THE CONTINUED FRACTION — a dart

THE CONTINUED FRACTION φ = [1; 1, 1, 1, …] — the slowest, most irrational number there is

Because φ = 1 + 1/φ, substituting into itself forever gives the simplest continued fraction possible: φ = 1 + 1/(1 + 1/(1 + …)) — all ones. Continued fractions with large terms converge fast (a big term is a good rational shortcut); φ's terms are the smallest they can be, so it converges as slowly as any number can. That is the precise sense in which φ is the “most irrational” number — the hardest to approximate by fractions. i13 generates it by iterating the map x → 1 + 1/x until it stops moving.

THE TECHNIQUE x → 1 + 1/x, iterated; the all-ones continued fraction

The demo iterates x → 1 + 1/x from a seed and watches it spiral into φ — the fixed point of the map: live demo

seed x =

HISTORY & CREDIT continued fractions · the all-ones expansion

“Every irrational is equally hard to pin down by fractions.” — not so: φ's all-ones continued fraction makes it the worst-approximable, the most stubbornly irrational number of all. cited

the identity · φ = 1 + 1/φ unrolled forever = [1;1,1,1,…].
slowest convergence · Hurwitz's theorem: φ (and its equivalents) are the extremal case of rational approximation.
a fixed point · x → 1+1/x converges to φ from any positive seed — the generative face of the golden ratio.

The simplest continued fraction is the hardest number to approximate. φ is extremal by being minimal. [1;1,1,1,…]

RECOMMEND FOR I-13 φ from the map x→1+1/x, on the compiler

On the canonical compiler, iterating x → 1 + 1/x twenty-five times converges to 1.6180339887… — ten digits, from nothing but a one and a divide:

$ i13 run g_continuedfraction.i13 # x -> 1 + 1/x, iterated RUN OK · 412 step(s) · peak stack 4 · call depth 26 phi = 1.6180339887802426 -- the all-ones continued fraction, converged

Recommend: the continued fraction is φ as a generator — the fixed point of x→1+1/x, reached from any seed. i13 lands ten digits deep in twenty-five steps. It is the cleanest statement of the batch's engine: a constant produced by iterating a rule until it stops changing. Not a keeper (convergence to a fixed point is a property every correct iteration shares — witnessed), but the purest picture of “generated, not stored,” and the reason φ is the most irrational number there is.