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    • Plus Advent Calendar Door #7: Tiling troubles

      7 December, 2016

      Out of all the regular polygons there are only three you can use to tile a wall with: the square, the equilateral triangle, and the regular hexagon. All the others just won't fit together.

      Pentagons

      Trying to fit pentagons around a point.

      It's quite easy to prove this. A regular polygon with n sides has interior angles of 180n−2n degrees. Suppose you try and make a tiling by fitting several copies, say k of them, around a point so that they all meet at a corner (see the image above). Then the k angles must add up to 360 degrees. If they add up to less there will be a gap, and if they add up to more then copies of the polygon will overlap. So we need k×180n−2n=360, which means that k=2nn−2. The term on the right hand side can be rewritten to give k=4n−2+2. Since k is a whole number (the number of copies of the polygon you are fitting together), this means that 4/(n−2) must also be a whole number. Therefore n−2 can only be equal to 4, 2, and 1, which means that n can only be equal to 6, 4, and 3.
      Pentagons

      Trying to fit a third polygon with two copies offset against each other.

      You could also try to make a tiling in which a corner of the polygon does not necessarily meet the corner of a neighbouring copy, but sits at some point x along the neighbouring copy's side. That neighbouring copy would therefore have an interior angle of 180 degrees at x (since x is in the interior of one of its sides). To make a tiling you would have to fill the remaining 180 degrees with k copies of the polygon, so you would need k×180n−2n=180. Using a similar argument as above you can convince yourself that this only works when n=3 or n=4.

      You can read more about tilings on Plus.

      Return to the Plus advent calendar 2016.

      This article comes from our Maths in a minute library.

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