Golden ratio explorer
Watch the golden rectangle build itself, divide a canvas at the golden section, and compare it honestly with the rule of thirds.
Result
5 squares removed — the leftover is still a golden rectangle (long : short = φ).
The orange piece is what's left — measure it: long side ÷ short side = φ again. That regress never ends; the spiral rides the squares inward.
How is this calculated?
Using the tool
The golden ratio is not magic — it is the one proportion that contains itself: cut a square off a golden rectangle and the piece left over is another golden rectangle, forever (that is literally what φ² = φ + 1 says). Watch the construction run, divide a real canvas at the golden section and compare it with the rule of thirds, and see how Fibonacci's whole numbers sneak up on φ — including why their famous spiral is a look-alike, not the real thing.
The math
φ is defined by a sentence, not a decimal: the whole is to the large part as the large part is to the small. Write that as algebra and you get φ² = φ + 1, whose positive root is (1+√5)/2 ≈ 1.618. That equation is why the rectangle regenerates itself — removing a 1×1 square from a φ×1 rectangle leaves φ−1 = 1/φ, the same proportion again. No other ratio does this, which is the honest reason φ shows up wherever self-similar growth does.
As a composition tool, the golden section is a way to divide consciously instead of centering by default: place the horizon, the eye, the doorway at 0.618 of the way across. The rule of thirds (0.667) is its rougher, easier cousin — photographers use thirds because viewfinders draw them, painters used the section because dividers construct it. The four crossings of either grid mark “off-center but anchored” — that part is craft, not mathematics.
And the honesty clause: Fibonacci's numbers really do converge to φ — each ratio lands on the other side and closer, forever — and sunflower heads and pinecones really do grow by them. But most “φ found in the Mona Lisa” overlays are drawn after the fact with generous rounding. Use the section as a deliberate instrument, not as a treasure hunt.