◄ WORLD V · SONNY 5DART 342 · a helldive at the net

THE GOLDEN RATIO the number a line divides into so its whole is to its greater as its greater is to its less

Cut a line so that the whole is to the larger part as the larger is to the smaller. That single condition — (a+b)/a = a/b — forces one number: φ = (1+√5)/2 ≈ 1.618…. It is the root of x² = x + 1, so it is the number that is its own reciprocal plus one (φ = 1 + 1/φ) and its own square minus one (φ² = φ + 1). Everything in this batch is a face of that self-reference: the Fibonacci recurrence converges to it, the continued fraction generates it, the spiral grows by it. It is the thread.

THE TECHNIQUE φ = (1+√5)/2, the root of x² = x + 1

i13 has no square root — so it generates φ instead, as a ratio of consecutive Fibonacci numbers. The demo shows the ratio closing on the golden number: live demo







  
HISTORY & CREDIT Euclid · extreme and mean ratio

  
“The golden ratio is an ancient mystical constant.” — it is a definition: the one proportion where the whole relates to the part as the part to the remainder. The mysticism is later; the ratio is Euclid. cited

  
c.300 BCE · Euclid — the “division in extreme and mean ratio” (Elements VI, def. 3).
1509 · Luca Pacioli — De Divina Proportione (illustrated by Leonardo).
the name · “golden ratio” (goldener Schnitt) is 19th-century — Martin Ohm, 1835; the φ notation is Mark Barr, ~1900 (for Phidias).

  
One condition, one number, and a thread that runs through every dart that follows. Euclid / Pacioli






  
RECOMMEND FOR I-13 φ generated by a Fibonacci ratio, on the compiler

  
On the canonical compiler, F(16)/F(15) = 987/610 = 1.618032… — and the Fibonacci identity F(16)−F(15)−F(14) = 0 holds exactly:

  
$ i13 run g_goldenratio.i13   # phi as a Fibonacci ratio (no sqrt)
RUN OK · 132830 step(s) · peak stack 4 · call depth 16
phi = 1.618032786885246     -- F(16)/F(15), closing on the golden number
check = 0                   -- F(16) = F(15) + F(14), the recurrence holds

  
Recommend as the batch's root: φ is the fixed point everything here orbits. i13 cannot take a square root, so it does the more revealing thing — it generates φ from the integer recurrence, as F(16)/F(15), and the value is already right to six places. That is the theme in one line: the golden ratio is not stored, it is produced — by a rule so simple (add the last two) that it needs nothing but counting. Not a keeper (a constant is not a mechanism); the keeper shot waits at dart 347, where the same Fibonacci numbers become a canonical form.