Projective Geometry

Girard Desargues, an architect and engineer from Lyon, turned the painters' practice into mathematics in 1639. Project a circle from a point onto a tilted plane and you get an ellipse, a parabola or a hyperbola. So all conic sections are one curve seen from different angles, and a theorem proved for the circle transfers to all of them. To make this work he treated parallel lines as meeting at a point "at an infinite distance", completing the plane.

A year later Blaise Pascal, aged sixteen, published his hexagon theorem. For any six points on a conic, the three pairs of opposite sides of the hexagon they form meet in three points on one straight line. Then the subject nearly vanished. Desargues' book was printed in a tiny run and lost for two centuries, and Descartes' coordinates, published two years earlier, carried geometry in a different direction.

The revival came from a prisoner of war. Jean-Victor Poncelet, a young French engineering officer, was captured during Napoleon's retreat from Moscow and held at Saratov on the Volga in 1813–14. Without books, he rebuilt geometry from what he remembered and pushed it further. His Traité des propriétés projectives des figures (1822) made projective properties the subject itself. He added points at infinity systematically, and even admitted "imaginary" points so that, for instance, every line meets every circle.

In 1825–27 Joseph Gergonne noticed the field's deepest symmetry. In the projective plane, swap "point" with "line" in any theorem and you get another true theorem. He printed dual theorems side by side in two columns. Poncelet insisted the idea was really his, and the two quarrelled in print for years.

By mid-century projective geometry had grown ambitious. In 1859 Arthur Cayley showed that even distance and angle, the things projection destroys, can be recovered inside projective geometry by singling out one special conic, the "absolute". He concluded that metrical geometry is part of projective geometry, which he called descriptive geometry, and that "descriptive geometry is all geometry".

Felix Klein took the next step in 1871. Choose the absolute one way and you get Euclidean geometry, another way hyperbolic, a third way elliptic. The non-Euclidean geometries, found through the long struggle over the parallel postulate, sat inside projective geometry all along. His Erlangen Program the following year made the hierarchy official, with projective geometry near the top.

Projective space also became the natural home for curves defined by equations. In the projective plane, allowing complex points, two curves of degrees $m$ and $n$ always meet in exactly $mn$ points, counted properly. That clean count fails in the ordinary plane, where intersection points can escape to infinity. This is where algebraic geometry takes up the story.

Desargues' theorem is the signature result of projective geometry, a statement about points and lines only, with no lengths or angles.

Take two triangles $ABC$ and $A'B'C'$ in perspective from a point: the lines $AA'$, $BB'$ and $CC'$ all pass through one point $O$, as if one triangle were a shadow of the other cast by a lamp at $O$. Now extend corresponding sides until they meet: $AB$ with $A'B'$, $BC$ with $B'C'$, and $CA$ with $C'A'$. The theorem says these three meeting points always lie on a single straight line. (If two sides are parallel, they meet at a point at infinity, and the statement still holds, one reason projective geometry adds those points.)

The slickest proof leaves the plane. Suppose the two triangles lie in different planes in space. Each pair of corresponding sides, like $AB$ and $A'B'$, lies in the plane through $O$, $A$ and $B$, so they meet. All three meeting points lie in the plane of triangle $ABC$ and in the plane of triangle $A'B'C'$. Two different planes meet in a line, so the three points are collinear. The flat case follows by viewing a flat drawing as the shadow of the three-dimensional one.

The converse also holds, and by duality, swapping "point" and "line", the theorem proves its own converse. Stranger still, there exist projective planes where Desargues' theorem fails. Hilbert showed that it holds exactly when the plane can be given coordinates from a number system in which multiplication is associative. A theorem about lines in a drawing is secretly a theorem about algebra.

Nothing in the axioms of a projective plane requires infinitely many points. The smallest possible one has seven points and seven lines, each line holding three points: the Fano plane. More generally, finite planes exist with $n^2 + n + 1$ points whenever $n$ is a prime power. Whether any other orders are possible is one of the oldest open questions in combinatorics. Order 10 took a supercomputer search in the 1980s to rule out, and order 12 is still unknown.