We have all seen interesting patterns of tiling the plane with interlocking shapes, known as a tessellation. The process of producing a complete inventory of all possible tessellation has resisted solution for over a century, until now.

The honor goes to Michael Rao of the Ecole Normale Superieure de Lyon in France. He has completed a computer-assisted proof to complete the inventory of pentagonal shapes, the last remaining holdout. He identified 371 scenarios for how corners of pentagons might fit together, and then checked, by means of an algorithm, each scenario. In the end, his computer program determined that the 15 known families of pentagonal tilings is a complete set.

A team of researchers led by Casey Mann of the University of Washington, Bothell had been working on a similar effort, and conceded that Rao had beaten them to the finish.

Rao’s effort must still be subjected to peer review, but Thomas Hales of the University of Pittsburgh, who recently proved the Kepler conjecture (that the supermarket scheme for stacking oranges is the optimal method) by means of a computer-assisted algorithm, has independently reconstructed much of Rao’s proof, and so researchers are relatively sure that Rao’s proof will hold up.

Additional details about Rao’s proof and the tessellation problem can be found in a very nice Quanta Magazine article by Natalie Wolchover.