Phi φ

Phi is the twenty-first Greek letter; unlike the earlier letters it is a Greek innovation with no Phoenician model, and its value is 500. Phi's most famous symbolic role is modern: in the early twentieth century the mathematician Mark Barr chose phi to denote the golden ratio, after the sculptor Phidias; the letter also stands for Euler's totient function. These are modern mathematical conventions, not ancient mystical meanings.

The Greek letters doubled as numerals in the Milesian (alphabetic) system, in which each letter carried a fixed value; the practice of summing a word's letters into a number, called isopsephy, is the Greek counterpart of Hebrew gematria and appears in inscriptions, the Greek Magical Papyri and early Christian texts. Phi is one of four supplementary letters, together with Chi, Psi and Omega, added at the end of the Greek alphabet once its Phoenician-derived series had run out; unlike the borrowed letters it has no Semitic ancestor and, unusually, no inherited name of its own either, its name simply being its sound with a vowel attached. Its isopsephy value of 500 belongs to the Milesian alphabetic numeral system, which also needed three further, otherwise obsolete letters, digamma (also called stigma), koppa and sampi, to cover every value from 1 up to 900; none of the three survives among the 24 letters in ordinary use today.

Phi's best-known modern association, the golden ratio, an irrational number close to 1.618 with notable mathematical properties, is a far more recent naming than many readers assume: the ratio itself was studied by ancient Greek mathematicians including Euclid, but the convention of labelling it with the letter phi dates only to the early 20th century, when the American mathematician Mark Barr proposed the symbol around 1909, reportedly to honour the sculptor Phidias, whose initial in Greek is phi and whose work at the Parthenon was believed, not entirely reliably, to embody the ratio's proportions. The golden ratio itself has attracted persistent, often exaggerated claims of appearing throughout art, architecture and nature, many of which later scholarship has questioned or rejected as invented pattern-matching rather than deliberate design.

Outside this mathematical association, physicists use phi for magnetic flux and for phase angles in wave equations, and astronomers, following the Bayer star-naming convention of 1603, use Phi for one of the fainter numbered stars in a constellation. None of these modern uses connects to phi's ancient Greek sound value, covered elsewhere in this atlas.