| n | aₙ | pₙ | qₙ | pₙ/qₙ | error |α−p/q| | q·|α−p/q| |
|---|
Continued fraction: α = a₀ + 1/(a₁ + 1/(a₂ + …)) — the best rational approximations to α.
Golden ratio φ = [1;1,1,1,…] has all partial quotients = 1 — it is the most irrational number, hardest to approximate by rationals. This is why Fibonacci numbers appear in plant phyllotaxis: they are denominators of convergents to φ.
Hurwitz theorem: |α − p/q| < 1/(√5 · q²) for infinitely many p/q; the constant √5 is sharp for φ. Three-distance theorem: {nα mod 1} for n=1…N creates exactly 2 or 3 distinct gap sizes.
Golden ratio φ = [1;1,1,1,…] has all partial quotients = 1 — it is the most irrational number, hardest to approximate by rationals. This is why Fibonacci numbers appear in plant phyllotaxis: they are denominators of convergents to φ.
Hurwitz theorem: |α − p/q| < 1/(√5 · q²) for infinitely many p/q; the constant √5 is sharp for φ. Three-distance theorem: {nα mod 1} for n=1…N creates exactly 2 or 3 distinct gap sizes.