Dynamic point-line configurations

Abstract

A set of points and lines is called a configuration if every line contains the same amount of points, and every point lies on the same amount of lines. A configuration with π points and λ lines with every point lying on m lines and every line contains n points is called a (πm, λn) configuration. The first question one could ask about them is if and when they exist. When m = 1 or n = 1, this is just points on lines or lines through points, so nothing interesting happens. When m = 2 or n = 2 it still concerns just intersection points of two lines or lines through two points, which is also not special. When m, n ≥ 3, the situation becomes more interesting. It turns that configurations exist for arbitrarily large m and n. In a recent paper ([5]), Marco Bramato, Lorenz Halbeisen and Norbert Hungerbühler found configurations where all the points lie on a cubic. These configurations further- more have the interesting property that the points can be moved along the cubic and the lines can be moved accordingly such that all the incidences are preserved. This raises the question if there are more curves that have dynamic configurations. It also raises the question when a configuration with a circumscribed curve is dynamic. One can also ask the dual question whether there are configurations where all the lines are tangent to some curve and if they are dynamic. In this thesis we are mostly interested in combining these two ideas. In Chapter 2 we will first give an overview of some background knowledge. Then in Chapter 3, we will go over some elementary results about configurations in general and tools to study them. In Chapter 4, we will discuss configurations with inscribed and circumscribed curves. Most of the results here were found independently of the existing literature, and we give full proofs. We will first prove some results about Poncelet polygons. With these results, we will prove a theorem, Theorem 4.1.5, which states that (π2, λd) with an in- scribed conic and a circumscribed curve of degree d that consists only of lines and conics is always dynamic. We also show that these exist when 2π = dλ and d ≤ λ − 1. This resembles a theorem of Darboux, Theorem 2.2.6, but it is different and seems to be entirely new. Then we show that the conjecture given in [10],Conjecture 2.9, regarding the Grünbaum- Rigby configuration is true. The lemma that we use to show this, is a special case of a much more general theorem. This theorem also appeared in [4], Theorem A, but we give a different, completely el- ementary proof. This more general theorem is used to prove that there are (m2π, n2λ) configurations with inscribed curves of degree 2(π+λ−1 choose λ ) and circumscribed curves of de- gree 2(π+λ−1 choose π ), assuming Conjecture 4.4.3. We use group theory for this, which also seems to be new.