Tiles made of regular pentagons – shapes with equal sides and angles – cannot completely cover an infinite plane, no matter how you arrange them.
Once you start distorting the shapes, however, interesting things begin to happen.
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In 1918, German Mathematician Karl Reinhardt described five irregular pentagons that could each tile a plane. His work raised some questions, such as ‘How many of these pentagons could be exist?’. In July of this year, Michael Rao, french computer scientist at the Ecole Normale Superieure in Lyon posted a solution online.
Using a computer algorithm to digital hunt for every possible irregular pentagon, he showed that only 15 types would actually tile a plane – two of those were only discovered in 2015. Rao’s process itself took two years, but without computers it may have taken him much longer.
“There are some problems which resist a nice, short proof” – Michael Roa
Since mathematicians have already spent decades figuring out all possible tiling arrangement for every other convex polygon, many believe the subject to now be completely closed, thanks to Rao work.
There’s more stories from 2017. Check out the Best of 2017 series here