cracking pattern" of surfaces as different in substance as paint, dried mud, and ceramic glazes.

Horizontal and vertical lines lend character and proportion to each other. They describe the essential two-dimensionality of the surface upon which they are drawn. When they interact, they create open and closed areas associated with tectonics, systems, and structures of all kinds.

OPEN STRUCTURE
It would be difficult to determine how a principle such as open structure gains acceptance at a particular time. To have continued as a vital factor in the work of almost every important artist and architect of the present era, it must have sprung from the deepest reservoirs of intuition and feeling. Its emergence through form rather than theory alone gives support to the idea that art is able to embody in its very fabric the first principles of a new vision. Certainly, from about 1890 to 1918 there was evident in the work of several artists an obsession with order and expression. Seurat's atomic color units, points; his reinstatement of Golden Section proportions in pictorial design; his reduction of forms to their simplest structural equivalents; and Cézanne's interest in dynamic synthesis and universal geometrics (sphere, cylinder, cone, cube) are but two indicators of creative regeneration. Others could be sought in architecture, music, poetry, and the sciences.

Transcultural influences are often fleeting and unreliable, but the interest shown in Japanese prints by major European artists--Manet, Monet, van Gogh, Gauguin--from the 1860s, and in Japanese houses by Western architects from about the turn of the century, seems to have been more than a passing flirtation. The Japanese use of line, plane, pattern, and texture, their handling of space in their paintings and prints, and their modular partitioning of areas in their dwellings inspired Western artists and architects over a number of decades. Some kind of basic language of design, bridging time, place, and culture, appeared to be evolving in the twentieth century. Museums, the camera, photographic reproductions in books and magazines (André Malroux's "museum without walls"), scholarship, and travel were pressing all art together for comparison.

The discipline of straight lines--horizontals and verticals--in relationship to themselves has brought us to a consideration of the more technical applications of lines . . . .

The earliest examples of right-angle lines are drawings from the caves of Southern France and from Mesopotamia. Graphic representations and a rudimentary geometry (Tectiforms) seem to have sprung up as twins in the household of humanity. This is not surprising, since both partake of people's eagerness to understand their surroundings and their need to build their own defenses against its seen and unseen powers.

It would be difficult to determine how a principle such as open structure.

If fear, hunger, and a need for ritual communion with a spirit-world led to the drawing of animal images or shamanistic images, the building of snares, corrals, and shelters, the technique of weaving, and the making of tools led to the second type of imagery. In the primal society, where acquisition of food by hunting or foraging and protection of the social unit against its foes were all-consuming responsibilities, both of these ^social-symbolic functions existed in intimate and vital conjunction.

Techniques of weaving and plaiting, of binding saplings to form rectangular pens, and, later, the making of bricks provided a repertory of forms that, in turn, served as models for thinking. They were aids to a simple arithmetic and, ultimately, to geometry and standardized measures; for instance, the knotted cord of the Egyptians. We assume that the brick, which was invented in Summeria in about 3500 BC or earlier, led immediately 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.

Tools and techniques altered the human environment even as they transformed the potentiality of humanity itself. Symbols, both drawn and spoken, helped people grasp the things of their environment; for to draw or name a thing is, in a sense, to domesticate it. Therefore, image-making, abstraction, language, and conceptual thinking are of the same origin, or lead to one another by a process of cultural growth and development.

Geometry as we understand it, goes back to the ancient Egyptians. The word itself provides a clue to its origin and use: geo = land; metry = measure. The problem of "land-measure" arose every year because of the flooding of the Nile valley and the Egyptian practice of levying taxes according to the extent of land ownership. Egyptian surveyors evolved the ingenious device of the knotted cord, which led to a way of finding a right angle as a component of a right-angle triangle, making it possible to calculate nearly all sizes and shapes of arable land. But the Egyptians had more complex uses for geometry and the right-angle triangle in the great building and engineering projects of the Phaoraohs and in the priestly art of numbers and astrology.

The knotted-cord-and-triangle idea was brought to Greece in the sixth century BC. Geometry was to the Greeks a divine exercise, an absolute and perfect way to create designs that were applicable in the building of temples, making exquisite pottery, creating statues of gods and heroes, and speculating on the essence of matter or the movements of the celestial bodies.

ANTHROPOCENTRIC MEASURES
Measure must be understood, first of all, not in terms of arithmetic, but in terms of the dimensions 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.

It should be natural to suppose that this unit of measure was taken from some portion of the human body at some unrecorded moment in the distant past. This would appear to be true if one considers that, while being a unit of length, it was also a unit of energy. The prime Egyptian unit, the cubit, was the length of the forearm from elbow to the middle finger extended, and we can assume that it was chosen because it was/is the preeminent working unit of the body. Similarly, the English foot rule is not merely the length of the average foot, but the distance between the rungs of a ladder; as such, it relates sensibly to the amount of energy required by both legs and arms in the act of climbing. The yard is the distance from the center of the body (the tip of the nose pointing forward) to the tip of the outwardly extended arm and thumb and is (or was) associated intimately with the measuring of cloth or rope.

Until the invention of the metric system in France at the end of the eighteenth century, all measurements of length, plane surfaces, weights, volumes, and time were related to human functions and capacities, to belief, concept and theory, invention and commerce. We refer to some of these casually but not incorrectly as measurement by "rule of thumb."

The fiercely rationalistic spirit of the French Revolution was so pervasive that even the foot-and-inch system, the old anthropocentric measures, were deemed inadequate, and scientific measurements were adopted. In 1799 a decision was taken as to the length of the meter. In the same year, the value of the meter and the kilogram, the weight of a liter of water, became law. The metric system is now obligatory in most non-English-Speaking countries. There are, then, two radically opposing systems of measure in use in this age of science-technology, world trade, and communication. Clearly each system has its own merits.

The metric system is a masterpiece of mathematical absurdity, the meter being the one ten-millionth part of a meridional quadrant of the earth, but it is remarkable in its clarity, unity, and versatility. By comparison, the Anglo-Saxon inch-foot-and-yard system is awkward and dated--yet, because of it relation to the human body and human imaging, it remains a far better visual measure than the indifferent meter. According to some authorities, this explains why reasonably good proportions in the designing of architecture and furniture have prevailed in countries where the old system has been kept, and why poorer proportions are sometimes observed in countries where the metric system is used exclusively and mechanically . . . .

. . . . [Japanese] system of proportioning in all things that pertain to practical, everyday use, such as household objects. Their traditional houses for living are constructed on an ancient module that is very much to the measure of man in a congested world. It is embodied most clearly in the ^tatami, the thick durable straw mat, measuring approximately 3 x 6 feet, so many units of which are used to cover the entire floor, and are both for walking upon (with special house slippers) and sleeping upon. The dimensions of all rooms in the traditional Japanese house are a multiple of this replaceable mat . . . .

Scholarly inquiry and analysis have determined that all important architecture of the ancient world was modular in plan and construction. Architecture was a priestly art or an art practiced by a few men who, while being artists of the highest order, were also initiates into the mysteries of mathematics and secret harmonies. It was also clearly related to sculpture, in that form and mass often assumed greater importance than space. The great sacred buildings of the Egyptians and Greeks were intended as shrines or sanctuaries, not as general places of worship for large numbers of people. Their spaces were small and dominated by thick walls and great cylindrical columns. Therefore, it is reasonable that the module should have been one of solid physical substance rather than of space. It would also seem logical that the one circular element of the plan, the column, deriving undoubtedly from the trunk of a tree in the case of Greek architecture, should have contained the basic unit for each temple. The column was the one element of structure with which the human individual could identify immediately by sight, touch, and embrace. He or she could, in more than a mathematical sense, "take its measure" and appreciate the scale and rhythmic unity of the whole building . As Rasmussen explains it: "The basic unit was the diameter of the column and from that were derived the dimensions not only of shaft, base, and capital but also of all the details of the entablature above the columns and the distances between them . . . . Where small columns were used, everything was correspondingly small; when the columns were large, everything else was large too."

This provides an excellent clue to the marvelously human scale of the temples of the Acropolis, the Parthenon and the Erectheum and, conversely, to the overpowering, superhuman size of many buildings erected during the Roman period and again after the Renaissance--railway stations, government buildings, banks, and churches--where columns in "large orders" were used for grandiose effects.

THE GOLDEN SECTION
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 us 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 [Visuals not included here] . 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 [Visuals not included here] . 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 [Visuals not included here] . 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 [Visuals not included here] . 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 , 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.

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