Mathematicians find a 15th way to tile your floor with pentagons

mathematicians find 15th pentagon to tile a plane 15
Casey Mann via The Guardian

If you’re thinking of re-tiling your bathroom, you might go with large squares that make a regular alternating pattern, but what if you want a shape with more sides — like a pentagon? If all five sides are equal, you can’t tile the plane, meaning they won’t fit together without overlapping or leaving gaps. It was a mathematical breakthrough in 1918 when German mathematician Karl Reinhardt found five nontraditional pentagons that could cover a flat surface. Fifty years later, R.B. Kershner found three more, followed by a ninth from Richard E. James III in 1975. Amateur mathematician Marjorie Rice found three more, and the 14th type was discovered by Rolf Stein in 1985. All’s been quiet on the pentagonal front ever since, until now.

Three researchers at the University of Washington Bothell recently found a 15th type. Mathematics associate professors and research co-directors Casey Mann and Jennifer McLoud-Mann, along with undergraduate researcher David Von Derau, made the discovery.

“We discovered the tile using using a computer to exhaustively search through a large but finite set of possibilities,” Mann tells The Guardian. “We were of course very excited and a bit surprised to find the new type of pentagon.”

The researchers say the finding could impact the fields of biochemistry and structural design. “Aside from the practical uses of this new knowledge, which would include a whole different way to tile a floor, 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,” Mann said in a statement.

When you step back and look at the pattern the pentagons make, it looks a little like the netting on a lacrosse stick. Maybe it’s not a giant leap to think this tessellation could be coming to a tile store near you someday.