THE METALLIC RATIO — a dart

THE METALLIC RATIOS φ is the first of a family — silver, bronze, and on — each x = k + 1/x

The golden ratio solves x = 1 + 1/x. Change the 1 to any positive integer k and you get the metallic ratios: x = k + 1/x. k=1 is golden (≈1.618); k=2 is silver (1+√2 ≈ 2.414, the ratio of a square's diagonal-plus-side); k=3 is bronze. Each has its own continued fraction (all ks) and its own Fibonacci-like sequence (the “k-bonacci” via a(n)=k·a(n−1)+a(n−2)). φ is not unique in kind — it is the smallest, the k=1 door into a whole metallic family.

THE TECHNIQUE x = k + 1/x ; k=1 golden, k=2 silver (1+√2), k=3 bronze

The demo iterates x → k + 1/x for several k — golden, silver, bronze — each converging to its metallic ratio: live demo

k =

HISTORY & CREDIT Vera de Spinadel, 1990s

“The golden ratio is one-of-a-kind.” — it is the first of an infinite metallic family; silver and bronze are its siblings, each a k + 1/x fixed point. cited

the family · x = k + 1/x → x = (k + √(k²+4))/2.
1990s · Vera W. de Spinadel — named and unified the metallic means.
silver · 1+√2, the Pell numbers' ratio (1,2,5,12,29…).

One template, a metal for every integer; φ the humble first. The golden ratio, generalized. de Spinadel 1990s

RECOMMEND FOR I-13 golden and silver, from x=k+1/x, on the compiler

On the canonical compiler, iterating x → k + 1/x gives golden (k=1) = 1.618 and silver (k=2) = 2.414 = 1+√2:

$ i13 run g_metallic.i13 # x -> k + 1/x RUN OK · 875 step(s) · peak stack 4 · call depth 26 golden = 1.6180339887802426 -- k=1 silver = 2.414213562373095 -- k=2 = 1 + sqrt(2)

Recommend: the metallic ratios place φ in its family — the k=1 member of x=k+1/x, with silver (1+√2) and bronze its siblings — and i13 forges golden and silver by the same iteration. Not a keeper (a parameterized fixed point is the continued-fraction mechanism again), but the dart that dissolves the golden ratio's mystique: it is the first and smallest of an endless metallic ladder.