Or to simplify (or maybe confuse):

a----------------------b--------------c

when ab:ac = bc:ab, we have our Golden ratio. This is part of the basis of fractals, and there can be no doubting that these are beautiful.

Although a bit long, I came across something a while ago which has quite an interesting thesis built on the foundation of the golden ratio – I offer it for your consideration!

The golden section is a mathematical ratio of (very) approximately 8:5… […]The ratio can be derived from the Fibonacci series […] each pair of digits slowly converges – after a bit of wobbling – more and more accurately on the ratio. Both the golden rule and the Fibonacci series itself are omnipresent in nature: in the whorls of the pinecone or the seedhead of the sunflower, in the number of the daisy’s petals, in the spirals of the nautilus or the snail’s shell.

The golden section casts its spell over all the arts. The ratio, especially in the form of the ‘golden rectangle’, was often applied to the proportions of classical architecture, and was employed in the construction of the pyramids. In painting the division is often used to position the horizon line or significant detail within the picture, lending the composition an intrinsic rightness. Artists from Leonardo to Seurat used the division in a highly complex way […] It naturally marks the main point of dénouement in many films, dramas and operas.

Perhaps the most significant appearance of the golden section is in music. If you divide the thirteen notes of the chromatic scale from C to C at the golden section, you land on the eighth, the dominant – G. This is the first new note we encounter in the harmonic series. To oversimplify grossly: musical notes are rarely pure, and the dominant, the fifth, is the other note we tend to hear singing above the main note in a musical sound. The relationship between tonic and dominant is the basis for practically the whole of Western music – and this is not dry theory, it’s just the way we hear things […] It seems reasonable to conclude that this vibrational relationship – and we’re talking about all sounds here, not just music – must have penetrated to every part of our spiritual, mental and physical constitution.

In short the golden section is a division we often can’t help making. In poetry its most explicit manifestation – there are plenty of others – is in the division of the sonnet […] evolved organically … as with our horizon line, there’s just a rightness to it… if sonnets had broken smack in the middle, they just wouldn’t have been any fun to write and people wouldn’t have written them.”

from Don Patterson’s Introduction to 101 Sonnets - from Shakespeare to Heaney
Faber & Faber, ISBN 0-571-19732-9