Nature

Mathematics is widespread in nature, and mathematical concepts are essential to understanding the biosphere, the rocks and oceans, and the atmosphere. This article explores a few examples.

The Fibonacci Series

In 1202 a monk in Italy, by the name of Leonardo Pisano Fibonacci, wanted to know how fast rabbits could breed in ideal circumstances. Suppose a newly born pair of rabbits, one male, one female, are put in a field. Rabbits are able to mate at the age of 1 month. So at the end of its second month, a female can produce another pair of rabbits. Suppose that these rabbits never die and that the female always produces one new pair (one male, one female) every month from the second month on. The puzzle that Fibonacci posed was: How many pairs would there be after 1 year?

At the end of the first month, they mate, but there is still only one pair.
At the end of the second month the female produces a new pair, so now there are two pairs of rabbits in the field.
At the end of the third month, the original female produces a second pair, making three pairs in the field.
At the end of the fourth month, the original female has produced yet another new pair, the female born two months ago produces her first pair also, making five pairs.

The resulting series of numbers, 1, 1, 2, 3, 5, 8, 13, 21, 34, …, is known as the Fibonacci series. Fibonacci's experiment is not very realistic, of course, because it implies that brothers and sisters mate, which leads to genetic problems. But the Fibonacci series is puzzlingly common in nature.

Bees. The Fibonacci series is evident in generations of honeybees. For instance, in a colony of honeybees there is one special female called the queen. There are many worker bees who are female too, but unlike the queen bee, they do not produce eggs. Then there are drone bees who are male and do no work. Males are produced by the queen's unfertilized eggs, so male bees have only a mother but no father. In contrast, females are produced when the queen has mated with a male, and so females have two parents. Females usually end up as worker bees but some are fed with a special substance, called "royal jelly," which makes them grow into queens ready to start a new colony when the bees form a swarm and leave their hive in search of a place to build a new nest.

Let's look at the family tree of a male drone bee ("he").

He had one parent, a female.
He has two grandparents, since his mother had two parents, a male and a female.
He has three great-grandparents: his grandmother had two parents but his grandfather had only one.
How many great-great-grandparents did he have?

Here is the sequence:

Number of	parents:	grand parents:	great-grand parents:	great, great grand parents:	gt, gt, gt grand parents:
of a male bee:	1	2	3	5	8
of a female bee:	2	3	5	8	13

Flowers and Other Plants. Another example of the Fibonacci series is the number of petals of flowers: lilies and iris have three petals; buttercups have five petals; some delphiniums have eight; corn marigolds have thirteen petals; some asters have twenty-one whereas daisies can be found with thirty-four or fifty-five petals. The series can also be found in the spiral arrangement of seeds on flowerheads, for instance on sunflowers, and in the structure of pinecones. In both cases the reason seems to be that this forms an optimal packing of the seeds (or cone studs) so that, no matter how large the seed-head (or cones), they are uniformly packed, and about the same size.

The Fibonacci series also appears in the position of a sequence of leaves on a stem. It should be noted that among plants there are other number sequences and aberrations. In other words, the Fibonacci series is really not a universal law, but only a fascinatingly prevalent tendency in nature.

The Golden Number (Phi)

If we take the ratio of two successive numbers in a Fibonacci series (1, 1, 2, 3, 5, 8, 13, …), dividing each number by the number before it, we will find the following series of numbers:

The ratio seems to be approaching a particular value known as the golden number, or Phi (ϕ). It has the value of ≈ 1.61804. The golden number is an amazingly universal constant. It turns out that ϕ = 1 + 1/ϕ, or ϕ² = ϕ + 1.

Plants grow from a single tiny group of cells right at the tip of any growing plant, called the meristem. There is a separate meristem at the end of each branch or twig and it is here that new cells are formed. Once formed, they grow in size. Cells earlier down the stem expand and so the growing point rises. These cells grow in a spiral fashion, as if the stem turns by an angle and then a new cell appears, turning again and then another new cell is formed and so on. These cells may then become a new branch, or perhaps on a flower become petals and stamens.

The amazing thing is that a single fixed angle can produce the optimal design no matter how big the plant grows. If this angle is an exact fraction of a full turn, for example, ⅓ (120°), then leaves of a vertical branch will be on top of each other. The fraction needs to be an irrational number. It turns out that if there are ϕ (or approximately 1.6) leaves per turn, then each leaf gets the maximum exposure to light, casting the least shadow on the others. This also gives the best possible area exposed to falling rain so the rain is directed back along the leaf and down the stem to the roots. For flowers or petals, it gives the best possible exposure to insects to attract them for pollination. And this angle optimizes the seeds on a sunflower. The Fibonacci numbers merely form the best whole number approximations to the golden number, ϕ.

Nature

The Fibonacci Series

The Golden Number (Phi)

Bibliography