I had already talked on √2 based on vilvam /bael twig and its leaves.
The continued fraction was
[1,1,2,1,2,1,2, . . . . . . ]
It takes an expert to say that √3 is due to wind on √2 leaves.
What will happen when wind blows on a bael twig. The two leaves will make a namaste and come back to the original state.
Therefore the continued fraction of √3 is
[1,1,1,1,2,1,1,1,2, . . . . . .]