General Facts About Tilings

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

All Triangles and Quadrilaterals Tile the Plane

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.

Conway's Criterion

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.

A Couple of Tesselation Paradoxes

The upper figure is a square 13 units on a side, with an area of 169. The lower figure, made by rearranging the pieces, measures 8 by 21 units, for an area of 168. Where did the missing unit of area go?
At least in the puzzle above the pieces were rearranged into a rectangle of different shape. Here both figures are squares 12 units on a side. The lower figure, made by rearranging the pieces, is missing the gray square. Where did the missing unit of area go?

The solution to both puzzles is the slanting line. If we were to draw the figures accurately on graph paper, the slanting line would not pass exactly through the ends of the adjoining pieces. The "missing" unit of area is actually spread out along the slanting line as a very narrow gap or overlap too small to be easily noticed.

Return to Symmetry Index
Return to Professor Dutch's home page

Created July 1, 1999, Last Update July 9, 1999