The word Geometry is of the Greek origin; it derives from the Greek geo (earth) and metron (measure). Originally, the subject of Geometry was earth measurement. With time, however, both the subject and the method of geometry have changed. From the time of Euclid's Elements (3rd century B.C.), Geometry was considered as the epitome of the axiomatic method which itself underwent a fundamental revolution in the 19th century. Revolutionary in many other aspects, the 19th century also witnessed metamorphosis of a single science - Geometry - into several related disciplines.
The subject of Projective Geometry, for one, is the incidence of geometric objects: points, lines, planes. Incidence (a point on aline, a line through a point) is preserved by projective transformations, but measurements are not. Thus in Projective Geometry, the notion of measurement is completely avoided, which makes the term - Projective Geometry - an oxymoron.
In Projective Geometry, conic sections (circles, ellipses, parabolas, hyperbolas) are indistinguishable. Perhaps, for this reason alone, some would forego the study of Analytic Geometry in favor of its Projective cousin. However, even without measurements, Projective Geometry is neither simple nor lacking in content. The applet below illustrates one of the most surprising geometric results probably discovered by Pappus of Alexandria (3rd century A.D.) who is considered to be the last of the great Greek geometers.
Let three points A, B, C be incident to a single straight line and another three points a,b,c incident to (generally speaking) another straight line. Then three pairwise intersections
(A point and a line are said to be incident if the line passes through the point, or, equivalently, if the point lies on the line.)
The configuration of 9 points and 9 lines is quite remarkable. Firstly, the triples of the points
The second remarkable feature of the configuration is that it is self-dual. Duality is germane to Projective Geometry. Two statements that only deal with incidence of points and lines are called dual if one is obtained from the other by simply swapping the words point and line. For example, the dual of Pappus' theorem reads
Let three lines A, B, C be incident to a single point and another three lines a,b,c incident to (generally speaking) another point. Then three pairwise intersections
(By convention, intersection of two (distinct) points is the straight line that passes through these points.)
The Duality Principle states that if one of the two dual statements is a theorem, so is the other one. (The applet allows you to verify that this is true for Pappus' theorem and its dual.) Indeed, the configuration in the theorem is such that by naming lines in a certain way and following the prescription of the dual theorem we again get the same 9 lines and 9 points as we originally had in the direct theorem.
The Duality Principle is a handy feature of Projective Geometry: you prove one theorem and get another one for free. The principle is quite simple to prove. Usually, one lists all the axioms of Projective Geometry and verifies that their duals are either provable or are stated as other axioms. The latter case is highlighted by the following pair:
- Axiom 1: Any two distinct points are incident with exactly one line.
- Axiom 2: Any two distinct lines are incident with exactly one point.
Although simple, the duality principle was not conceived until the 18-19 centuries. Pappus' theorem has been generalized by B. Pascal (1623-1662) who proved at the age of 16 that the points A,B,C and a,b,c may be taken on a conic section instead of two straight lines which is a real generalization since a plane through the apex of a cone cuts out a conic section which is the set of two straight lines. The dual of Pascal's theorem has been proven by Charles Julien Brianchon (1783-1864) in 1810 and is known as Brianchon's theorem.
The Duality Principle, along with the emergent non-Euclidean geometries, had a major effect on mathematical thinking and formalization of mathematics. The geometric axioms may be dealing with points and lines, but since they are interchangeable due to the Duality Principle, it's hard to relate to their "physical" prototypes. It's in this sense that the famous B. Russell's grumble must be understood.
Pappus' theorem and its dual admit slightly different formulations. They are given in the "customary" geometric terms:
(Consider the hexagon AbCaBc.)
The points of intersection of the opposite sides of the hexagon whose vertices lie alternatively on two straight lines, lie on a straight line.
(consider the hexagon BC1cb3)
The diagonals of a plane hexagon whose sides pass alternatively through two fixed points, meet at a point.
The proof, as in the case of Pascal's theorem, employs the theorem of Menelaus. Presently, however, we ought to consider five transversals [Coxeter, pp. 67-70]. Below, for the convenience sake, I bring up the notations in line with those in Pascal's theorem.
Assume the configuration is as depicted above. We are to prove that the points J, K, L are colinear. We shall apply the theorem of Menelaus to ΔGHI and its five transversals: DKC, AJB, ELF, ACE, and DFB:
|DKC||HK/GK · GC/IC · ID/HD = 1|
|AJB||HA/GA · GB/IB · IJ/HJ = 1|
|ELF||HF/GF · GL/IL · IE/HE = 1|
|ACE||HE/IE · IC/GC · GA/HA = 1|
|DFB||HD/ID · IB/GB · GF/HF = 1|
Multiply the five identities and rearrange the terms:
HK/GK · GL/IL · IJ/HJ · (GC/IC · ID/HD · IE/HE · HF/GF · HA/GA · GB/IB · HE/IE · IC/GC · GA/HA · HD/ID · IB/GB · GF/HF) = 1.
Since all terms in the parentheses cancel out:
HK/GK · GL/IL · IJ/HJ = 1,
which by (the converse of) Menelaus' theorem means that the three points K, J and L lie on a transversal, I. e., they are colinear. Which supplies the proof of Pappus' theorem - almost. The problem is the proof depends on the existence of point G, the intersection of AF and BC. What if it does not exist? I.e., what if the lines AF and BC are parallel?
From the broad geometric view point the simplest reply is that the nature of Pappus' theorem is projective. The theorem only depends on the notion of incidence. If it is true for one configuration of lines and points it is bound to be true for projective images of the latter. For G lying at infinity, just find a projective mapping that makes G finite and also keeps finite all other points in the proof. Apply the proof to the new configuration. It is done.
However, some readers may wish to devise a more "elementary" proof. Still, this may not be necessary. The choice of ΔGHI, where G lies at the intersection of AF and BC is quite arbitrary. There are other pairs of lines that could serve the same purpose, for example the pairs
To furnish a completely "elementary" proof, observe that a point at infinity is located at the same distance to all the finite points. Heuristically, we may expect then that, say, GL/GF in the third equation will be 1 for G at infinity. In each of the five equations above the letter G appears exactly twice: once in a numerator and once in a denominator. Let's try to exclude those terms. The actual justification now comes not from the Menelaus theorem, but from considering pairs of similar triangles:
|HDK, IDC||HK/HD · ID/IC = 1|
|HAJ, IBJ||HA/HJ · IJ/IB = 1|
|HFE, ILE||HF/HE · IE/IL = 1|
|HEA, IEC||HE/HA · IC/IE = 1|
|HDF, IDB||HD/HF · IB/ID = 1|
After simplification, the product of the five identities becomes,
HK/HJ · IJ/IL = 1,
which says that the triangles HKJ and ILJ are similar. Therefore, the points J, K, L are collinear also in this case.
(Pappus' theorem is generalized by a theorem of Pascal. Pappus' theorem has numerous applications. In particular, it can be used to establish the Minimax Principle for two-person zero-sum games. There is a straightforward proof of the theorem in the framework of projective geometry which just paraphrases that of Pascal' theorem.)
- H.S.M. Coxeter, S.L. Greitzer, Geometry Revisited, MAA, 1967