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Pack it in with pentagons

By David, 23 January 2018 News

Image of a grid, each square is a differrent colour and divided up by different shaped pentagons.

All fifteen pentagonal tilings! Image: Wikimedia commons/EdPeggJr

What kinds of shapes pack a flat surface with no gaps? Squares are great for floor tiles, and bees stick hexagons together. With a bit of work, you can get any triangle, no matter how stretched, to fit together without gaps. But what about pentagons? These five-sided shapes are awkward, but some types fit together.

Mathematicians love a challenge, which is why over the past 50 years, we’ve discovered many new types of pentagons that fit together. That includes four types found by Marjorie Rice, who never went to university but loved doing geometry. The most recent new pentagon, the 15th to be found, was discovered just two years ago.

Recently, the mathematician Michaël Rao attacked the problem. Michaël started by working out a huge list of possible ways that the corners of pentagons could meet. Maybe four of them meet at one point, or two corners meet the side of a third pentagon. In the end, Michaël had 371 different possibilities to check.

It was far too many possibilities to check by hand, but just right for a computer. So Michaël wrote a program to check each of his possible examples, and see what tilings came out.

When the program finished, it had discovered 15 types of pentagons that could tile a flat surface. All 15 had been discovered before, and Michaël had not found any new tilings. Now we know there are no more to be found.

There is one gap in his proof, though. Michaël only checked convex pentagons. That is, he didn’t check any pentagons with angles greater than 180 degrees – corners that go in instead of out. So maybe there’s still a tiling pentagon for you to discover!

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