I recently saw a question here on mathoverflow: «For what n and t can a square be partitioned into n similar rectangles in t congruence classes?», where Joseph Gordon gave a proof that, indeed, a square can be partitioned into n non-congruent similar rectangles for any $n\ge3$. His method involves the use of Fibonacci-numbers. For n=3, the aspect ratio,r, of the similar rectangles is the square of the plastic number (also known as the Padovan constant).
Using his method I did some calclulations for a few n (stopped at n=16, where I found r approximately 1.6180355) and the results seem to suggest that as n grows larger, the aspect ratio of the similar rectangles tends towards the golden ratio. Can this be proved/disproved in any way?