Projective Geometry Leads to the Unification of All Geometries

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Projective Geometry Leads to the Unification of All Geometries

Overview

The nineteenth century witnessed a great change in the nature of geometry. From beginnings in perspective drawing of artists in the eighteenth century, mathematicians developed projective geometry, and with the work of Jean-Victor Poncelet geometry became the study of properties of figures that remain unchanged under families of transformations. These transformations now became the objects of study, and different geometries would eventually be seen as parts of one unified whole.

Background

Since ancient times, geometry has always been at the heart of mathematics; mathematicians in ancient Greece were referred to as "geometers." During the Renaissance, when Europe had recovered from the Black Death (bubonic plague), dynastic rule became the standard form of government, and economies were again growing, there was a renewed interest in the classical works of the ancient Greeks. Thus, geometry returned to take its place in the curriculum of the newly established European universities. Classical works in geometry from Euclid, Archimedes, Apollonius, and Pappus were returned to Europe via Byzantium and the Arab states, and these works were restored, translated, and studied. This set the stage for the later development of new branches of geometry.

Also contributing to the rebirth of geometry was royal patronage of mathematics and mathematicians. Of utmost importance to sovereigns were map making, shipping, trade, and war. It was imperative, then, for these rulers to lend royal support to the study of cartography, hydraulics and hydrostatics, astronomy, and ballistics. Royal scientific societies were chartered, and mathematicians had the means to devote themselves to the development of the discipline. The arts also enjoyed royal patronage, and artists now had the means to work.

The growth of commerce would lead to the emergence of practical mathematics outside the university curriculum, such as reckoning with the Hindu-Arabic place value system and algebra. Geometry itself would be put to practical use as well. Our story begins with the use of geometry in art.

Perhaps for the first time, Renaissance painters began to try to give the appearance of depth on a flat canvas. What size should a given object be in an attempt to make the picture seem more realistic? In mathematical terms, how can the artist project a three-dimensional object on a two-dimensional canvas? The answers came from the geometry of perspective. We have all seen pictures of a road vanishing in the distance or perhaps of a body falling from a great height. Objects further away must be drawn smaller. The Italian Filippo Brunelleschi (1377-1446) studied the theory of perspective, and later Leon Battista Alberti (1404-1472) wrote his text on the subject, Della Pittura ("On Painting"; 1435). In it he describes the ideas of vanishing point and vanishing or horizon line. All groups of parallel lines intersect at a point on the horizon line. The set of parallel lines that are meant to be seen as perpendicular to the plane of the picture intersect at the vanishing point. The notions of "ideal" points where parallel lines "intersect" and project a three-dimensional object onto a twodimensional canvas would be exploited further in the nineteenth-century works of Jean-Victor Poncelet (1788-1867) and others.

Girard Desargues (1591-1661) was a French engineer with a professional interest in perspective as used by artists and architects. Desargues studied the works of the classical Greeks and wanted to include the theory of perspective into the larger framework of geometry. His innovation was to study the properties of figures that remain invariant (constant or unchanging) under projection. Since the property of being a conic section is such an invariant, Desargues could attempt to unify the study of conics. He introduced the notion of points at infinity where parallel lines meet. His new terminology included "ordinance" for a collection of parallel lines or of a collection of lines intersecting at the same point (what today is called a pencil of lines) and "butt" for the point of intersection of the lines in the ordinance. Desargues had the unfortunate luck to be working in synthetic geometry (geometry of figures without recourse to formulas) just as analytic geometry was being born. Analytic geometry flourished, while Desargues' work was largely ignored. It was forgotten until being rediscovered by French geometer Michel Chasles (1793-1880) in 1845.

The French Revolution (1789-1799) helped make possible the further development of geometry. As mentioned earlier, royal patronage and the establishment of universities helped revive the study of classical geometry. In France military schools were also established to provide a scientific education for military engineers. Naturally, these schools supported the monarchy during the Revolution, and they were subsequently closed. The revolutionary government deemed it necessary to create its own schools, and it did so, creating both an engineering school (the École Polytechnique in 1794) and a school for teachers (the École Normale Superieure in 1795). In keeping with the romanticism of the time, these schools led to the expanding of literacy and opportunity for many, and this in turn led to the successful mathematical careers of many men who may not have had the same opportunity under the old regime. This period is also marked by the rejection of the Age of Enlightenment's emphasis on reason and the coming of the Age of Romanticism, which favored the use of intuition and imagination, two qualities necessary to challenge the dominance of Euclidean geometry.

Gaspard Monge (1746-1818) was instrumental in organizing these new schools, which had as their instructors such prominent mathematicians as Joseph Louis Lagrange (1736-1813), Pierre Simon Laplace (1749-1827), and Monge himself. When Napoleon took over France, many mathematicians, including Monge and Poncelet, accompanied him on his excursions. As we shall see, Poncelet's trip with Napoleon was to be particularly eventful.

Impact

In 1822 Poncelet published Traité des propriétés projectives des figures ("Treatise on the Projective Properties of Figures"). This work set the stage for a reformulation and unification of geometry that was to culminate in the work of Felix Klein (1849-1925) in 1872. Poncelet was in Napoleon's army as an engineer and accompanied him during the Russian campaign, when he was taken prisoner. During his imprisonment Poncelet began reworking geometry for himself (having no books in prison), and the result is his Traité. One of the ideas discussed in this work is the theory of polars in conic sections. However, the most important study that Poncelet undertook in Traité was the study of central projections. Two figures in two distinct planes are related by a central projection if corresponding points in the two figures can be joined by concurrent lines all passing through a fixed point (much like the vanishing point in perspective drawing or the eye when looking at an object). While this sounds complicated, if you have seen a light cast a shadow, you have seen a central projection. You can make a shadowbox to see the idea. Hang a light in a thin, transparent "box" and draw a picture on the side of the box. The figure on the box and the shadow on the table are related by a central projection.

Poncelet's work caused many mathematicians to take a look at projective geometry. The analytic geometry of Descartes and Fermat had led the way towards calculus, but now analytic geometry would return to its roots as mathematicians employed analytic methods in the study of projective geometry. The non-Euclidean geometry of Nikolai Lobachevsky (1792-1856) was published in the early nineteenth century (1829), and it too would eventually find its way back to converge with Poncelet's study of invariants.

In Traité Poncelet arrived at a notion of duality using poles and polars. Joseph-Diaz Gergonne (1771-1859) is in fact responsible for the term "polar," and the principle of duality was first explicitly stated (without need of reference to polars) by Gergonne in 1826, four years after Poncelet's groundbreaking work. In the principle of duality a true proposition about "points" and "lines" remains true of the words "points" and "lines" are interchanged. For example the statement "Not all points are on the same line" can be paired with a dual statement that "Not all lines contain the same point." This notion of duality is a valuable tool in projective geometry for discovering new properties. Gergonne would go on to use duality in investigating algebraic curves. He studied the relationship between points lying on such a curve and the dual notion of tangents to such a curve. Although he made some missteps (not allowing for vertical tangents, for example), Gergonne did further the use of duality and projective techniques beyond elementary geometry and conic sections.

A final note on the impact of the French work on projective geometry was the founding of the journal Annales de Mathématiques by Gergonne in 1810 to publish the works of former students of the École Polytechnique. One cannot underestimate the importance of the emergence of mathematical journals in contributing to the growth of mathematics in this century.

Perhaps the last of the great French geometers of this period was Michel Chasles. Recall that it was Chasles who in 1845 found a copy of Desargues' work and rescued it from obscurity. Chasles was also the co-discoverer (with August Möbius) of the most important projective invariant, the cross ratio of four points on a line. With the notable exception of the work of Jacob Steiner (1796-1863), analytic methods would come to dominate future work in projective geometry. Chasles himself was forced to admit that leadership in projective geometry had passed to Germany by 1837, with the majority of German geometers favoring tools from analytic geometry. Not coincidentally, the passing of the torch to Germany coincides with the fall of Napoleon (and his supporters), the restoration of the monarchy and the conservative order in France, and the growth in power of the German states.

An axiomatic system for projective geometry was established in 1847 by Karl Georg Christian von Staudt (1798-1867) in his work Geometrie der lage ("Geometry of Position"). This axiomatic system did not make use of distance; instead, von Staudt was able to create within the axiomatic system a notion of coordinates intrinsic to geometry, as opposed to the extrinsic notion of coordinates based on distance. This allowed the notions of addition and multiplication to be defined via geometrical constructions; also, this intrinsic introduction of coordinates would eventually lead to generalized projective geometry using abstract algebraic systems as coordinates instead of the real numbers. A simple example of such an abstract algebraic system would be the set {0, 1, 2, 3, 4} with arithmetic module 5 (for example 3 + 4 = 2 and 2 × 3 = 1). Using this system as coordinates leads to finite geometries, combinatorics, and the theory of codes. Von Staudt's axiomatization of projective geometry also paved the way for new axiomatizations of Euclidean and non-Euclidean geometries.

Side by side with the synthetic projective geometry of Steiner and von Staudt, analytic projective geometry also flourished in the nineteenth century, with Augustus Ferdinand Möbius (1790-1868) and Julius Plücker (1801-1868) as its main adherents. These men independently invented what are now known as homogeneous coordinates for the projective plane, that is, the Euclidean plane augmented by points at infinity. The use of homogeneous coordinates helped to crystallize the relationship between geometry and algebra. Transformations between geometric objects can now be seen as substitutions. From these very types of transformations the theory of matrices was developed by James Joseph Sylvester (1814-1897) and Arthur Cayley (1821-1895).

With all the pieces falling into place, Klein could unify and classify the geometrical studies of the nineteenth century. He did so in his famous Erlanger Programm of 1872. As was the custom in German universities, a professor would give an inaugural lecture upon his appointment. Klein's appointment to the University of Erlangen was the occasion for his program for the unification of geometry. In this lecture Klein defined what a "geometry" meant, noting essentially that new geometries could be defined by starting with different groups of transformations; thus, he showed that projective geometry does indeed serve as the all encompassing geometry.

The trip from Poncelet's question about projective invariants to a new definition of geometry took most of the nineteenth century. Our fellow travelers' influence continues into the twentieth century as well. Higher dimensional geometry grew by taking the idea of coordinates beyond three dimensions. This higher dimensional geometry, combined with the study of electricity and magnetism, led to the development of vector analysis. Loosening the notion of transformation even further to include stretching, shrinking, and bending (but not tearing or puncturing) leads to topology, so-called "rubber sheet geometry." If one considers looking beyond the family of conic sections (which are invariant under projections) to other higher degree curves, one enters the realm of algebraic geometry, which developed around the attempt to classify curves of a certain degree and the search for invariants under transformations. Projective methods are very useful in algebraic geometry, where curves are simplified when viewed projectively. Also, the behavior of a curve at infinity can be studied in the same way as the behavior of the curve at an ordinary point. Plücker and Cayley were very much involved in this study. One should also note that algebraic geometry was an important tool in Andrew Wiles's (b. 1953) proof of Fermat's last theorem in 1995.

Euclidean geometry had been considered by philosophers such as Descartes, Hume, Spinoza, and Kant to be the paradigm of logical thought and certainty. With the discovery of non-Euclidean geometries and the reduction of Euclidean geometry to a sub-discipline of other wide-ranging mathematical theories, this paradigm was shattered. Anyone who looked for absolute right or wrong was forced to look elsewhere. This had a profound impact on philosophy and what people think about human knowledge. If the geometric nature of the universe is open to experiment, there are no necessary truths about space and time.

GARY S. STOUDT