Iterated point-line configurations in projective planes.

摘要

In this Ph.D. dissertation we analyze the behavior of iterated point-line configurations defined by the following process: begin with a set of four points in the real plane in general position. Add to this collection the intersections of all lines through pairs of these points. Iterate. Ismailescu and Radoicic (2004) showed that the limiting set is dense in the plane. Let IRP denote the Ismailescu/Radoicic plane found in this manner. We show that the number of points in IRP grows doubly exponentially. Next, we analyze the relationship between Pappus's Theorem and Desargues's Theorem in projective geometry and give a result that characterizes IRP in terms of Pappus. Through an analysis of cross products, we provide a simple convention to represent this iterative process. We also discuss the growth rate of this construction in the free projective plane. We prove that the lower bound is asymptotic to the trivial upper bound, which pins down this growth rate. Finally, we address some unanswered questions and conclude with a result in additive number theory that was proved along the way.