FabR[Senior Patek Moderator]
26166
Yep, that is precisely the number! ;-) The most important time I used this constant $\phi$ in a mathematical paper was probably when I was studying the growth of certain "differential posets" (I wrote quite a few things in this area while at MIT)....
....and after performing some miracles, $\phi$ explains precisely the largest growth of the posets!
Anyways, art and nature are full of examples as well, and fortunately for us, also horology...and it's always great when all these areas intersect! 
Cheers.