There are so many websites dedicated to the golden ratio in art (perhaps one of the best is this one) that I don't want to bore you with more explanations.
But for those who don't know what is the golden section, perhaps it suffice to say that
Probably since the making of this BBC documentary starring John Cleese, The Human Face the concept of a magical number (PHI =1.618 0339 887) in Nature and Art has been growing in interest even more.
(And a little diversion on "beauty" in the following video clip it's almost a must to add).
To see many example of golden ratios in art and Nature, simply go here.
You will see over and over again Leonardo's work, Greek architecture, the Nautilus (see the picture above, how curious they are!) , Botticelli's, Piero della Francesca's paintings and fresco's (by the way Piero della Francesca he was born in Sansepolcro, Tuscany---oh, the link is a surprise of mine...), Palladio, Mondrian, Seurat, and many more examples.
Actually, there are not many example of Golden Ration in Art* because this hypothetical mathematical way of creating beauty is more a myth. You can check many paintings, from Leonardo da Vinci's Mona Lisa to Picasso and you will see that the focus of the painting (eg. an eye, the head for the portrait, the particular scene or bridge) is rarel exactly at the right spot....
How to do that? Well, simply multiply each dimension of the painting or image by 0.62 and see what there is at the intersections of the two lines.
See? The "rule" of the golden ratio in art for the Mona Lisa does not work. The centre of interest(s) (the face, the eye, the hands) are not even nearly near any of the intersections of the red lines. This does not make the painting a failure as you know...Conclusion: good composition is not just framed into mathematical rules as good and nice as these can be. There is no golden rule but perhaps one: variety.
In case you want it, here is a vector grid (in zip format)in Adobe Illustrator format (ai file) to play around on pictures that you see. Overlap it, resize it to the painting size and see...
Download the Rule of Thirds Grid (ZIP file) to be used in your photo-editing programm
Download the Golden Section Grid (ZIP file) to be used in your photo-editing programm
In photoshop I open a picture, then I "File>place..." the grid over it and adjust it to the picture ("edit>free transform...", or CONTROL + T)
*The Golden Rule can be seen much more in Nature than in fine art.
Mona lisa picture from wikipedia