Apparently finding pentagons that can form a seamless plane (think tiling) is a bit of a mathematical sport. Alex Bellos of The Guardian reports:
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.
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.
The newly discovered pentagon that can tile the plane is the first to be discovered in 30 years. A team from the University of Washington Bothwell discovered the 15th tileable pentagon using a computer program.
The problem of classifying convex pentagons that tile the plane is a beautiful mathematical problem that is simple enough to state so that children can understand it, yet the solution to the problem has eluded us for over 100 years.
It’s a beautiful problem, and all of the preceding solutions have beautiful results: