Was puzzeling with this question:
What is the most sided regular n-polygon that can be made with lego?
It has to be sturdy (the polygon should stay in shape when pushed around)
made with the normal lego bricks (not bend or angled bricks )
no use should be made from non- obvious lego limitations (like the angle that bricks make under tension, the minimal angle two bricks make because bouncing against nobs no one of them.
mathematicly precise
For people who don't know lego:
What is the most sided regular n-polygon where :
all lengths are natural or rational or decided by some construction
all angles are right or decided by some construction
The square is the only one.
To make any other regular polygon that's rigid, the sides would have to be exactly integer length to line up with another peg of a base of legos, which is arrived at by going an integer number of pegs to the right (say) and an integer number of pegs up (say). This is a Pythagorean triple with integer sides.
This means that $\sin \theta$ is rational, where $\theta$ is the exterior angle of the regular polygon. But we know that the exterior angles of regular polygons are $\theta = 2 \pi / n$ where $n \geq 3$ is the number of sides. We also know from Niven's Theorem that if $\sin ax = b$, where $a,b$ are rational, then $b = 0, \pm 1/2, \pm 1$.
The only case which gives rational $\sin \theta$ is $n=4$, which is a square.