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 Modular is 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.

DIAGONALS
The horizontal and vertical scheme, the right-angle construct, while being the clearest and, in so many ways, the most useful, is limited in at least one respect: It does not admit of much distinction between things standing at rest, maintaining themselves in position, and being in motion.

At some point in the child's development, it becomes very important that he or she be able to draw figures walking, running, falling, playing; branches reaching upward; birds flying; and the like. The increasing experience of motion, of control, in himself/herself, in others and in the environment, is something he or she must be able to put down graphically. Something similar may be said of image-making in tribal societies: There comes a time when worrisome restrictions and ambiguities must be overcome. The limitations of certain formulas, used for untold generations and accepted as the ritual form-language, may have to give way under pressure of new circumstances and socio-psychological necessities--witness the changes in the art of the Oglala Sioux that occurred with the coming of the day of the horse. We observe this kind of development in sculpture as well: A hierarchic stiffness is succeeded by movements of every kind and degree of suppleness.

It seems too simple from our vantage point, but the child's discovery of oblique lines is an important moment in his or her "career" as artist. In the art of evolved Western and Oriental societies, the oblique line plays a dual role at times: It not only conveys action, but it is one of the principal means of creating pictorial space, as we have seen. What is Renaissance perspective, central perspective, in essence and especially in its original form, but a graphic mathematical formulation of oblique lines relating to a particular understanding of optics and to a linear perception of time and space? With or without rules or principles, the diagonal line becomes for each of us a gradient of location in pictorial depth. Often it serves two purposes in the same drawing, design, or painting; not infrequently brisk, even violent, motion and depth are rendered simultaneously by the same diagonal thrust. Works as different in style and character as Paolo Uccello's famous The Flood, c. 1445-1447, or his panel Battle of San Romano, c. 1445, or Tintoretto's Crucifixion of 1565, or his Last Supper of 1592-94, or indeed one of Toulouse-Lautrec's circus or cabaret scenes of the 1880s-1890s, employing radical foreshortening, uptilted floor planes, and oblique projection, as seen in Japanese prints, provide examples of the use of diagonals to express conflict of physical or psychological forces or swift movements in and out of deep space.

The diagonal is at variance with the pull of gravity, as well as with the sense of equilibrium in the spectator and the parallel sides of the usual pictorial format. It signifies things in the throes of change, acting or being acted upon; and, as change must occur in time and space, both time and space, depending on how they are conceived (intellectually, spiritually, psychologically, culturally), find expression in works of art and architecture.

Kenneth Clark, in his excellent book The Nude, cites the relief sculptures from the frieze of Mausoleum (Mausolus of Caria in Asia Minor, c. 350 B.C.) as "an admirable example of heroic energy, and in particular, of the means through which it was so long expressed, that pose, or rhythmic accent, which I may call the "heroic diagonal." Clark traces the use of the heroic diagonal in the paintings of Antonio del Pollaiuolo (1433-1498) and the works of Michelangelo (1475-1564).

It is interesting to compare this development with that of Frank Lloyd Wright, starting sometime in the 1930s, when he based the design of certain houses on the 120Á angle of the hexagon, instead of the more conventional 90Á of the rectangle. The result was a much easier flow of space, a destruction of corners, facilitating both physical and psychological movement throughout the building. Then, during the early 1940s, Wright's ideas about "plasticity and continuity of space and structure" led him toward the circle. Finally, as though following the logic of organic and esthetic geometry, Wright favored more and more the principle of the spiral--for example, in his Guggenheim Museum of New York, 1957-1959.

Cubism, because it placed so much importance on the straight line and articulation of movements in space by means of directional planes, led directly to a kind of painting in which diagonal action would be of unique importance. Movement and countermovement would return to art in the new idiom--in the prismatic, spiraling action of Robert Delaunay's paintings of the Eiffel Tower, 1910-1912, in the cinematic movement of Marcel Duchamps's painting Nude Descending a Staircase, No. 2, 1912, in Futurist Paintings, or in the sensitively balanced weights, the multiple perspectives and cadences of Paul Klee's drawings and paintings.

Other developments that owed something to Cubism were Suprematism and Constructivism, 1910-1930, in Russia. Their interests centered forcibly upon a utopian purity of line, color, and plane. These were meant to apply immediately to a revolutionary age of mass communication, machine production, unprecedented efficiency, of "godlike plentitude" and self-sufficiency.

The camera, Japanese prints (from the 1850s), popular entertainment, the yearly triumphs of the Industrial Age, the rise of the modern city--Paris itself!--had more than a little to do with the ^use of diagonals in modern art and design. Degas and Toulouse-Lautrec discovered in each of these a new way of seeing things. Japanese prints, from the moment they were "discovered" by artists in the early 1860s, had suggested a more casual way of locating things, people, and events in shallow space. Photography, invented around 1839, was advancing to the point of being able to capture bodies and movements in even more unconventional perspectives. The spectacles of the music halls, cabarets, fairs, and circuses of the 1880s and 1890s, where much of the high spirits of the age were concentrated, would provide wholly new subjects and situations. Views from high, low, "candid" angles of vision, resulted in paintings full of unfamiliar, stolen "takes" on people and events.

Diagonals and curves are deceptive. They often convey the very essence of randomness and disorder, as in the case of the scattered matches, or of the prodigal disarray of nature following a storm or a flood, or of the great junk heaps of the industrial age. They are associated as often with constructive energies as with destructive energies; but, because they are associated in some way or another with energy, they provoke us to look beyond the accidental to underlying patterns.

The familiar checking patterns of concrete, ceramic glazes, paint, or of a field of mud that has dried under a hot sun would qualify in the minds of many persons as the most accidental or capricious patterns in nature. Yet, these patterns are all remarkably alike and conform closely to what is known as the orthogonal net (involving right angles or perpendiculars). Most lines (Cracks), despite their curvature and directional orientation, form the simplest of all nets. The cracking is sequential. When a fissure occurs, it joins an existing fissure by creating the three-rayed intersection.

We encountered the notion of kinetic energy earlier in this chapter in connection with the line, the broken line that is the result of a trail of points, lines that "go for a walk", and rhythmical patterns We understand what Paul Klee meant when he said:

"All figuration is movement, because it begins and ends somewhere."

The problem is to gain some knowledge of the energies--potential and kinetic--that "create and articulate forms in nature," in order that they may "serve as a basis for the creation of free and composite forms." Knowledge, not only of these energies as separate types, but of their synthesis in various forms, patterns, and organisms in nature, must accompany individual developments. Figuration must represent something definable in the life history of the object and its function.

We could hardly do better than study Chinese landscape paintings, starting with the great Li Ch'eng or Fan K'uan of the tenth or early eleventh century, masters of the early Sung Dynasty, for examples of forms that were conceived largely in terms of energy or vital force. Cézanne's landscapes would yield similar evidence, and Leonardo's nature studies would convince us of the need to adjust our looking and thinking beyond even the cultural framework of his time--he was certainly ahead of his time in most things. Kenneth Clark makes these comments on Leonardo and his preoccupations:

"I remember that the Virgin with St. Anne was painted at a period in his life when his mind was absorbed by three scientific studies, anatomy, geology and the movement of water. The movement of water symbolized for him the relentless continuum of natural force; anatomy the complexity of life and its power of renewal; and from his geological studies he had formed the concept that the whole world was breathing and renewing itself like a living organism. In one of his manuscripts he says: "The earth has a spirit of growth. Its flesh is the soil, its bones the stratifications of the rock which forms the mountains, its blood the springs of water; and the increase and decrease of blood in the pulses is represented in the earth by the ebb an flow of the sea."

Everything in nature, even the solid-seeming earth, was in a state of flux." [Kenneth Clark, Looking at Pictures (NY: Holt, Rinehart & Winston, Inc. 1960), p. 164]

We no longer find it difficult to accept the idea that motion is the norm. But, from our limited human perspective, some things move, change more rapidly than others. Some things seem hardly to change at all. Therefore we characterize some forms as static--fixed or stationary--and others as ^dynamic. The former often consists of verticals (upright figures withstanding the pull of gravity) and horizontals (fallen figures overcome by gravity) based on our visual and physical experience of gravitation from early childhood. The latter consist mainly of diagonals and curves.

CURVED LINES
We look to curves of several types for secrets of natural or organic form-development. Since only the perfect circle, or absolute centric symmetry (which does not exist in nature), expresses equal tensions on all sides and in all directions, we recognize from the start that unequal tensions are the true order of nature, and that all of natureÍs forms are the result of dynamic--static synthesis.

Stress and strain, tension and compression, follow curves. Growth follows a curve, liquids follow lines of flow that are curved, objects hurtling through space follow curves (according to Einstein, space itself is curved), erosion creates curved forms and surfaces. Therefore, whether in the living event, the happening, or in the forms themselves, we recognize curves as the expression of the dynamic principle. We experience this in our own bodies as we gain more and more motor control in work and play.

As Arnheim says: "The lever construction of the human body favors curved motion. The arm pivots around the shoulder joint, and subtler rotation is provided by the elbow, the wrist, the fingers." He points out that animals reveal mastery of circular movements in the paths they make through woods and fields. A colony of pigeons will practice the most beautiful banking or three-dimensional slalom movements, aerial choreography. The curve is associated with joy, with consummate skill.

The purest , most universal form of motion is the spiral: the counterclockwise spiral, the levogyre, is the one found most often in nature. We discover it in the growth of trees and in other members of the vegetable kingdom, in the majestic sweep of the great spiral galaxies, in the long bones of animals, and in seashells. There are also many instances of double spirals--the two intersecting curves in the seed heads of ripe sunflowers, in the centers of daisies., the seed cones of fir trees. Like curves, spirals are not all the same. The equiangular or logarithmic spiral of the elegant chambered nautilus is one type, seen also in ram's horns. It is interesting to note that the double spiral of the sunflower corresponds to the ratio of the Fibonacci Series. If you count the number of seeds in a clockwise spiral and in an intersecting counterclockwise spiral, the two figures will be that of a sequence in Fibonacci's magic chain.

Wright, in his later work, emulates the spiral of a seashell. Buckminster Fuller, the architect/engineer-mathematician-utopian, used thousands of equilateral triangles conjoined as icosahedrons to form a geodesic dome of great economy of means and strength. The geodesic lines take the place of straight lines of plane geometry to form the shortest paths across the dome's surface. Le Corbusier moved deliberately toward the use of large curved forms in his later works; he said that the inspiration for the roof of his chapel Notre-Dame-du-Haut, Ronchamp, France, came from a crab shell that he had found on a beach. His experience as sculptor and painter must have contributed to his sensitivity to curved relationships, to color, light, and decoration in the chapel. These men show a deepening interest in the geometry of energy, the geometry of growth, worthy of comparison with that of Leonardo. The language and imagery of biology were used with greater frequency in talks and writings by Wright and Le Corbusier--their increasing references to "Nature" and the "organic."

Algebraic geometry and the geometry of growth appear to have some things in common. We can identify curves in living forms by those that pertain to the realm of pure mathematics--the parabola, the hyperbola, the ellipse, the spiral, and so forth. But we have to take into account another important factor: the role of nature as sculptor. The curve of the seashell or crab shell--of each seashell and crab shell--is the result of the give and take between biological geometry (replication and the structure of heredity via the DNA helix) and the environment; That is, forces working from within against forces, pressures from without; one kind of physical substance or system against another. It would be difficult to think of living form, except submicroscopic forms (viruses and the like), perhaps, that would serve as a model of mathematical perfection throughout. The surface configuration of most forms would reveal curves of far greater variance, having been sculpted by erosion, by action of wind and water by a process of subtraction in the case of earth formations. Or an erosion pattern would have been built in by nature over perhaps millions of years, as in the forms of most aquatic creatures--fish, shellfish, seals, squids, and the like, their "pre-eroded" surfaces having been designed by nature so as to offer the least resistance to the force and density of water.

These curves--the serpentine lines of flow, the analytic curves or conic sections (hyperbola, parabola, ellipse), the logarithmic spiral, catenary curves, flat curves and banking curves--are the repertory of curves with which the artist-designer works, consciously or not. We discover by analysis that lines and form profiles of particular beauty and strength are those that reveal considerable contrast in the kinds of curves that follow one another, that flow in and out of one another. They consist of deep or slow curves (ellipses or parabolas) and shallow or fast curves (hyperbolas), long and short curves. They flow into one another with an inevitability and naturalness usually absent from the pure curves of mathematics, especially that of the circle.

Diagonals and curves, expressed either two-dimensionally as lines or three-dimensionally as flat and curved planes--that is, in drawings or in pieces of sculpture--are the artist's form-equivalents of kinetic energy. Graham Collier writes of overt or kinetic energy in terms of thrust and identifies three common manifestations:

(1) Point Thrust, seen in the arrow, the column or shaft, the steeple;

(2) Centripetal Thrust, [helical] seen in the clock spring, the spiral seashell, in all spiral forms, natural or man-made, where energy uncoils from a center; and

(3) Pressure or Pneumatic thrust, seen in a balloon, in ripe fruit and vegetables, eroded earth forms, sea creatures, the back of the human skull, the head of the femur, the egg, and all forms that are the result of tensions fairly evenly distributed.

(4) Radial Thrust. I believe he could have included a fourth type of thrust, seen in the wheel, in certain seed heads, in explosions, the Universal Mode System, and the like: radial thrust.

These kinds of thrust are often seen in combination with elements of potential energy, as in a tent, a clothesline, or a suspension bridge, where the point thrust of pole or nylon is complemented by the catenary arc of the canvas, the line or the cables, or contrariwise in the association of continuous and discontinuous patterns in the formations of mountains.

JACKSTRAW LINES
Proceeding now to what would seem the lowest order of plastic figuration, bordering on chaos, we come to a kind of linear agitation often called jackstraw lines. Even though they may appear to be the very denial of formal sense, they do have a place in the visual language. They correspond to things seen and are conspicuously expressive. Also, they differ greatly in quality and function. Some may suggest a network of form and space, a microcosm or a macrocosm; some may resemble a dance of energized particles, like a swarm of microbes, while others may fluctuate on the surface as a kind of allover texture.