# Attack on the pentagon results in discovery of new mathematical tile

Joy as mathematicians discover a new type of pentagon that can cover the plane leaving no gaps and with no overlaps. It becomes only the 15th type of pentagon known that can do this, and the first discovered in 30 years.

Alex Bellos in The Guardian:

In the world of mathematical tiling, news doesn’t come bigger than this.

In the world of bathroom tiling – I bet they’re interested too.

If you can cover a flat surface using only identical copies of the same shape leaving neither gaps nor overlaps, then that shape is said to tile the plane.

Every triangle can tile the plane. Every four-sided shape can also tile the plane.

Things get interesting with pentagons. The regular pentagon cannot tile the plane. (A regular pentagon has equal side lengths and equal angles between sides, like, say, a cross section of okra, or, erm, the Pentagon). But some non-regular pentagons can.

The hunt to find and classify the pentagons that can tile the plane has been a century-long mathematical quest, begun by the German mathematician Karl Reinhardt, who in 1918 discovered five types of pentagon that do tile the plane.

(To clarify, he did not find five single pentagons. He discovered five classes of pentagon that can each be described by an equation. For the curious, the equations are here. And for further clarification, we are talking about convex pentagons, which are most people’s understanding of a pentagon in that every corner sticks out.)

Most people assumed Reinhardt had the complete list until half a century later in 1968 when R. B. Kershner found three more. Richard James brought the number of types of pentagonal tile up to nine in 1975.

More here.