While I was trying to make an analogous question to Select $n^2 + 1$ points in the unit square. Show that at least two points are no more than a distance $\frac{\sqrt{2}}{n}$ apart, using equilateral triangles, I ended up with : Select $2^n + 1$ points in a unit equilateral triangle. Show that at least two points are no more than a distance $\frac{1}{2^n}$ apart.
But then I got stuck! there was no other shape that the trick of decomposing a shape to smaller version of itself can be applied (short of fractals that is).
So I had to ask this question: Are Square and equilateral triangle the only convex regular ngons that can be composed to smaller versions of themselves?
Yes. To decompose a regular n-gon into smaller versions of itself, the interior points have to have the surrounding angles add up to $360^\circ$. The only regular polygons that have angles that are an integer division of $360^\circ$ are the triangle, the square, and the hexagon. The hexagon has angles of $120^\circ$, but that makes places that only two meet be concave, so you can't make the edges straight.