Pine cones, sunflowers, artichokes, daisies... A great number of plants exhibit spirals that appear in two sets according to their orientation, clockwise and counterclockwise. The study of these arrangements is called Phyllotaxis. Most often, when counting the spirals in each set, we get a pair of consecutive Fibonacci numbers. (These are 1, 1, 2, 3, 5, 8, 13, 21,... each the sum of the previous two.) We will talk about a simple mathematical model of meristem growth based on Dynamical Systems theory, which might give us insights into this remarkable phenomenon observed for centuries.