Digits of the golden ratio

Euclid wrote the ratio down without a flattering name. The Elements, around 300 BCE, calls it division in "extreme and mean ratio" — cutting a line so that the whole is to the longer part as the longer part is to the shorter — and uses it to construct the regular pentagon and two of the five Platonic solids. Greek writers before him may have known it through the pentagram, but the surviving record starts with Euclid.

Luca Pacioli gave it a reputation. His De divina proportione of 1509, illustrated by Leonardo da Vinci, treated the ratio as theologically significant; Leonardo called it the sectio aurea, the golden section. Michael Mästlin produced what appears to be the first decimal value in a letter of 1597, and in 1608 Kepler noticed that ratios of consecutive Fibonacci numbers close in on it — the result behind the Kepler triangle he also described.

The modern vocabulary is nineteenth-century, and sources disagree on its origin: the German goldener Schnitt is variously traced to Johann Gehler's 1789 dictionary and to textbooks of the 1830s. Mark Barr proposed the symbol φ around 1910, after the sculptor Phidias.

What came with the name was a great deal of invention. The claims that the Parthenon, the Great Pyramid and the Mona Lisa were laid out on φ, and that people reliably prefer golden rectangles, are not supported by the evidence; George Markowsky's 1992 paper in The College Mathematics Journal, "Misconceptions about the Golden Ratio," works through the arithmetic and shows that most of these rest on selective measurement. The genuine appearances — pentagonal symmetry, Penrose tilings, phyllotaxis — are less decorative and more interesting.

Key properties

• φ = (1 + √5)/2 is the positive root of x² = x + 1, so φ² = φ + 1 ≈ 2.618 and 1/φ = φ − 1 ≈ 0.618.
• φ is irrational but algebraic of degree 2 — a root of the rational polynomial x² − x − 1 — so, unlike π and e, it is not transcendental.
• Its continued fraction is [1; 1, 1, 1, …], every term the smallest possible. That makes φ the hardest real number to approximate by fractions, the extremal case of Hurwitz's theorem.
• The ratio of consecutive Fibonacci numbers F(n+1)/F(n) converges to φ, alternating above and below it.
• φ = 2·cos(36°) = 2·cos(π/5), and in a regular pentagon the ratio of a diagonal to a side is exactly φ.
• The golden angle, 360°/φ² ≈ 137.508°, is the angle whose repeated rotation spaces points most evenly around a circle.
• The convergents of the continued fraction are exactly the Fibonacci ratios 1/1, 2/1, 3/2, 5/3, 8/5, 13/8, …

Where they turn up

• Pentagonal geometry: the pentagram, the regular pentagon's diagonals, and the vertex coordinates of a regular icosahedron, which are the cyclic permutations of (0, ±1, ±φ).
• Penrose tilings, which Roger Penrose developed in 1973–74: φ sets both the ratio of the two rhombs' areas and their long-run relative frequency.
• Phyllotaxis — the arrangement of seeds, leaves and florets at the golden angle, which packs them more evenly than any rational fraction of a turn would.
• Continued-fraction and lattice algorithms use φ as the worst case, which is why the Fibonacci numbers show up in bounds for the Euclidean algorithm.
• Fibonacci retracement levels on price charts, drawn at 61.8% and 38.2%. This is a convention of technical analysis, not a result with demonstrated predictive value.
• Claims that the Parthenon, the Great Pyramid, the Mona Lisa or the proportions of the human body were designed around φ are widely repeated but rest on selective measurement; they are beliefs about the ratio rather than findings about it.