Put any length through 1.618

φ = 1.6180339887498948, the positive solution of x² = x + 1. Two consequences follow from that equation and they are the reason the number is useful: subtracting 1 gives its own reciprocal (0.6180339887498948), and adding 1 gives its square (2.618033988749895). No other positive number behaves that way, which is why a golden rectangle keeps producing golden rectangles when you cut squares off it.

Multiply the width by 0.618034 for the larger region and subtract that from the total for the smaller one. A 1200px container gives 741.64px and 458.36px, so a two-column layout with a 32px gap would be 725.64px of content beside 442.36px of sidebar. Taking 0.618 and 0.382 separately and rounding both is the usual way to end up a pixel short of the container.

Mostly no, and the nautilus is flatly wrong. A nautilus shell is a logarithmic spiral whose radius grows by roughly 1.33 per quarter turn, not 1.618, and the measurement has been repeated often enough that the claim survives only in design blogs. The Parthenon claims depend entirely on which rectangle you choose to draw over the photograph — include the steps and you get one number, exclude them and you get another. Gustav Fechner's 1876 experiments on rectangle preference are the origin of the aesthetic story, and later replications found the preference weak and heavily dependent on how the question is asked.

Because consecutive Fibonacci ratios converge on φ rather than equalling it, alternating above and below as they go. 21 ÷ 13 is 1.615385, which is 0.16% low; 34 ÷ 21 is 1.619048, 0.06% high; 89 ÷ 55 is 1.618182, and by then the error is 0.009%. If you are laying out on an 8-point grid, Fibonacci numbers are often the better choice precisely because they are whole numbers — the ratio between them is close enough that nobody will see the difference.

Only if the page has a genuine hero, because the steps get large very quickly. From a 16px base, the fourth step up is 109.67px, which on a 375px-wide phone fits about three words per line. Ratios between 1.2 and 1.333 give a usable hierarchy across five or six steps where 1.618 gives you three; the typography scale generator renders every step so you can see where the ratio stops being practical.

The rule of thirds puts the dividing line at 33.333% and the golden section puts it at 38.197%, a gap that is invisible on a business card and obvious on a billboard. On a 1920px frame the two lines sit 93.4px apart — 640px against 733.4px. Photographers reach for thirds because a viewfinder can draw it as a grid; layout tools reach for φ because it keeps working when you subdivide the result again.

In the golden angle, 360° ÷ φ² = 137.5078°, which is the most useful of the lot. Stepping that far around a circle each time never repeats a position, so it is how sunflower florets pack without lining up and how a palette generator can hand out an arbitrary number of distinguishable hues — walk the color wheel in 137.5° increments and the twentieth hue is still nowhere near the first.