Repertory of Curves

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.