## New tessellation discovered

By David Shaw, 31 August 2015

*Image: Wikimedia commons/Tomruen, by permission from www.jaapsch.net*

**If you wanted a really cool bathroom, what shape would you use for the tiles? Squares will work, or rectangles, or even hexagons. But can we use five-sided tiles? You may not know them, but there are a whole range of pentagonal tiling shapes. And a few weeks ago, mathematicians found a new one to join them.**

Mathematicians have been interested in tiling surfaces for hundreds of years. When looking for a tiling pentagon, they follow a few simple rules. The shape has to have five sides. None of the angles can be bigger than 180 degrees. And copies of the shape have to fit together with no overlaps or gaps in an endless pattern that mathematicians call a tessellation.

In 1918, Karl Reinhardt found five different pentagons that tessellated. Between then and 1985, another nine were found. For 30 years, no-one discovered any more tiling pentagons. Maybe we had discovered them all.

Last month, a group of mathematicians investigated the problem. They wrote a computer program to check different types of pentagons and see if any of them might work. Their program found lots of already discovered examples, and some pentagons that didn’t quite work. Then one day, it found a solution. It was a pentagon shape that followed all the rules, and a new tessellation was discovered.

The researchers aren’t stopping there. They are looking for another example and have booked time on a supercomputer to find solutions even faster. But the computer can’t tell us if we’ve found all the tiling pentagons that could exist. So maybe a person with a pencil and paper will one day discover another new tessellating pentagon.

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