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φ, the golden ratio
φ (phi) is the number (1 + √5) / 2 = 1.6180339887… It is irrational, its digits never repeat, and it turns up in several shapes and sequences that are built from smaller copies of themselves. This page sets out what it is, where it genuinely appears, and which well-known claims about it are not supported.
The definition
Cut a line into a longer part a and a shorter part b so that the whole is to the longer part as the longer part is to the shorter: (a + b) / a = a / b. That ratio is φ. Euclid described this cut around 300 BC in the Elements (Book VI, Definition 3) as dividing a line in "extreme and mean ratio". The name "golden" is much later, and the Greek letter φ for it was suggested in the early twentieth century.
Solving the definition gives φ² = φ + 1, so φ = (1 + √5) / 2. Two small facts follow at once:
- Its square is itself plus one: φ² ≈ 2.618.
- Its reciprocal is itself minus one: 1 / φ ≈ 0.618. That is why "0.618" and "1.618" travel together.
The digits go on without end: OEIS A001622 lists them.
Fibonacci numbers
The Fibonacci sequence starts 0, 1 and adds the last two numbers to get the next: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55… (OEIS A000045). Divide each number by the one before it and the ratios close in on φ: 8/5 = 1.6, 13/8 = 1.625, 21/13 ≈ 1.615, 34/21 ≈ 1.619. The link is exact, not a coincidence. Binet's formula gives every Fibonacci number directly from φ:
F(n) = (φn − ψn) / √5, where ψ = (1 − √5) / 2 ≈ −0.618.
Because ψ is smaller than 1 in size, its powers shrink toward zero, so for large n the Fibonacci numbers are simply φn / √5, rounded.
A number made of ones
Written as a continued fraction, φ is 1 + 1 / (1 + 1 / (1 + 1 / (1 + …))): nothing but ones. Fractions built from continued fractions are the best approximations a number has, and with nothing but ones they improve as slowly as possible. In that precise sense φ is the irrational number that fractions approximate worst, which is why mathematicians sometimes call it "the most irrational number".
Where it really appears
- The regular pentagon. In a regular pentagon, a diagonal divided by a side is exactly φ, and the diagonals of a pentagram cut one another in the golden ratio.
- The golden angle. Divide a full turn in the golden ratio and the smaller part is 360° / φ² ≈ 137.5°. Many plants place each new leaf or seed about this far round from the last, which packs them evenly; the spirals in a sunflower head are counted in neighboring Fibonacci numbers such as 34 and 55.
- Self-similar shapes. Remove a square from a golden rectangle and the rectangle left over is golden again, so the cut can repeat forever.
Claims that don't hold up
Many popular statements say the Parthenon, the pyramids, the Mona Lisa or the human body are built on φ, or that golden rectangles are the shapes people find most pleasing. Measured carefully, these mostly fail: the result depends on which points are chosen to measure between, and a ratio near 1.6 is easy to find in almost anything. George Markowsky's paper "Misconceptions about the Golden Ratio" (The College Mathematics Journal, vol. 23, no. 1, 1992) works through the best-known examples. φ doesn't need the myths: the mathematics above is exact.