There are only 15 possible pentagonal tiles

Tiling the plane with a single pattern is a mathematical problem that has interested humans since Antiquity, notably for the aesthetic quality of tiles in mosaics or tiling. One of the unresolved problems in this field that has been puzzling the scientific community since 1918 has now been definitively resolved thanks to Michaël Rao of the Laboratoire d'informatique du parallélisme (CNRS/Inria/ENS de Lyon/Université Claude Bernard Lyon 1). Using computing tools he was able to demonstrate that there are only 15 five-sided patterns that can tile the plane. The research is now available on the Arxiv.org website. There are a number of solutions for covering a floor with a single form, such as triangles, squares, rectangles, hexagons, etc. The exhaustive search for all of the convex forms that can tile the plane—a form with angles smaller than 180° that can cover an entire wall without overlapping—was initiated by Karl Reinhardt during his thesis in 1918. He showed that all triangles and quadrilaterals can tile the plane, but that there were only 3 types of hexagons that could do so, and that a polygon with seven sides or more could not do so.