Turn a sunflower one way and you count thirty-four spirals in one direction and fifty-five in the other, and the two numbers are consecutive Fibonacci numbers — which is why the count is always the same. The reason is a single angle: 137.5 degrees. It is the only angle at which a plant can lay out its seeds so that no two of them ever line up radially.
The golden angle is 360 times the reciprocal of the golden ratio, and it comes out at 137.507764 degrees. It is the only angle that makes the spiral close on itself without ever repeating a position, and the only angle at which consecutive seeds never line up in a straight line back to the centre.
Set the angle to anything else and the same failure appears: at 120 degrees the seeds form three rigid spokes, at 90 degrees they form a square grid, at 180 degrees they form a straight line. Drag the slider away from the golden angle and the head develops scars, exactly as a real one does.
The number of spirals in each direction is the number of seeds in a row of the spiral lattice, and for the golden angle those are consecutive Fibonacci numbers: 21 and 34, 34 and 55, 55 and 89. The pairs are not random; they are the ones that keep the gap between neighbouring seeds even.
The overlap is the point. Each floret is pressed against the ones around it, and the packing that results is the densest arrangement of equal circles known — and it is the same packing that shows up in the seed head of a daisy, the arrangement of scales on a pinecone, and the packing of seeds in a pomegranate.