The principal conclusion is that Fibonacci phyllotaxis follows as a mathematical necessity from the combination of an expanding apex and a suitable spacing mechanism for positioning new leaves. I have considered an inhibitory spacing mechanism at some length, as it is a plausible candidate. However, the same treatment would apply equally well to depletion of, or competition for, a compound by developing leaves, and could no doubt accommodate other ingredients. The mathematical principles involved are clear when it is assumed that only two leaves (the contacts) position a new leaf. There is some experimental evidence for this assumption. Nonetheless, it is not a precondition for Fibonacci phyllotaxis, since a computer model shows that this pattern is generated even when many leaves contribute to inhibition at a given point. Indeed, the Fibonacci pattern seems to be a robust and stable mathematical phenomenon, a finding which goes some way to explaining its widespread occurrence throughout the plant kingdom.