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Oriented Matroids: The Power of Unification
4. Arrangements of pseudolines
(This photograph is used with the kind permission of H. Begehr, (and assistance of Sven Guckes) and appears in H. Begehr (Ed.): Mathematik in Berlin, Geschichte und Dokumentation, Erster Halbband. Shaker Verlag, Aachen, 1998, p. 332, and was provided by Hochschularchiv Freie Universitaet Berlin, Math.-Nat. Fak., Dekanat, Photosammlung).
If one picks out a pseudoline one can try to "stretch" it so that it does not change the cell structure into which the pseudolines divide the plane, but straightens out the pseudoline until it becomes a straight line. An arrangement of pseudolines is called stretchable if all the pseudolines in the arrangement can be made into straight lines in such a way that at each stage of the transformation, the combinatorial type (as indicated by the faces that are formed) of the pseudolines is maintained. What is your intuition? Do you think that all pseudoline arrangements arestretchable into an arrangement of the same type with lines? The German mathematician Gerhart Ringel conjecturedthat a specific arrangement of pseudolines could not be transformed by "stretching" intoa collection of straight lines. Eventually he was proven right.
The most natural approach to Pappus' Theorem involves conic sections but here I consider a special case, where we have two lines which are shown parallel but could be intersecting. There are three points on each line: a, b, c on one, and A, B, C on the other. Join a to B and C; b to A and C; c to A and B. Pappus' Theorem states that the lines aB and aC, bA and bC, cA and CB intersect at points, respectively, p, q, and r which are collinear. Thus, in our picture, p, q, and r are collinear. Put somewhat differently, the points p, q, and r, which by their definition might not seem forced to be on a line, in fact, must be on a line.
Where do the pseudolines come in?
The diagram above shows how the octagonal polygon and a tiling of it gives rise to 4 pseudolines! This ingenious idea allows one to unify ideas about tilings and the enumeration of tilings with ideas about arrangements of lines and pseudolines. Mathematics can make a big leap forward when such an unexpected unification of ideas occurs.
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