Steven Dutch, Professor Emeritus, Natural and Applied Sciences, University of Wisconsin - Green Bay

Pentagon Types That Tile the Plane

At one time the issue of tiling the plane with pentagons was simple. K. Reinhardt identified five types of tiling pentagon in 1918. In 1967, Richard Kersher discovered three new classes and the problem was considered solved. Martin Gardner described these types of pentagons in the July 1975 issue of Scientific American and to the surprise of everyone, a number of amateur mathematicians turned up a number of other types. Now thirteen classes of tiling pentagons are known and it is not known if the list is complete.

Hexagon Types That Tile the Plane

Reinhardt also enumerated the types of hexagons that tile the plane and this result has endured. There are three types, shown below.

Aperiodic Tilings

It's easy to construct tilings that are aperiodic. Isoceles 36-72-72 triangles can be joined into 10 pyramids that radiate away from a common center, for example. The challenge is to find tilings that are only aperiodic. Not only must the polygons tile the plane aperiodically, but no subset may tile periodically. One can easily devise a tiling with a single pentagon surrounded by triangles. While the complete set of tiles is aperiodic, the triangles can tile the plane periodically, so this example is not a valid aperiodic tiling.

A number of truly aperiodic tilings are known. The most strinking are the so-called Penrose Tiles, which consist of only two shapes. So far it is not known if there is a aperiodic tiling consisting of only a single shape of tile.