◄ WORLD V · SONNY 5DART 353 · a helldive at the net

THE GOLDEN SPIRAL a spiral that grows by φ every quarter turn — the same shape at every scale

A logarithmic spiral that widens by φ each quarter turn: after a full turn it is φ⁴ ≈ 6.854× larger, and it looks exactly the same at every scale — self-similar, the shape scaling can never change. Draw squares whose sides are Fibonacci numbers, nest them, and a quarter-circle in each traces the approximation; the true curve is the limit. It is φ's self-reference made geometric: φ² = φ + 1 says one turn's growth contains the last's plus a whole. Nautilus shells, spiral galaxies, and low-pressure systems approximate it — growth that keeps its proportion.

THE TECHNIQUE r grows by φ per quarter turn; φ⁴ per full turn

The demo grows the radius by φ each quarter turn — and after four, the full-turn factor φ⁴ ≈ 6.854: live demo







  
HISTORY & CREDIT the logarithmic spiral · Bernoulli's spira mirabilis

  
“Spirals get wider in a complicated way.” — the golden (logarithmic) spiral grows by a constant factor per turn, so it is the same shape at every magnification. Jacob Bernoulli called it the ‘marvelous spiral’ and had it on his grave. cited

  
the curve · a logarithmic spiral with growth factor φ per quarter turn.
self-similar · Jacob Bernoulli's spira mirabilis — unchanged by scaling (eadem mutata resurgo).
the squares · nested Fibonacci squares approximate it; the limit is the true spiral.

  
Growth that never changes shape, only size — φ's self-reference drawn as a curve. spira mirabilis






  
RECOMMEND FOR I-13 the per-turn growth φ⁴, on the compiler

  
On the canonical compiler, growing by φ four times (a full turn) gives φ⁴ = 6.854 — the self-similar scale factor, from the Fibonacci-ratio φ:

  
$ i13 run g_goldenspiral.i13   # phi^4 : growth per full turn
RUN OK · 59415 step(s) · peak stack 4 · call depth 16
phi = 1.618032786885246
per_turn = 6.8540816015509645     -- phi^4, the full-turn scale factor

  
Recommend: the golden spiral is φ as a self-similar curve — growth that keeps its proportion, the same at every scale — and i13 grounds the per-turn factor φ⁴ = 6.854. Not a keeper (a shape is not a mechanism), but the dart that makes φ's defining identity φ²=φ+1 visible as geometry, and closes the visual arc from angle (352) to spiral.