Euclid (fl. 300 BCE) — “the father of geometry” — called it extreme and mean ratio, and Luca Pacioli (1445-1517), an Italian mathematician and Franciscan friar, called it the divine proportion. It was Pacioli’s book De divina proportione, illustrated by Leonardo da Vinci (1452-1519), which really gave the concept wings (and which, one could argue, led to centuries of what sometimes appears to be dilettantish and coquettish claims around its application). If you have heard of it, you probably know it as the golden ratio.
It has been represented by the Greek letter τ (τ — tau — is the first letter of the ancient Greek word τομή, which mean ‘cut’ or ‘section’). But nowadays the golden ratio is commonly referred to by the Greek letter φ — phi. One good reason to represent it with a symbol like that is that the golden ratio is an irrational number — i.e. a number that never ends. Written in decimals it looks something like this: 1.618033988749894…
Euclid’s definition of this particular ratio is as follows:
A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the lesser.
Or in other words, using my illustration above: The line AB is to the line AG as the line AG is to GB. And yet another way to say it: If I take the distance AB and divide it by the distance AG, I will get 1.618033988749894... . And if I take the distance AG and divide it by the distance GB I also get 1.618033988749894...
But, it’s just circles and lines on a piece of paper, isn’t it? Yet, quasi-religious babble aside, when I get my straightedge and pair of compasses out to construct a representation of φ — or it could be something else, it could be a circle that intersect another circle in any mathematically thought-provoking way, or even just one that does so in an aesthetically pleasing manner, because constructing with a pair of compasses is in itself bliss — when I get my straightedge and pair of compasses out to make some circles and lines on a piece of paper, then I know I’m about to kiss the sky.