The geometry of line arrangements: exploiting symmetry
Moshe Cohen (Mathematics and Statistics)
In high school you used equations to determine the point of intersections of two lines. For this project, we generalize to more lines, and we study the various intersection points that arise. These collections of points inform us about the geometry of the lines; for example, three lines with a single triple point looks like pizza for six, while three lines with three double points gives an extra triangle. With enough lines, our intuition betrays us, and a single collection of intersection points can lead to different geometries: this behavior is what we will investigate. The mathematical objects we'll be playing with are called "arrangements" (or collections) of lines. This topic is an access point for many difficult questions in much deeper areas of mathematics, but we will play in the shallow end of the pool. For more information, see this video.
Prerequisites: Facility with high-school-level geometry and algebra; attention to detail and care with work; diligence for challenging problems; communication skills (or interest in improving them). The following will be helpful but are not necessary: MATH 263 Discrete Mathematics, MATH 361 Modern Algebra, MATH 331 Geometry, other experience with proof-writing in mathematics, and any familiarity with computer programming.
How should students express their interest in this project?
1. Check out the YouTube video linked above and then its much more challenging sequel here. [Don't be intimidated if you don't understand some things; the second talk is aimed at graduate students.]
2. Spend at least 30 minutes thinking about -- and attempting to solve -- one of the following two problems. The Orchard Problem (stated below) or the problem stated just after 1:45 in the first YouTube video (above): How many "different" arrangements of lines are there for n lines? (Try n=3,4,5, and maybe even 6.) Furthermore, how can you assure me your answer is correct? [If these questions are unclear to you, you are invited to schedule an appointment with me to get clarification.] .
3. Come to my office to share with me what you've learned and whatever PROGRESS you've made. [You are NOT expected to have finished solving anything.] Email me to set up an appointment at a convenient time for you. If Monday afternoons work for you, you can sign up here.
"The Orchard Problem": We must plant trees (points) in an orchard (the plane) in such a way to maximize the number of paths (lines) through the orchard. The condition is that every path (line) must contain exactly three trees (points) -- no more, no less. Given a particular number p of points (try p=3,4,5,6, and 7), what is the maximum number l(p) of lines that can be drawn? Furthermore, how can you assure me your answer is correct? [SPOILER: No one knows an exact formula!]
[This URSI project is about struggling with problems that haven't been solved before, even when things get tough, and sometimes even when the answer is impossible to find, so you shouldn't let this frustrate you! Your enjoyment should come from exploring the darkness of the cave, whether or not you find any treasure.]