Originally Posted by

**tonio** I suppose point A is the intersection point of both rays...then, after choosing point B on ray 1 and drawing from it a perpendicular to ray 2 you get a straight-angle triangle 30-60-90, sometimes aka golden triangle, with hipotenuse d, and thus the leg opposite to the 90 deg. angle is half the hipotenuse ==> the first part of the polygonal line is $\displaystyle \frac{1}{2}d$, and the other leg's length is, by Pythagoras, $\displaystyle \frac{\sqrt{3}}{2}d$. Repeat the process, again you get a 30-60-90 triangle but with hipotenuse $\displaystyle =\frac{\sqrt{3}}{2}d$, so this time the polygonal line's length is $\displaystyle \frac{\sqrt{3}}{4}d$ , and etc.

The whole polygonal line's length is thus $\displaystyle \sum\limits_{k=0}^\infty \frac{1}{2}\left(\frac{\sqrt{3}}{2}\right)^k=\frac {\frac{1}{2}}{1-\frac{\sqrt{3}}{2}}$ $\displaystyle =\frac{1}{2-\sqrt{3}}=2+\sqrt{3}$

Tonio