Well this is cool. Apparently, a group of mathematicians have recently found a new type of pentagon that can tile the plane—meaning that it can cover the plane leaving no gaps and having no overlaps. It is the 15th type of pentagon known to be able to tile the plane and the first discovered in 30 years.
Apparently searching for tiling pentagons has been a thing for about a century now. Karl Reinhardt, back in 1918, discovered five classes of pentagons that tile the plane. These five were considered all the possible tiling pentagons until 1968, when three more were found. The list continued to grow until about 30 years ago, when it stalled at 14 types of pentagons. Until now!
The discovery was made by Casey Mann, Jennifer McLoud, and David Von Derau. The three, working at the University of Washington Bothell, used a computer to search through a finite set of possibilities.
Why so much interest in tiling pentagons? Well, it turns out that pentagons are the only one of the “-gons” that isn’t completely understood. For example, all triangles and quadrilaterals have been classified as being able to tile the plane, exactly three types of (convex) hexagons tile the plane, and no other –gon is able to tile the plane. But pentagons haven’t been fully classified yet.
So the research must press on! There may end up being more tilings that are discovered, but for now, have a picture of the 15 pentagon shapes known to tile the plane (with the most recent one in the bottom right corner) (picture from article linked above):