Counting Triangles with Combinatorics

Abstract

SummaryInspired by a problem presented in a New York Times essay, we consider the number of triangles formed by a finite collection of lines in the plane, under the restrictions that no more than two lines intersect at any point and that no two are parallel. We explore the ramifications of relaxing each of these requirements, and we derive a ‘unified triangle counting formula’ for any arrangement of finitely many lines in the plane.MSC: 05B30 Additional informationNotes on contributorsMatthew GlomskiMATTHEW GLOMSKI earned his Ph.D. at the University at Buffalo and joined the mathematics faculty of Marist College. In his spare time he enjoys hiking in the nearby Catskill Mountains.K. Peter KrogK. PETER KROG earned his Ph.D. at the University of Connecticut and joined the mathematics faculty at Marist College in 1996. His mathematical interests include group theory, statistics, and combinatorics.Mason NakamuraMASON NAKAMURA is an applied mathematics student at Marist College and plans to pursue his doctorate. He enjoys golfing, hiking, and snorkeling during his time away from mathematics.Elizabeth M. ReidELIZABETH M. REID is a member of the mathematics faculty at Marist College. She earned her Ph.D. at the University at Buffalo and enjoys hiking in her spare time.