Q:

Equilateral polygon plane tiling

From playing around with some toothpicks and peas, I think that it should be possible to prove that the plane cannot be tiled by a possibly infinite set of equilateral polygons with the same number of sides whenever $n>6$, though I am not sure regarding $n=5$. 
Is there a simple proof / contradiction to the above?

A:

Just crinkle the edges of a square. This will at least let you form an equilateral $2n$-gon for any $n\geq 2$.

For $n$ an odd numbers of edges just have a church with $(n-3)/2$ `steeples'.
You can clearly tile the plane with such shapes.

