It's not everyday that the mathematics community gets rocked, but that's exactly what happened after a University of Washington Bothell team of mathematicians recently discovered a new kind of pentagon capable of tiling a plane, according to a press release by the university.
Discovering a new tiling pattern might not sound so exciting to the uninitiated, but this pentagon is only the 15th ever discovered capable of completely tiling a plane, or covering a flat surface using only identical copies of the same shape with neither gaps nor overlaps. It is the first such pentagon discovered in the last 30 years.
The team made the discovery using a computer program engineered by undergraduate researcher David Von Derau. Here is the breakthrough shape itself, in all of its glorious dimensions:
The discovery goes beyond just offering designers a new way to tile a floor.
“Aside from the practical uses of this new knowledge, which would include a whole different way to tile a floor,” explained Casey Mann, one of the team members that discovered the new shape, “the impact of this new tile moves us one step closer to having a complete understanding as to how shapes can fit together on a plane.”
“Many structures that we see in nature, from crystals to viruses, are comprised of building blocks that are forced by geometry and other dynamics to fit together to form the larger scale structure,” Mann told the Guardian.
The new shape is also dynamic for having an aesthetic appeal that can be difficult to quantify. Mathematicians are perhaps most curious about designs that tile a plane because they find them alluring. For a mathematician, this is art.
"Mainly we study them for pure pleasure," admitted Dr. Steven Strogatz, a Cornell University mathematician who was not involved in the discovery, to the Huffington Post.
The problem does have a particular appeal for puzzle lovers. Every triangle has been proven capable of tiling a plane, as has every four-sided shape. But things get interesting with five-sided polygons, or pentagons. It turns out that the regular pentagon cannot tile a plane. Only misshapen, non-regular pentagons are up to the task, perhaps adding to their charm.
Exactly three, and only three, hexagons have been proven capable of tiling a plane. But after that, there are no polygons with more than six sides capable of it. So the plentitude of discovery lies with the pentagon, at least as far as tiling a plane is concerned. The 15 pentagons now discovered that can pull it off are probably not the complete list. In fact, there is no proven limit to the number of pentagons that could do it, so the avenue for future discoveries remains open.