stoutfellow

It is proverbial that "the fox knows many things, but the hedgehog knows one big thing". I must admit that, when it comes to the history of mathematics, I have a constitutional preference for foxes, but there have been a fair number of notable hedgehogs. One of my favorites of these is the unfortunate Girard Desargues. Desargues developed a new and revolutionary way of thinking about geometry, known as "projective geometry". It has come to be recognized as fundamental by modern geometers; in fact, a strong case could be made that projective geometry is the center around which most of today's geometries revolve. Unfortunately, Desargues' work went largely unregarded for nearly two centuries. Among his contemporaries, only Blaise Pascal paid attention, and himself discovered a number of interesting results using Desargues' ideas. There are, I think, two principal reasons for this disregard. One is that Desargues, like many mathematicians, had a penchant for coining new words for the concepts he discovered, but he rather overdid it, with an efflorescence of terms, largely biological in origin, all but one of which have been long since forgotten. (The exception is "involution", which, ironically, was the term one reviewer singled out for the harshest criticism.) The other is that another French mathematician, at about the same time, came up with yet another new way of doing geometry, one more immediately useful and flexible, and that method, quite justifiably, received far more attention - but that's a story for another day. Some comments on what Desargues did are under the cut.

The story actually begins in Renaissance Italy, in the studios of the great painters. The discovery of perspective and its potential for vivid representation of three-dimensional space, inspired many thinkers, most notably Leonardo da Vinci, to investigate its geometric properties. The fact that shapes and sizes must be distorted to look real, and most notably the fact that lines actually parallel should be represented as intersecting - think of railroad tracks heading to the horizon - required much thought and explanation.

Envision a photographic slide, being projected onto a screen. Under normal circumstances, the images on the slide and on the screen differ only in size, the screen images being larger by a specifiable factor. This is true, however, only because the slide and screen are parallel to one another. If the screen is canted somehow, rotated about an axis (and this may be any axis, not merely the horizontal or vertical ones!), the screen images are greatly distorted. Images projected onto nearer parts of the screen appear smaller, proportionately, than images projected farther away. Square images become rectangles, parallelograms, trapezoids, or other, nameless, quadrilaterals. Circles turn elliptical, parabolic, even hyperbolic. Yet one major thing does not change: lines continue to be lines. Here is rock-bottom geometry; whatever features are unchanged under this sort of transformation are surely the most fundamental.

(How does this relate to painting? The painter transforms a largely horizontal scene onto a vertical one; it is a sort of reverse-projection, with the painter's eye as the light source. Which of the painting or the outer world should be regarded as the slide and which as the screen is, perhaps, moot.)

Desargues' projective geometry is a careful investigation of this sort of transformation, this "perspectivity" as it is called. It requires that the Euclidean plane be regarded as supplemented by a set of "points at infinity", where parallel lines intersect. Consider, again, the train-tracks. That they seem to converge at the horizon has nothing to do with the horizon; if one were to paint a number of train-tracks, running in different directions, they would each appear to converge at a different point; those points would all lie on a line, and the plane containing that line and the viewer's eye would be horizontal. In general, given a perspectivity between two planes, there will be (finite) points on plane #1 which do not correspond to any (finite) points on plane #2; the line they form corresponds to the "line at infinity" of plane #2. Conversely, certain points on plane #2 correspond to points at infinity on plane #1. (In each case, consider the plane through the perspective point parallel to one or the other plane; where it intersects the other plane yields the anomalous line in question.)

There is more. (In this regard, Desargues was actually partially anticipated by Johannes Kepler, whose astronomical research led him to study conic sections more intensively than anyone since Apollonius of Perga.) A perspectivity does not, generally, transform circles to circles, but it does transform conic sections of all types into other conic sections. (Envision an ordinary flashlight. The beam that emerges from it seems circular, but cast upon the ground at different angles it takes on other shapes. Note that the light actually forms a cone, and the image on the ground is, precisely, a cross-section of that cone.) Indeed, any conic section may be transformed into any other by an appropriate perspectivity.

What good is all this? There are numerous theorems of Euclidean geometry which appear in pairs; the general flavor is, "Let l and m be two parallel lines; then blah-blah-blah. Let l and m be two nonparallel lines; then blee-blee-blee." Projective geometry, by conflating parallel lines with nonparallel ones, makes it possible to unify these theorems, and this revelation of an underlying unity is a thing greatly to be desired. More, if a theorem is of a projective flavor - if the situation it describes is one that is preserved by perspectivity - then one may prove it by considering a special case, which may be easier to handle. For example, if a theorem involves three lines intersecting in a point, an appropriate perspectivity can move that point off to infinity, making the lines parallel and, quite possible, simplifying the necessary computations. If a theorem involves conic sections in general, one may prove it for the special case of a circle, and then extend it by applying a perspectivity. (Pascal did this for his wonderful Mystic Hexagram theorem, for example.)

Let me end by citing the theorem for which Desargues is most famous, and which bears his name. Let ABC and A'B'C' be two triangles, and suppose that the lines AA', BB', and CC' intersect in a single point. (Recall that this includes the possibility that they are parallel.) Let A" be the intersection of BC with B'C', and define B" and C" similarly; then A", B", and C" lie on a single line. (More briefly, and more jargon-ridden: if two triangles are perspective from a point, then they are perspective from a line.) This is a beautiful and consequential theorem, and it is fundamental to the modern study of triangles - which is, as I may have mentioned, one of my primary interests. For this theorem, even if there were nothing else, Desargues has my love.

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