Fritjof Capra (30 page)

This meaning of “mathematical” is quite different from the one understood by scientists during the Scientific Revolution and the subsequent three hundred years. However, it is not unlike the understanding of some of the leading mathematicians today. The recent development of complexity theory has generated a new mathematical language in which the dynamics of complex systems—including the turbulent flows and growth patterns of plants studied by Leonardo—are no longer represented by algebraic relationships, but instead by geometric shapes, like the computer-generated strange attractors or fractals, which are analyzed in terms of topological concepts.
⁵⁶

This new mathematics, naturally, is far more abstract and sophisticated than anything Leonardo could have imagined in the fifteenth and sixteenth centuries. But it is used in the same spirit in which he developed his “geometry done with motion”—to show with mathematical rigor how complex natural phenomena are shaped and transformed by the “necessity” of physical forces. The mathematics of complexity has led to a new appreciation of geometry and to the broad realization that mathematics is much more than formulas and equations. Like Leonardo da Vinci five hundred years ago, modern mathematicians today are showing us that the understanding of patterns, relationships, and transformations is crucial to understand the living world around us, and that all questions of pattern, order, and coherence are ultimately mathematical.

EIGHT

Pyramids of Light

L
eonardo’s scientific method was based not only on the careful and systematic observation of nature—his much-exalted
sperienza
¹
—but also included a detailed and comprehensive analysis of the process of observation itself. As an artist and a scientist, his approach was predominantly visual, and he began his explorations of the “science of painting” by studying perspective: investigating how distance, light, and atmospheric conditions affect the appearance of objects. From perspective, he proceeded in two opposite directions—outward and inward, as it were. He explored the geometry of light rays, the interplay of light and shadow, and the very nature of light, and he also studied the anatomy of the eye, the physiology of vision, and the pathways of sensory impressions along the nerves to the “seat of the soul.”

To a modern intellectual, used to the exasperating fragmentation of academic disciplines, it is amazing to see how Leonardo moved swiftly from perspective and the effects of light and shade to the nature of light, the pathways of the optic nerves, and the actions of the soul. Unencumbered by the mind-body split that Descartes would introduce 150 years later, Leonardo did not separate epistemology (the theory of knowledge) from ontology (the theory of what exists in the world), nor indeed philosophy from science and art. His wide-ranging examinations of the entire process of perception led him to formulate highly original ideas about the relationship between physical reality and cognitive processes—the “actions of the soul,” in his language—which have reemerged only very recently with the development of a post-Cartesian science of cognition.
²

THE SCIENCE OF PERSPECTIVE

Leonardo’s earliest studies of perception stand at the beginning of his scientific work. “All our knowledge has its origin in the senses,” he wrote in his very first Notebook, the Codex Trivulzianus,
³
begun in 1484. During the subsequent years he embarked on his first studies of the anatomy of the eye and the optic nerves. At the same time, he explored the geometries of linear perspective and of light and shadow, and demonstrated his profound understanding of these concepts in his first master paintings, the
Adoration of the Magi
and the
Virgin of the Rocks
.
⁴

Leonardo’s interest in the mathematics underlying perspective and optics intensified in the summer of 1490, when he met the mathematician Fazio Cardano at the University of Pavia.
⁵
He had long discussions with Cardano on the subjects of linear perspective and geometrical optics, which together were known as “the science of perspective.” Soon after these discussions, Leonardo filled two Notebooks with a short treatise on perspective and with numerous diagrams of geometrical optics.
⁶
He returned to the study of optics and vision eighteen years later, around 1508, when he explored various subtleties of visual perception. At that time, Leonardo revised his earlier notes and summarized his findings on vision in the small Manuscript D, which is similar in its brevity and elegant compact structure to the Codex on the Flight of Birds, composed around the same time.

Linear perspective was established in the early fifteenth century by the architects Brunelleschi and Alberti as a mathematical technique for representing three-dimensional images on a two-dimensional plane. In his classic work
De pictura (On Painting)
,
⁷
Alberti suggested that a painting should give the impression of being a window through which the artist looks at the visible world. All objects in the picture were to be systematically reduced as they receded into the distance, and all sight lines were to converge to a single “central point” (later called the “vanishing point”), which corresponded to the fixed viewpoint of the spectator.

As architectural historian James Ackerman points out, the geometry of perspective developed by the Florentine artists was the first scientific conception of three-dimensional space:

As a method of constructing an abstract space in which any body can be related mathematically to any other body, the perspective of the artists was a preamble to modern physics and astronomy. Perhaps the influence was indirect and unconsciously transmitted, but the fact remains that artists were the first to conceive a generalized mathematical model of space and that it constituted an essential step in the evolution from medieval symbolism to the modern image of the universe.
⁸

Leonardo used Alberti’s definition of linear perspective as his starting point. “Perspective,” he states, “is nothing else than seeing a place behind a pane of glass, quite transparent, on the surface of which the objects behind that glass are to be drawn.”
⁹
A few pages later in the same Notebook, he introduces geometric reasoning with the help of the image of a “pyramid of lines,” which was common in medieval optics.
¹⁰
The first statement about perspective, too, continues with a reference to visual pyramids. “These [objects],” Leonardo explains, “can be traced through pyramids to the point of the eye, and the pyramids are intersected on the glass pane.”
¹¹

Figure 8-1: The geometry of linear perspective, Codex Atlanticus, folio 119r

To determine to what extent exactly the image of an object on the glass pane diminishes with the object’s distance from the eye, Leonardo conducted a series of experiments, in which he methodically varied the three relevant variables in all possible combinations—the height of the object, the distance from the eye, and the distance between the eye and the vertical glass pane.
¹²
He sketched the experimental arrangements in several diagrams; for example, as shown in Figure 8-1, where the object is kept stationary and the observer’s eye, together with the glass pane in front of it, is placed in two different locations. The corresponding “pyramids” (isosceles triangles) with the two different visual angles are clearly shown.

With these experiments, Leonardo established conclusively that the height of the image on the glass pane is inversely proportional to the object’s distance from the eye, if the distance between the eye and the glass pane is kept constant. “I find by experience,” he recorded in Manuscript A, “that, if the second object is as far from the first as the first is from the eye, although they are of the same size, the second will seem half the size of the first.”
¹³
In another entry he records a series of distances with the corresponding reductions of the object’s image, and then concludes: “As the space passed through doubles, the diminution doubles.”
¹⁴

These results, obtained during the late 1480s, mark Leonardo’s first explorations of arithmetic, or “pyramidal,” progressions. To establish them, he did not really have to perform all these experiments, because the inverse linear relationship between the distance of the object from the eye and the reduction of its image on the glass pane can easily be derived with elementary Euclidean geometry. But it would be almost another ten years before Leonardo would acquire those mathematical skills.
¹⁵

Figure 8-2: Section of the human skull, Anatomical Studies, folio 43r

Leonardo demonstrated his thorough understanding of linear perspective not only in his art, but also in his scientific drawings. While he was conducting his experiments on the geometry of perspective, he also investigated the anatomical connections between the eye and the brain.

He documented his findings in a series of magnificent pictures of the human skull, in which the foreshortening of visual perspective is employed to great effect (see Fig. 8-2). Leonardo combined this technique with delicate renderings of light and shade to create a vivid sense of space within the skull, in which he exhibited anatomical structures that had never been seen before and located them with complete accuracy in three dimensions.
¹⁶
He used the same mastery of visual perspective and subtle renderings of light and shade in his technical drawings (see, for example, Fig. 8-3), depicting complex machines and mechanisms with an elegance and effectiveness never seen before.
¹⁷

While he skillfully used Alberti’s rules of perspective to produce radical innovations in the art of scientific illustration, Leonardo soon realized that for his paintings, these rules were too restrictive and fraught with contradictions.
¹⁸

Alberti had suggested that the geometric horizon of a painting should be at the eye level of the painted figures so as to create the illusion of a continuity between the imaginary space and that of the spectators. However, frescoes and altarpieces were often placed quite high up, which made it impossible for the spectators to look at them from a viewpoint that would make the illusion work. Moreover, Alberti’s system assumed a fixed viewpoint in front of the vanishing point, but most spectators were likely to move around and look at the painting from different angles, which would also destroy the illusion. In
The Last Supper
, Leonardo, well aware of the internal contradictions of linear perspective, played around with Alberti’s rules to enhance the presence of the human figures and create elaborate illusions,
¹⁹
but after that he no longer painted any architectural motifs and went far beyond the linear perspective of the quattrocento.

To refine the theory of perspective, Leonardo questioned Alberti’s simplistic assumption that the lines of all visual pyramids meet in a single mathematical point within the eye. Instead, he studied the actual physiology of visual perception. “Perspective,” he noted, “is nothing else than a thorough knowledge of the function of the eye.”
²⁰
He took into account that natural vision is binocular—produced by two moving eyes rather than the single fixed eye of Alberti’s geometry. He carefully investigated the actual pathways of the sensory impressions, and he also considered the effects of atmospheric conditions on visual perception.

From his studies of the anatomy of the eye and the physiology of vision,
²¹
Leonardo derived a theory of perspective that went well beyond Alberti, Piero della Francesca, and other leading artists of the early Renaissance. “There are three kinds of perspective,” he declared. “The first is concerned with the reason for the diminution [of] things as they recede from the eye. The second contains the way in which colors vary as they recede from the eye. The third and last encompasses the declaration of how objects should appear less distinct the more distant they are.” He specified that the first, traditional kind was called “linear perspective”
(lineare)
, the second “perspective of color”
(di colore)
, and the third the “perspective of disappearance”
(di spedizione)
.
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