Fifteenth, convex, tiling pentagon found

You cannot tile a floor only with regular, identical, convex pentagons.


In 1918 the German mathematician Karl Reinhardt discovered five types of convex pentagon that can tile the plane. (The pentagons that belong to a particular type all share a common feature — see this paper for a description of the types.) Then there was a slow trickle of discoveries through the century, with Rolf Stein eventually bringing the number of types up to fourteen in 1985. (You can read more about the discoveries in Alex Bellos’ Guardian article.) And now, thirty years later, Casey Mann, Jennifer McLoud and David Von Derau of the University of Washington Bothell have announced that they have found another convex pentagon that can tile the plane:

New tiling pentagon

All the fifteen known, convex, tiling pentagons are shown below with the new one at the bottom right.

fifteen known tiling pentagons

fifteen known tiling pentagons


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