Why is it impossible for a regular polygon with more than six sides to tesselate the plane?
I know it has something to do with the angles, but I'm not sure what to write for this. 3 hexagons tesselate the plane and they have angles at 120, but why can't an octagon? Why is it impossible? Thanks.
An octagon has an internal angle of 135 degrees.
If we put 2 octagons side-by-side, that makes 2*135=270°, leaving a gap of 90° which does not fit another octagon. Therefore we cannot tessellate with an octagon.
Use the same logic with other regular polygons and you will find that only three regular polygons can tessellate.
To understand why regular polygons with more than six sides cannot tessellate the plane, let's consider a few properties of tessellations and regular polygons.
A tessellation is a tiling of the plane using one or more shapes, with no overlaps or gaps. In a regular tessellation, the same shape is used to completely cover the plane without any overlaps or gaps. Regular polygons are polygons with equal side lengths and equal internal angle measures.
To create a regular tessellation with a regular polygon, the polygon's internal angles must add up to a multiple of 360 degrees. In other words, the sum of the interior angles of the regular polygon must evenly divide into 360 degrees.
For a regular hexagon, each internal angle is 120 degrees, and the sum of the internal angles is 720 degrees (120 x 6). This allows the hexagons to fit together seamlessly and fully cover the plane.
But for an octagon, each internal angle is 135 degrees, and the sum of the internal angles is 1080 degrees (135 x 8). Since 1080 is not a multiple of 360 degrees, it is impossible to create a regular tessellation with octagons alone. If you were to attempt to fit octagons together, gaps or overlaps would inevitably occur, violating the definition of a tessellation.
In summary, regular polygons with more than six sides cannot tessellate the plane because the sum of their internal angles does not evenly divide into 360 degrees, preventing them from fitting together seamlessly without gaps or overlaps.