There is no ratio or particular number for division of the frame or placement of points which has any demonstrated special power. That includes both the rule of thirds and the golden ratio. However, the basic idea that centered subjects tend towards a more staid composition while off-center provides dynamic interest is fundamentally sound. Just don't get caught up a certain number. I've written extensively on this at:
- What is the "Rule of Thirds"?What is the "Rule of Thirds"?
- What is the 'Golden Ratio' and why is it better than the 'Rule of Thirds?'What is the 'Golden Ratio' and why is it better than the 'Rule of Thirds?'
- What is the 'Diagonal Method' and should I use it instead of 'The Rule of Thirds?'What is the 'Diagonal Method' and should I use it instead of 'The Rule of Thirds?'
- When is it OK to place the subject in the middle of a picture?When is it OK to place the subject in the middle of a picture?
- When taking a picture, where should the horizon be?When taking a picture, where should the horizon be?
The golden ratio was invented by minor landscape artist John Thomas Smith in 1797, and his reasoning for it is incredibly weak — it seems mostly that he wanted to become famous for inventing an artistic concept. While the math of the golden ratio was known to the ancient Greeks, the idea of using it in aesthetics dates to around 1854, put forth by German intellectual Adolph Zeising. (The Parthenon isn't based on the golden ratio, nor is any other ancient art we know of, and despite repeated claims, there's no evidence that it was used in the Renaissance either.) And, if you look at What historic reasons are there for common aspect ratios?What historic reasons are there for common aspect ratios?, note that 5:8 never became popular, while 5:7 certainly did — if the golden ratio were more appealing, you'd think we'd naturally gravitate to a ratio very close to it.
All of this comes about, I think, because art feels difficult and mysterious. It would be very comforting for their to be secret yet repeatable mathematical rules which, when followed, would lead to beauty. That particularly fit with the mindset of the Enlightenment, when these ideas became popular, and that explains why the rules became so popular even though their actual application didn't. The world just plain doesn't work that way, and there's no magic numbers for art.