The one large, well-established appearance of phi in the living world is phyllotaxis, the arrangement of leaves around a stem and of seeds, scales and florets in a head. Count the visible spirals in a sunflower head, a pine cone or a pineapple and you generally get two counts that are consecutive Fibonacci numbers: 34 and 55, or 55 and 89, and so on up.
The underlying quantity is the divergence angle between successive primordia as they are laid down at the growing tip, which converges on roughly 137.5 degrees, the golden angle, a full turn divided in the ratio phi.
The reason is the irrationality result. If the divergence angle were a simple fraction of a turn, say one third, then every third element would line up radially and the arrangement would leave large empty wedges. The worse the angle can be approximated by a fraction, the longer anything takes to line up, and phi is the worst-approximable number there is. The golden angle is therefore the one that never repeats and packs most evenly.
This is not folklore. Helmut Vogel in 1979 gave a simple model, place element n at angle n times 137.5 degrees and at radius proportional to the square root of n, which reproduces sunflower packing convincingly. Stephane Douady and Yves Couder in 1992 showed the mechanism experimentally, dropping magnetised droplets into a rotating dish of silicone oil and finding that a system in which new elements appear at intervals and are repelled by existing ones settles onto the golden angle by itself, as a dynamical attractor. No plant needs to know any mathematics. The angle falls out of local repulsion and steady growth.
The honest caveats: not every plant shows Fibonacci phyllotaxis, other patterns including Lucas-number spirals occur, and whether the golden angle is also optimal for light capture rather than only for packing is a separate question still argued over in the literature.
Sources A Better Way to Construct the Sunflower HeadHelmut Vogel with Phyllotaxis as a Physical Self-Organized Growth ProcessS. Douady and Y. Couder