Fibonacci Sequence in Nature

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Fibonacci's Rabbits
The original problem that Fibonacci investigated (in the year 1202) was about 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 one month so that at the end of its second month a female can produce another pair of rabbits. Suppose that our rabbits never dieand 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 will there be in one year?

1. At the end of the first month, they mate, but there is still one only 1 pair. 2. At the end of the second month the female produces a new pair, so now there are 2 pairs of rabbits in the field. 3. At the end of the third month, the original female produces a second pair, making 3 pairs in all in the field. 4. 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 5 pairs.

The number of pairs of rabbits in the field at the start of each month is 1, 1, 2, 3, 5, 8, 13, 21, 34, ...

Another view of the Rabbit's Family Tree:
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Both diagrams above represent the same information. Rabbits have been numbered to enable comparisons and to count them, as follows: * All the rabbits born in the same month are of the same generation and are on the same level in the tree. * The rabbits have been uniquely numbered so that in the same generation the new rabbits are numbered in the order of their parent's number. Thus 5, 6 and 7 are the children of 0, 1 and 2 respectively. * The rabbits labelled with a Fibonacci number are the children of the original rabbit (0) at the top of the tree. * There are a Fibonacci number of new rabbits in each generation, marked with a dot. * There are a Fibonacci number of rabbits in total from the top down to any single generation.

Dudeney's Cows
The English puzzlist, Henry E Dudeney (1857 - 1930, pronounced Dude-knee) wrote several excellent books of puzzles (see after this section). In one of them he adapts Fibonacci's Rabbits to cows, making the problem more realistic in the way we observed above. He gets round the problems by noticing that really, it is only the females that are interesting - er - I mean the number of females! He changes months into years and rabbits into bulls (male) and cows (females) in problem 175 in his book 536 puzzles and Curious Problems (1967, Souvenir press): If a cow produces its first she-calf at age two years and after that produces another single she-calf every year, how many she-calves are there after 12 years, assuming none die? This is a better simplification of the problem and quite realistic now. But Fibonacci does what mathematicians often do at first, simplify the problem and see what happens - and the series bearing his name does have lots of other interesting and practical applications as we see later.  So let's look at another real-life situation that is exactly modelled by Fibonacci's series - honeybees.

Honeybees and Family trees
There are over 30,000 species of bees and in most of them the bees live solitary lives. The one most of us know best is the honeybee and it, unusually, lives in a colony called a hive and they have an unusual Family Tree. In fact, there are many unusual features of honeybees and in this section we will show how the Fibonacci numbers count a honeybee's ancestors (in this section a "bee" will mean a "honeybee"). First, some unusual facts about honeybees such as: not all of them have two parents! 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 produce no eggs.  There are some drone bees who are male and do no work.  Males are produced by the queen's unfertilised eggs, so male bees only have a mother but no...
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