Every curve here is scaled so that it ends on the same circle, which leaves only one thing to compare: how it got there. The logarithmic spiral meets every radius at the same angle, so it can grow without changing shape — which is why the shell, the horn and the galaxy are all drawn with that one. The Archimedean and Fermat spirals cannot: their angle creeps toward a right angle and the outer turns flatten into circles. The Fibonacci spiral, meanwhile, is a chain of quarter circles doing an impression of the golden one.
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All of them on one center, each scaled to end on the dotted circle. The three of nature are dashed. Every curve starts on the ray going right.
Each turn, crossing the ray
A tick wherever the curve comes back round to the ray it started on, from the center on the left to the rim on the right. Even ticks, bunching ticks, ticks that spread apart: that is the whole difference between these curves, in one line each.
The logarithmic family holds the same angle to the radius everywhere — that is what equiangular means, and the growth per turn is the same anywhere along the curve. For the Archimedean and Fermat spirals the figures are read at the outer end, and both keep changing: wind them out further and the angle climbs toward 90° while the growth per turn falls toward 1. The Fibonacci chain has no single angle at all, so the range along its last turn is given instead. Pitch is simply 90° less the angle to the radius; it is the way astronomers quote a galaxy.
What tells them apart
Draw the radius out to a point on a spiral and then the tangent there, and measure between them. For the logarithmic spiral that angle is the same at every point, near the middle or far out, and it fixes everything else: the curve is a scaled copy of itself, and the radius multiplies by the same factor on every turn. Archimedes' spiral instead adds the same length each turn, so its angle starts near zero and climbs; Fermat's adds the same area, so its turns crowd together. Turn on the angle markers and watch which of the three reads the same number at all three radii.
The nautilus is not the golden spiral
Both are logarithmic, and there the resemblance stops. The golden spiral multiplies its radius by φ⁴ = 6.854 every turn. Measured nautilus shells come in near 3.2, which is roughly the value that makes each whorl sit against the one before it — a shell built to 6.854 would fly apart into a loose coil with nothing to hold it. Put the two cards next to each other, or overlay them, and the claim does not survive the picture.
Why living things keep picking one
A nautilus cannot rebuild its shell as it grows, so it adds to the open end instead; a horn and a tusk do the same thing, and so does a claw. If each new piece is a scaled copy of the last, the only curve the result can follow is the logarithmic one. A galaxy arrives at it from the other direction: the arms are a density wave in a disc whose rotation is nearly flat, and that winds a pattern up at a constant pitch. Same curve, different reason.