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Friday, June 10th 2016

22:47

Bring on the Rabbits!

  • STATE OF EXISTENCE: Cool
  • IN THE BACKDROP: The hum of a new air conditioning system

Well, I've played my little game long enough, and I thank you for your patience. But now the time has come to start counting lagomorphs.

I hope that in my preceding discussion I have made it clear that the so-called golden ratio number φ (phi), has a life of its own, apart from the Fibonacci series. If you have caught on to that point, we are now ready to talk about the Fibonacci numbers and how they are intertwined with phi. (If you still say that phi is derived from the Fibonaccis, you're probably only teasing me, or you didn't read the preceding posts on the topic.)

Leonardo Bonacci (1170?-1250?) made a number of important contributions to math, the most significant of which is undoubtedly his promotion of the Indian/Arabic numerals in Europe, which made life a lot easier to anyone having to undertake any calculations. If you've heard of Leonardo at all, it was probably under his nickname, Fibonacci, and most likely in connection with the series of numbers that is designated after him.

The series is embedded in a puzzle that Fibonacci posed to his reading audience (Livio, 96):

A certain man put a pair of rabbits in a place surrounded on all sides by a wall. How many pairs of rabbits can be produced from that pair in a year if it is supposed that every month each pair begets a new pair which from the second month on becomes productive?

Please follow along with the picture as I walk us through the beginning of solving this puzzle. I might clarify, by the way, that we are counting the mature pairs only, just because that's how it's presented. The outcome would be no different if we counted immature pairs alongside the mature ones, except we would have only one numeral 1 at the outset, rather than two of them.

Fibonacci's Rabbits

  1. During the first month we have only the 1 original pair.
  2. In the second month the original pair will have produced one pair of offspring, which is not yet mature enough to produce any baby rabbits, so we are still holding at 1 mature pair.
  3. By the the time of the third month, the original pair will have given us another immature pair, and the earlier pair will now have matured, but not yet yielded any new rabbits. Thus the number we can put down for the third month for mature rabbits is 2.
  4. Come month 4, there is more offspring: one pair each from the original pair and the second generations pair. Also, the offspring from the newer pair will have matured. So the talley of mature rabbits for that month is 3.
  5. In the fifth month we get 5 mature rabbits, and each of the three that had matured in the previous month have contributed an as-yet immature pair.
  6. In the sixth month, the three previously immature ones will have matured and can get added to the previous five. Thus we have 8 mature pairs and five pairs that are only a month old--but probably can't wait until they are grown sufficiently to add to the pool.

I trust that you can see the pattern that is developing. Month by month the collection of rabbits is increasing. The numbers of mature pairs that we have now for the first six months are:

1 1 2 3 5 8

Each new number is created by taking the last two and adding them. So, let's continue this pattern until we reach the twelfth month, which will give us the answer to the specific puzzle that Fibonacci posed:

1 1 2 3 4 8 13 21 34 55

So, there will be 55 pairs of rabbits (110 individuals) living in the man's enclosure after a year. That doesn't sound too bad, but keep in mind that we are now entering a phase when the numbers will grow quite rapidly. Give it another three months, and the number will be 233 pairs, which is to say, of course, 466 grown rabbits. Then there will also be another 144 immature pairs (288 young rabbits), giving us a total of 754 rabbit hopping around, looking for lettuce and carrots.

As fans of the Fibonacci series know, and as I have intimated already, there is a close relationship between the Fibonacci numbers and φ. Take a number in the series and divided by the previous one, and you'll get some number that is in the vicinity of phi. So, let's look at two of such ratios.

1) The number of the twelfth month (55), divided by that of the eleventh (34): 55/34 = 1.617647059 ...

2) Since I brought up the fifteenth month, let's see what we get when we divide 233 by 144. The answer is 1.618055556 ...

Let us recall the approximate numerical value of phi: 1.618033988 ... . We can see that the longer we go on with the process, the closer we will get to phi. But there's something else that may not jump out at you immediately because of the many digits. Let us put these three numbers into numerical order, lower to higher:

1.617647059 -- 1.618033988 -- 1.618055556

The ratio of the 12th number to the 11th was below phi, whereas the estimate of the 15th to the 14th was a bit higher. I knew that would be the case, even before I tested it, just to make double sure. This is one of the interesting aspects of the relation of the Fibonacci series to φ: the different ratios of one number to its predecessor will always be an "approximation," though they get extremely close. But the direction of the error will alternate. The ratios with even numbers will be slightly lower than an more accurate rendering of φ, while the odd ones will exceed it just a little. Needless to say, as you have seen for yourself if you did the little exercise at the outset of this series, even with non-Fibonacci numbers the approximation gets very close, how much more within the Fibonacci series!

Next time: some more cool stuff about the Fib. numbers, as well as some theologial reflections.

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