Take the Fibonacci sequence — each number the sum of the two before it. Tile squares with those side lengths, spiraling out. Connect the corners with quarter-arcs. The result is a logarithmic spiral whose growth factor per quarter-turn approaches the golden ratio:
$$ \varphi = \frac{1 + \sqrt{5}}{2} = \lim_{n\to\infty} \frac{F_{n+1}}{F_n} $$
The math is real. The "nature loves φ" claim is mostly post-hoc pattern-matching: nautilus shells aren't actually golden spirals, and the Parthenon's dimensions don't quite hit the ratio either. What is real is that the spiral solves a specific problem: if a plant adds new leaves at angle 360°/φ², no two leaves ever shadow each other for a very long time.
The growth-factor slider lets you walk through nearby spirals. At 1.20 you get the loose curl of an unfurling fern; at 2.4, the tight whorl of a snail. Only one of those is golden, and you will immediately see why people felt the need to invent the word.