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Century-Old Pentagon
Mystery Solved
❯
FACT: TILES MADE OF REGULAR PENTAGONS — equal angles,
equal sides — can’t completely cover an infinite flat plane, no
matter how you spin them. But squish and stretch the shapes a
little, and things get interesting. In 1918, German mathematician Karl
Reinhardt described five irregular pentagons that could each tile a plane.
His work raised the question: How many such pentagons could there be?
In July, French computer scientist Michaël Rao of the École Normale
Supérieure in Lyon, France, posted a solution online to that century-old
pentagon puzzler. Using a computer algorithm to digitally scour all the
possible misshapen pentagons, he showed that only 15 types work, with
the last discovered just two years ago. Rao’s process took two years, but
without computers, it would have taken more than a decade. “There are
some problems which resist a nice, short proof,” he says.
Because mathematicians had already figured out all possible tilings for
every other convex polygon — shapes where all vertices point outward —
this discovery effectively closes the book on the subject. STEPHEN ORNES