CURVILINEAR TRANSFORMATIONS
In today’s mathematical language, the concept of mapping can be applied also to Leonardo’s transformation of a circle into an ellipse, in which the points of one curve are mapped onto those of another together with the mapping of all other corresponding points from the square onto the parallelogram. Alternatively, the operation may be viewed as a continuous transformation—a gradual movement, or “flow,” of one figure into the other—which was how Leonardo understood his “geometry done with motion.” He used this approach in a variety of ways to turn rectilinear into curvilinear figures in such a manner that their areas or volumes are always conserved. These procedures are illustrated and discussed systematically in Codex Madrid II, but there are countless related drawings scattered throughout the Notebooks.41
Leonardo used these curvilinear transformations to experiment with an endless variety of shapes, turning rectilinear planar figures and solid bodies—cones, pyramids, cylinders, etc.—into “equal” curvilinear ones. On an interesting folio in Codex Madrid II, he illustrates his basic techniques by sketching several different transformations on a single page (see Fig. 7-6). In the last paragraph of the text on this folio, he explains that these are examples of “geometry which is demonstrated with motion” (geometria che si prova col moto).42
As Macagno and others have noted, some of these sketches are highly reminiscent of the swirling shapes of substances in rotating liquids (e.g., chocolate syrup in stirred milk), which Leonardo studied extensively. This strongly suggests once again that his ultimate aim was to use his geometry for the analysis of transformations of actual physical forms, in particular in eddies and other turbulent flows.
In these endeavors, Leonardo was greatly helped by his exceptional ability to visualize geometrical forms as physical objects, mold them like clay sculptures in his imagination, and sketch them quickly and accurately. “However abstract the geometrical problem,” writes Martin Kemp, “his sense of its relationship to actual or potential forms in the physical universe was never far away. This accounts for his almost irresistible desire to shade geometric diagrams as if they portrayed existing objects.”43
EARLY FORMS OF TOPOLOGY
When we look at Leonardo’s geometry from the point of view of present-day mathematics, and in particular from the perspective of complexity theory, we can see that he developed the beginnings of the branch of mathematics now known as topology. Like Leonardo’s geometry, topology is a geometry of continuous transformations, or mappings, in which certain properties of geometric figures are preserved. For example, a sphere can be transformed into a cube or a cylinder, all of which have similar continuous surfaces. A doughnut (torus), by contrast, is topologically different because of the hole in its center. The torus can be transformed, for example, into a coffee cup where the hole now appears in the handle. In the words of historian of mathematics Morris Kline:
Figure 7-6: Leonardo’s catalog of transformations, Codex Madrid II, folio 107r
Topology is concerned with those properties of geometric figures that remain invariant when the figures are bent, stretched, shrunk, or deformed in any way that does not create new points or fuse existing points. The transformation presupposes, in other words, that there is a one-to-one correspondence between the points of the original figure and the points of the transformed figure, and that the transformation carries nearby points into nearby points. This latter property is called continuity.44
Leonardo’s geometric transformations of planar figures and solid bodies are clearly examples of topological transformations. Modern topologists call the figures related by such transformations, in which very general geometric properties are preserved, topologically equivalent. These properties do not include area and volume, as topological transformations may arbitrarily stretch, expand, or shrink geometric figures. In contrast, Leonardo concentrated on operations that conserve area or volume, and he called the transformed figures “equal” to the original ones. Even though these represent only a small subset of topological transformations, they exhibit many of the characteristic features of topology in general.
Historians usually give credit for the first topological explorations to the philosopher and mathematician Leibniz who, in the late seventeenth century, tried to identify basic properties of geometric figures in a study he called geometria situ (geometry of place). But topological relationships were not treated systematically until the turn of the nineteenth to the twentieth century, when Henri Poincaré, the leading mathematician of the time, published a series of comprehensive papers on the subject.45 Poincaré is therefore regarded as the founder of topology. The transformations of Leonardo’s “geometry done with motion” are early forms of this important field of mathematics—three hundred years before Leibniz and five hundred years before Poincaré.
One subject that fascinated Leonardo from his early years in Milan was the design of tangled labyrinths of knots. Today this is a special branch of topology. To a mathematician, a knot is a tangled closed loop or path, similar to a knotted rope with its two free ends spliced together, precisely the structures Leonardo studied and drew. In designing such interlaced motifs, he followed a decorative tradition of his time.46 But he far surpassed his contemporaries in this genre, treating his knot designs as objects of theoretical study and drawing a vast quantity of extremely complex interlaced structures.47
Topological thinking—thinking in terms of connectivity, spatial relationships, and continuous transformations—was almost second nature to Leonardo. Many of his architectural studies, especially his designs of radially symmetrical churches and temples, exhibit such characteristics.48 So, too, do many of his numerous diagrams. Leonardo’s topological techniques can also be found in his geographical maps. In the famous map of the Chiana valley (Fig. 7-7), now in the Windsor Collection, he uses a topological approach to distort the scale while providing an accurate picture of the connectivity of the terrain and its intricate waterways.
The central part is enlarged and shows accurate proportions, while the surrounding parts are severely distorted in order to fit the entire system of watercourses into the given format.49
DE LUDO GEOMETRICO
During the last twelve years of his life, Leonardo spent a great deal of time mapping and exploring the transformations of his “geometry done with motion.” Several times he wrote of his intention to present the results of these studies in one or more treatises. During the years he spent in Rome, and while he was summing up his knowledge of complex turbulent flows in his famous deluge drawings,50 Leonardo produced a magnificent compendium of topological transformations, titled De ludo geometrico (On the Game of Geometry), on a large double folio in the Codex Atlanticus.51 He drew 176 diagrams displaying a bewildering variety of geometric forms, built from intersecting circles, triangles, and squares—row after row of crescents, rosettes and other floral patterns, paired leaves, pinwheels, and curvilinear stars. Previously this endless interplay of geometric motifs was often interpreted as the playful doodling of an aging artist—“a mere intellectual pastime,” in the words of Kenneth Clark.52 Such assessments were made because art historians were generally not aware of the mathematical significance of Leonardo’s geometry of transformations. Close examination of the double folio shows that its geometric forms, regardless of how complex and fanciful, are all based upon strict topological principles.53
Figure 7-7: Map of the Chiana valley, 1504, Windsor Collection, Drawings and Miscellaneous Papers, Vol. IV, folio 439v
When he created his double folio of topological equations, Leonardo was over sixty. He continued to explore the geometry of transformations during the last years of his life. But he must have realized that he was still very far from developing it to a point where it could be used to analyze the actual transformations of fluids and other physical forms. Today we know that for such a task, much more sophisticated mathematical tools are needed than those Leonardo had at his disposal. In modern fluid dynamics, for example, we us
e vector and tensor analysis, rather than geometry, to describe the movements of fluids under the influence of gravity and various shear stresses. However, Leonardo’s fundamental principle of the conservation of mass, known to physicists today as the continuity equation, is an essential part of the equations describing the motions of water and air. As far as the ever-changing forms of fluids are concerned, it is clear that Leonardo’s mathematical intuition was on the right track.
THE NECESSITY OF NATURE’S FORMS
Like Galileo, Newton, and subsequent generations of scientists, Leonardo worked from the basic premise that the physical universe is fundamentally ordered and that its causal relationships can be comprehended by the rational mind and expressed mathematically.54 He used the term “necessity” to express the stringent nature of those ordered causal relationships. “Necessity is the theme and inventor of nature, the curb and the rule,” he wrote around 1493, shortly after he began his first studies of mathematics.55
Since Leonardo’s science was a science of qualities, of organic forms and their movements and transformations, the mathematical “necessity” he saw in nature was not one expressed in quantities and numerical relationships, but one of geometric shapes continually transforming themselves according to rigorous laws and principles. “Mathematical” for Leonardo referred above all to the logic, rigor, and coherence according to which nature has shaped, and is continually reshaping, her organic forms.
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
Leonardo’s scientific method was based not only on the careful and systematic observation of nature—his much-exalted sperienza1—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 ey
e, 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
The Science of Leonardo: Inside the Mind of the Great Genius of the Renaissance Page 22