PhiProportionExtreme and mean ratioSelf similarityIrrational
Meaning
The ratio in which a line is cut so that the whole is to the greater part as the greater is to the lesser. Its value is about 1.6180339887 and it is irrational, which is why it can be approached but never reached by any whole-number division. What makes it remarkable is self-similarity: a rectangle of those proportions can be cut into a square and a smaller rectangle of the same proportions, indefinitely, and the same ratio governs the pentagon and the pentagram.
Sacred Ratio
phi = (1 + square root of 5) / 2, approximately 1.6180339887
Construction
Divide a segment so that the whole is to the greater as the greater is to the lesser (Euclid, Elements VI, definition 3, and the construction at VI.30). Equivalently, take the diagonal of a regular pentagon against its side, or the continued fraction of all ones, or the limit of the ratios of consecutive Fibonacci numbers. The golden rectangle and the logarithmic spiral are its two common visual forms.