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The Fibonacci Sequence
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Mathematics as a subject is said to have inspired different people in
different ways. Some of the world’s best paintings are said to have more
of a mathematical appeal than anything else. In the case of an Italian
mathematician, Leonardo Fibonacci, it led to the introduction of one the
best known series in mathematics. Determined to find out the breeding rate
of rabbits, the mathematician sat ruminating and mulling on the problem
and it eventually led an arithmetical sequence, now known as the Fibonacci
series, in the year 1225.
For the uninitiated, the series begins like this: 1, 1, 2, 3, 5, 8, 13,
21, 34, 55, 89, 144… Have you figured out the sequence? Each number in the
series is the sum of the previous two numbers. Amazingly simple, right?
Yet, this simple sequence has been found to have remarkable links with
other branches of mathematics and with the world of nature. How? Let’s
see.
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Try and examine a plant that sends out individual leaves from a single stem.
Find out two leaves directly above each other. Now count one of the pair and the
number of intervening leaves. The number will always be a Fibonacci one.
Similarly try this with a sunflower. Count the clockwise spirals of seeds on the
head of a sunflower. Next count the anti-clockwise spirals of the same flower.
The figures you determined will always be a Fibonacci number. What’s more,
they’ve been found to be two consecutive Fibonacci numbers. Someone has recorded
the fact that the largest number counted, as 144 and 233.
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Leonardo Fibonacci |
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Some experts have been so fascinated by the Fibonacci series that they have
calculated the ratio of successive numbers in the series. The two successive
numbers 144 and 233, for instance, are said to be in the ratio 1: 1.61805. And,
the ratio of 233 with its succeeding number 377 (remember 144 + 233) is 1:
1.6180257. Successive ratios were said to be close to the number 1.618033989.
What is interesting about this last fact is that this ratio is considered the
Golden Ratio known to mathematicians since 300 BC. The Golden Ratio number,
denoted by the Greek letter phi ( ),
is said to be a never-ending decimal. While constructing many of the world’s
buildings, the architects have made use of rectangular shapes whose height width
proportion has been that of the Golden Ratio or the Golden Mean.
The sides of each rectangle are in the proportion 1: 1.6180257.
Experts say that if a Golden Ratio rectangle is divided into a square and a
rectangle, the smaller rectangle repeats the same proportion! If the smaller
rectangle is further divided into a square and a rectangle, the same would be
true of the new small rectangle.
Yet
another uniqueness about the Golden Ratio number is that it is said to be the
only number which can be squared by adding one to it. That is, in mathematical
terms, 2
= (
+ 1). It is also said to be the only number that can be turned into its own
reciprocal by subtracting one from it. That is, 1/
=
- 1.
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