Alex Bellos gives us exciting news from the world of mathematics of the discovery of a new pentagonal shape that has the ability to “tile the plane”. i.e., to cover a flat surface using only that shape, leaving no gaps. Here it is.
All triangles and quadrilaterals can tile the plane and “It was proved in 1963 that there are exactly three types of convex hexagon that tile the plane. And no convex heptagon, octagon, or anything else-gon tiles the plane.” It turns out that pentagons are the only shape for which the number that can tile a plane is as yet unknown. Not all pentagons can tile the plane, and the familiar regular symmetric pentagon is an example of one that does not.
The search for the exhaustive list of these pentagonal tiles is quite fascinating. After five were discovered by German mathematician Karl Reinhardt in 1918, R. B. Kershner added three more in 1968, and Richard James added another in 1975. Then in the next few years an amateur, a San Diego woman in her 50s named Marjorie Rice who had came across an article in Scientific American about James’s discovery, added four more. (Her story can be read here.) Rolf Stein in 1985 found a 14th one and then things remained dormant until the most recent one was announced last month by Casey Mann, Jennifer McLoud and David Von Derau of the University of Washington Bothell.
So here is the current list of 15.
So how many possible pentagons are there that can tile the floor? This may be one of those problems, like the four-color map, that defies a clean analytical solution and we may need a computer-assisted proof in order to get a definitive answer.