Unsolved Problems in Geometry

Abstract

Draw a wiggly curve on a sheet of paper, without lifting your pencil or allowing the curve to cross itself, joining the end to the point of beginning. Now try to find four points on your curve that form the vertices of a That problem in plane geometry appears to be a little bit harder but not radically different from the well-known construc tion problems of high school geometry courses: Given a circle in the plane, con struct an inscribed square. Our problem just has a few more wiggles in it. It also happens to be unsolved: No one has yet been able to prove that every closed curve contains the vertices of a Ge ometry, despite its procrustean image as an ancient, completed subject, abounds in un solved problems that are still under active investigation. Many of these problems are easy to understand, and some of them are even being solved. The most famous of the long-unsolved problems of geometry is the four-color con jecture that every map can be colored with no more than four colors in such a way that adjacent regions are assigned different col ors. This conjecture, formulated in 1852, was first solved in 1976 with an innovative