If you divide two consecutive terms in the Fibonacci sequence (the later term ÷ the earlier term), the value jumps around at first, but as the terms get larger, it gradually approaches a single value. That value is the golden ratio, φ ≈ 1.618…. Move the slider to increase the term number and watch the ratio approach φ.
Why does it converge to the golden ratio? The Fibonacci sequence is defined by F(n+1) = F(n) + F(n-1). If we assume that the ratio r = F(n+1)/F(n) converges to some value, dividing the recurrence relation by F(n) eventually gives the equation r = 1 + 1/r. Solving this gives r² = r + 1, and the positive solution of this equation is exactly the golden ratio, φ = (1+√5)/2 ≈ 1.618.
The frequent appearance of spiral structures in nature, such as the arrangement of sunflower seeds and the shells of nautiluses, is also related to this convergence property.