Fritjof Capra (29 page)

Leonardo’s second method of squaring the circle is much more pragmatic. Again, he divides the circle into many small sectors, but then—perhaps encouraged by his intuitive grasp of the limiting process in the first method—he simply rolls half of the circumference on a line and constructs the rectangle accordingly, its short side being equal to the radius. Thus he arrives again at the correct formula, which he properly attributes to Archimedes.
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Leonardo’s second method, which greatly appealed to his practical mind, involves what we now call the mapping of a curve onto a straight line. He compared it to measuring distances with a rolling wheel, and he also extended the process to two dimensions, mapping various curved surfaces onto planes.
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On several folios of Manuscript G, he described procedures for rolling cylinders, cones, and spheres on plane surfaces to find their surface areas. He realized that cylinders and cones can be mapped onto a plane, line by line, without any distortion, while this is not possible for spheres. But he experimented with several methods of approximately mapping a sphere onto a plane, which corresponds to the cartographer’s problem of finding accurate plane maps of the surface of the Earth.

One of Leonardo’s methods involved drawing parallel circles on a portion of the sphere, thereby marking off a series of small strips, and then rolling the strips one by one, so that an approximate triangle is generated on the plane. The strips were probably freshly painted so that they left an imprint on the paper. As Macagno points out, this technique foreshadows the development of integral calculus, which began in the seventeenth century with various attempts to calculate the lengths of curves, areas of circles, and volumes of spheres.
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Indeed, some of these efforts involved dividing curved surfaces into small segments by drawing a series of parallel lines, as Leonardo had done two centuries earlier.
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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.
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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)
.
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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.”
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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.
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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.
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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.
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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.
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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.
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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.
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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,
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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.
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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.
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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.
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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 use 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.
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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.
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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.