Fibonacci numbers can also be seen in the arrangement of seeds on flowerheads. Here is a diagram of what a large sunflower or daisy might look if magnified. The centre is marked with a black dot.

You can see that the seeds seem to form spirals curving both to the left and to the right. If you count those spiralling to the right at the edge of the picture, there are 34. How many are spiralling the other way? You will see that these two numbers are neighbours in the Fibonacci series.

The same happens in real seed heads in nature. The reason seems to be that this forms an optimal packing of the seeds so that, no matter how large the seedhead, they are uniformly packed, all the seeds being the same size, no crowding in the centre and not too sparse at the edges.

If you count the spirals near the centre, in both directions, they will both be Fibonacci numbers. The spirals are patterns that the eye sees, "curvier" spirals appearing near the centre, flatter spirals (and more of them) appearing the farther out we go.

Here are some more pictures of 500, 1000 and 5000 seeds - click on them for the full picture:

Click on the image on the right for a Quicktime animation of 120 seeds appearing from a single central growing point. Each new seed is just phi (0-618) of a turn from the last one (or, equivalently, there are Phi (1-618) seeds per turn). The animation shows that, no matter how big the seed head gets, the seeds are always equally spaced. At all stages the Fibonacci

The same pattern shown by these dots (seeds) is followed if the dots then develop into leaves or branches or petals. Each dot only moves out directly from the central stem in a straight line.

This process models what happens in nature when the "growing tip" produces seeds in a spiral fashion. The only active area is the growing tip - the seeds only get bigger once they have appeared.

[This animation was produced by Maple. If there are N seeds in one frame, then the newest seed appears nearest the central dot, at 0618 of a turn from the angle at which the last appeared. A seed which is i frames "old" still keeps its original angle from the exact centre but will have moved out to a distance which is the square-root of i.]

Note that you will not always find the Fibonacci numbers in the number of petals or spirals on seed heads etc., although they often come close to the Fibonacci numbers.

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987 ..More.. E