# Fivefold tilings

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.

## Papers

All images are copyright 2008 by Craig S. Kaplan. You are
free to use them for **personal** and **non-commercial** purposes.
Please check with me about any other uses.