Arrangements of Lines with Only Double Points
Sarah Goodhill ’22 and Professor Moshe Cohen
A line arrangement is the finite collection of lines on a plane. Line arrangements have been studied since the ancient Greeks, but in modern times we often approximate real-life data using lines. We do not consider parallel lines, so we insist that every pair of lines intersect exactly once. When only two lines intersect at a point, we call it a double. Only one arrangement is possible created when considering each of three, four, and five lines. Perhaps surprisingly, this is not the case for larger numbers of lines. We find that for six lines there are 4 such arrangements and that for seven lines there are 11. To prove this, we investigate the number os sides of each region enclosed by the lines. For example, three lines with doubles enclose a triangle. Ultimately, we discovered that these results had already been previously found, by several people, including Vassar’s Professor of Mathematics Louise Duffield Cummings in 1932 in a paper called “Hexagonal systems of seven lines in a plane”.