what you're seeing
A convergent pₙ/qₙ is the truncation of the continued fraction after n terms. The
best-approximation theorem: it is the closest fraction to x of any denominator
up to qₙ. This bench checks that twice over — the recurrence vs. a brute-force search of every
denominator, and the mediant descent of the Stern–Brocot tree.
the punchlines
φ = [1;1,1,1,…] — every term is the smallest one can be, so its convergents (the
Fibonacci ratios 1,2,3,5,8,13…) crawl in slowest. That makes φ
the most irrational number — and the reason sunflower seeds sit at the golden angle:
it's the angle hardest to approximate by a tidy fraction, so the seeds never line up into gaps.