Alex's Adventures in Numberland - Alex Bellos [119]
A sunflower with 34 anticlockwise and 21 clockwise spirals.
The Liber Abaci set up the sequence like this: say that you have a pair of rabbits, and after one month the pair gives birth to another pair. If every adult pair of rabbits gives birth to a pair of baby rabbits every month, and it takes one month for the baby rabbits to become adults, how many rabbits are produced from the first pair in a year?
The answer is found by counting rabbits month by month. In the first month, there is just one pair. In the second there are two, as the original pair have given birth to a pair. In the third month there are three, since the original pair have again bred, but the first pair are only just adults. In the fourth month the two adult pairs breed, adding two to the population of three. The Fibonacci sequence is the month-on-month total of pairs:
An important feature of the Fibonacci sequence is that it is recurrent, which means that each new term is generated by the values of previous terms. This helps explain why the Fibonacci numbers are so prevalent in natural systems. Many life forms grow by a process of recurrence.
There are many examples in nature of Fibonacci numbers, and one of my favourites concerns the reproductive patterns of bees. A male bee, or drone, has just one parent: his mother. Female bees, however, have two parents: a mother and a father. So, a drone has three grandparents, five great-grandparents, eight great-grandparents, and so on. Plotting a chart of the drone’s ancestry (as in the diagram overleaf), we find that the number of relatives he has per generation is always a Fibonacci number.
In addition to its association with fruit, promiscuous rodents and flying insects, the Fibonacci sequence has many absorbing mathematical properties. Listing the first 20 numbers will help us see the patterns. Each Fibonacci number is traditionally written using an F with a subscript to denote the position of that number in the sequence:
A chart tracing the ancestral history of one male bee (shown at the bottom).
(F0
0)
F1
1
F2
1
F3
2
>
F4
3
F5
5
F6
8
F7
13
F8
21
F9
34
F
55
F11
89
F12
144
F13
233
F14
377
F15
610
F16
987
F17
1597
F18
2584
F19
4181
F20
6765
Upon closer examination, we see that the sequence regenerates itself in many surprising ways. Look at F3, F6, F9,…, in other words, every third F-number. They are all divisible by 2. Compare this with F4, F8, F12,…, or every fourth F-number – they are all divisible by 3. Every fifth F-number is divisible by 5; every sixth F-number, divisible by 8; and every seventh number by 13. The divisors are precisely the F-numbers in sequence.
Another amazing example comes from , or . This number is equal to the sum of:
.0
.01
.001
.0002
.000005
.0000008
.00000013
.000000021
.0000000034
.00000000055
.000000000089
.0000000000144
So, the Fibonacci sequence pops its head up again.
Here’s another interesting mathematical property of the sequence. Take any three consecutive F-numbers. The first one multiplied by the third one is always one different from the second one squared:
This property is the basis of a centuries-old magic trick, in which it is possible to cut up a square of 64 unit squares into four pieces and reassemble them to make a square of 65 pieces. Here’s how it’s done: draw a square of 64 unit squares. It has a side length of 8. In the sequence, the two F-numbers preceding 8 are 5 and 3. Divide the square up using the lengths of 5 and 3, as in the first image below. The pieces can be reassembled to make a rectangle with sides the length of 5 and 13, which has an area of 65.
The trick is explained by the fact that the shapes are not a perfect fit. Though it is not that obvious to the naked eye, there is a long thin gap along the middle diagonal with an area of one unit.