Fibonacci rectangles

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The series of numbers we now call the Fibonacci series were probably first discovered in the 2nd century B.C. in India. They take their current name from the great 12th century Italian mathematician, Leonardo of Pisa, better known by the abbreviation of the Latin for ‘son of Bonacci’, Fibonacci.

Fibonacci’s greatest achievement was the introduction to Western culture of something most of us use every day – the Arabic system of numbers, 0 – 9. Before that, calculations had to be performed using the cumbersome Roman numerals, I, II, III, IV etc.

But among many other contributions Fibonacci posed a rather strange problem. Imagine placing a baby pair of a very special breed of rabbits in a field. They are special, first of all, because they live forever but also because they are very orderly in their breeding habits. They begin to mate at the age of one month and two young are born a month later, after which they immediately mate again. At the end of any given month, how many pairs of these immortal rabbits will there be in the field?

The answer for the first month is easy – there is one pair and they now mate. At the end of the second month there are two pairs, because the first pair have given birth. The first pair mate again but their offspring are too young. At the end of month three there are three pairs, two of which are old enough to mate. So far, we have 1 pair in month 1, 2 pairs in month 2, and 3 pairs in month 3, but things are about to get interesting. At the end of month four, there at 5 pairs:  two pairs have been born and one is newly ready to breed, making three breeding pairs. Month 5, there are the 5 pairs we had already, plus the offspring of the three breeding pairs, that makes eight.

All this gets mind boggling very quickly, but Fibonacci noticed a pattern. In any one month, let’s call the number of pairs ready to breed x and the number of newborn pairs y. The total number of pairs is (x + y). Now think about the next month. The x breeding pairs give birth to another x pairs, who are too young to breed. So now the total number of pairs is (x + y) + x, with (x + y) of them ready to breed. Don’t worry, we only have to stay sane for another month.
In the following month, the (x + y) breeding pairs will give birth to (x + y) pairs so the total is now (x + y) + ((x + y) + x). And we can stop because we now have enough information to see the pattern. Start in any month, call it month A:-

Month A: x + y pairs
Month B: (x + y) + x pairs
Month C: (x + y) + ((x + y) + x) pairs.

Can you see it? No matter which month we call A, if we label the number of breeding pairs x and the immature pairs y, then the total in month C is just the sum of the two previous months. Starting with that one pair we have 1, 2, 3, 5, 8, 13, 21, 34… and so on forever.

In fact, to make things consistent, the series is usually written as 0, 1, 1, 2, 3, 5… (sometimes without the zero), so that the rules apply from the very beginning.

All this would be a serious waste of time and mental effort if it weren’t for the fact that the Fibonacci series crops up all over the place – in science, in nature, even (so it is claimed) in the stock market. There is something about this series that the universe loves.
In this design, which comes in a number of variants, the lengths of the sides of each successive square follow the rules of the Fibonacci series to produce this wonderfully structured rectangle.

ddx site catalogue - Fibonacci rectangles

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