This mathematically
simple sequence was first presented arithmetically by Leonardo Fibonacci of
Pisa at the turn of the 12th century and is known as “the Fibonacci Series” in
which each successive figure is the sum of the previous two; that is, 1, 2, 3,
5, 8, 13, 21, 34, 55, 89, 144, and so on. It has wide practical applications
and is also a fundamental principle in nature underlying the structure of
animal and plant organisms.
When he was young,
Fibonacci spent years in North Africa learning mathematics from Arab scholars.
His book, Liber Abaci, which was published in 1202, introduced the decimal
system to Europe. Roman numerals had been used in Europe to that point and
there was no sign for zero.
Interestingly,
Fibonacci used the theoretical reproduction of rabbits under ideal
circumstances to demonstrate how the series worked. Beginning with two new-born
rabbits able to mate at one month and with the female able to produce another pair
of rabbits (always assuming a male and a female) at the end of the each month,
it would work in rabbit terms over twelve months as given above; that is, 1, 1,
2, 3, 5, 8, 13, 21, 34, 55, 89, 144. Theoretically, the rabbit farmer could
calculate in advance how many rabbits he would have in a year.
A couple of examples
of calculations follow:
–
6 months = 23 = 8
rabbits
–
12 months = 24 x
32 = 144 rabbits
–
18 months = 23 x
17 x 19 = 2,584 rabbits
As mathematicians
experimented with the Fibonacci Series, other patterns began to emerge,
including phi or 1.618 (also known as the “golden mean” or the Divine
Proportion); that is, the ratio of each successive pair of numbers in this
series approximates 1.618. After the fortieth number in the series, the ratio
is accurate to 15 decimal places: 1.618033988749895. This had actually been known at the time of Euclid but had
to wait until the 20th century to be given the name phi.
Many Renaissance
artists, Leonardo da Vinci among them, made use of this calculation. Da Vinci
constructed many of his models and paintings with phi and the Fibonacci Series
in mind. His famous drawing of the Vitruvian Man was designed according to
these principles, as was what is perhaps the most famous of all paintings, the
Mona Lisa. The name “Vitruvian” honours Marcus Vitruvius, the renowned Roman
architect to the Emperor Augustus who wrote about the Divine Proportion in his
De Architectura.
Phi pops up everywhere in the universe from the way that galaxies spiral, pine cones are patterned, petals arrange themselves, and shells are spiralled, to the structure of the double helix of the DNA. Musical scales are based on it and, beginning with R N Elliott in 1927, people have even developed theories to anticipate stock exchange movements based on Fibonacci ‘waves’.
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