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.
Trying to fit pentagons around a point.
It’s quite easy to prove this. A regular polygon with sides has interior angles of
Suppose you try and make a tiling by fitting several copies, say of them, around a point so that they all meet at a corner (see the image above). Then the 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
which means that
The term on the right hand side can be rewritten to give
Since is a whole number (the number of copies of the polygon you are fitting together), this means that must also be a whole number. Therefore can only be equal to 4, 2, and 1, which means that can only be equal to 6, 4, and 3.
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 along the neighbouring copy’s side. That neighbouring copy would therefore have an interior angle of 180 degrees at (since is in the interior of one of its sides). To make a tiling you would have to fill the remaining 180 degrees with copies of the polygon, so you would need
Using a similar argument as above you can convince yourself that this only works when or
You can read more about tilings on Plus.
This article comes from our Maths in a minute library.