Alex's Adventures in Numberland - Alex Bellos [122]
Nautilus shell.
Falcons descend on their prey in a logarithmic spiral.
Plants, as well as birds of prey, move to the music of phi. When a plant grows, it needs to position its leaves around the stem in such a way as to maximize the amount of sunlight that falls on each leaf. That’s why plant leaves aren’t directly above each other; if that were the case, the bottom ones would get no sunlight at all.
As the stem goes higher, each new leaf appears at a fixed angle around the stem from the previous leaf. The stem sprouts a leaf at a predetermined rotation, as in the diagram on the following page.
What is the fixed angle that maximizes sunlight for the leaves, the angle that will spread out the leaves around the stem so that they overlap as little as possible? It is not 180 degrees, or a half turn, because the third leaf would be directly above the first. The angle is not 90 degrees, or a quarter turn, because if this were the case, the fifth leaf would be directly over the first – and also the first three leaves would be using only one side of the stem, which would be a waste of the sunlight available on the other side. The angle that provides the best arrangement is 137.5 degrees, and the diagram above shows where the leaves would be positioned if successive leaves are always separated by this angle. The first three leaves are positioned well apart from each other. The next two, leaves four and five, are separated by more than 50 degrees from their nearest leaves, which still gives them a good amount of room. The sixth leaf is at 32.5 degrees from the first. This is closer to a leaf than any previous one, which it has to be since there are more leaves, yet the distance is still a pretty wide berth.
How leaves spiral up a stem.
The angle of 137.5 degrees is known as the golden angle. It is the angle we get when we divide the full rotation of a circle according to the golden ratio. In other words when we divide 360 degrees into two angles such that the ratio of the larger angle to the smaller angle is phi, or 1.618. The two angles are 222.5 degrees and 137.5 degrees, to one decimal place. The smaller one is known as the golden angle.
The mathematical reason why the golden angle produces the best leaf arrangement around a stem is linked to the concept of irrational numbers, which a those numbers that cannot be expressed as fractions. If an angle is an irrational number, no matter how many times you turn it around a circle you will never get back to where you started. It may sound Orwellian, but some irrational numbers are more irrational than others. And no number is more irrational than the golden ratio. (There’s a brief explanation why in the appendix.)
The golden angle.
The golden angle explains why you generally find on a plant stem that the number of leaves and number of turns before a leaf sprouts more or less directly above the first one is a Fibonacci number. For example, roses have 5 leaves every 2 turns, asters have 8 leaves for every 3 turns and almond trees have 13 leaves every 5 turns. Fibonacci numbers occur because they provide the nearest whole-number ratios for the golden angle. If a plant sprouts 8 leaves for every 3 turns, each leaf occurs every turn, or every 135 degrees, a very good approximation for the golden angle.
The unique properties of the golden angle are most strikingly seen in seed arrangements. Imagine that a flower head produces seeds from the centre point at a fixed angle of rotation. When new seeds emerge, they push the older seeds further out from the centre. These three diagrams show the patterns of seeds that emerge with three different fixed angles: just below the golden angle,