Every real number has best rational approximations: fractions p/q that are closer to x than any fraction with smaller denominator. The denominators q₁, q₂, q₃, … grow, but at what rate? Paul Lévy proved in 1935 that for almost every real number, qₙ^(1/n) converges to e^β ≈ 3.27582, where β = π²/(12 ln 2).
The partial quotient 292 in π's continued fraction causes the jump at n=5 (giving 33102 as denominator). This makes 355/113 exceptionally good: the reason it approximates π so well for its size. Over many terms the average growth rate approaches e^β.
The golden ratio φ = [1;1,1,1,…] has Fibonacci denominators 1, 1, 2, 3, 5, 8, 13, … growing at rate φ ≈ 1.618 per step. This is far slower than e^β ≈ 3.276, which is why φ is the "most irrational" number: its approximations improve the most slowly. Most numbers have denominators growing much faster, at rate e^β.
The value β = π²/(12 ln 2) emerges from integrating the Gauss-Kuzmin distribution. The ln 2 comes from working in base 2 (binary), and π² arises from the same sources as ζ(2) = π²/6. Lévy's constant: 1.1865691104156254… e^β = 3.275822918721811159787681882…
π's continued fraction [3; 7, 15, 1, 292, 1, 1, ...] has a famous large entry: 292. This makes 355/113 an exceptionally accurate approximation (correct to 6 decimal places). Despite such jumps, Lévy's theorem says the long-run average growth rate is always e^β ≈ 3.276.
Levy's constant beta = pi^2/(12 ln 2) ≈ 1.18657. For almost every real number, the nth convergent denominator qn satisfies qn^(1/n) to e^beta ≈ 3.27582. Proved by Paul Levy in 1935. The golden ratio, with Fibonacci denominators growing at rate phi ≈ 1.618, is far below average, confirming it as the hardest number to approximate. The formula combines pi and ln 2, connecting circle geometry with logarithms through the Gauss-Kuzmin distribution.