Hamilton Institute Seminar

Wednesday, April 29, 2020 - 13:00 to 14:00
on Zoom

Speaker: Professor Michael Rao, ENS Lyon, France

Title: Pentagons tiling the plane

Abstract:  In 1918, Karl Reinhardt asked which convex polygons tile the plane, and pentagons were
the only case that remained open. Fifteen types of such pentagons have been found between 1918 and
2015. I present an exhaustive search of all families of convex pentagons which tile the plane. This
proof is in two parts. First we show that if a pentagon tiles the plane, then it is in one of 371
families. Then a computer program tries, for each family, all possible tile arrangements, and shows
that there are no more than the already known fifteen types. In particular, this implies that there
is no convex polygon which allows only non-periodic tilings.