Does there exist a tiling of the plane in which each individual tile has fivefold symmetry? This simply stated question turns out to have surprising depth, and remains entirely unsolved. Strangely, closely related questions have simple solutions. The corresponding threefold, fourfold, and sixfold tiling problems are trivially solved by the regular tilings of the plane. Fivefold tilings are immediate in both spherical and hyperbolic geometry. There seems to be something peculiar about the interaction between fiveness and flatness.
Over the centuries, this problem has caught the attention of a few researchers. While nobody has managed to produce a fivefold tiling, the search has led to the discovery of many tilings that are fascinating and wonderful. I am interested both in the mathematical problem itself, and in the aesthetic possibilities of these tilings.
This article needs to be fleshed out. In the meantime, please see the references below. In June 2008, I led a public workshop on the fivefold tiling problem, in which I made use of hundreds of small plastic tiles that I had had manufactured via laser cutting. I'll post photos from that event here.
|Craig S. Kaplan||Last updated:|