Puzzling Pentagons—How Many Ways Can We Cover a Flat Surface?

Abstract

Convex polygons, shapes bounded by straight lines in which all of the corners point outward, are the simplest of shapes, but despite their simplicity, there are many unsolved problems concerning polygons. Some polygons have the nice property that they fit together espcially well, so that you can use lots of copies of them to cover a large surface to form what is called a tiling. For some polygons, such as triangles, it is easy to see how to form tilings. For others, such as seven-sided convex polygons, it is impossible to form a tiling. This article discusses the history of a long-standing open question in geometry: which convex pentagons give rise to tilings of the plane? The authors also discuss their contribution to the solution of this problem, which involved developing a computerized algorithm to help them search for a new kind of convex pentagon that can form tilings, and the basic idea of this algorithm is discussed.