Fibonacci Series and the Golden Ratio

Fibonacci Series

How was the term ‘Fibonacci’ derived? Are you aware of the Fibonacci series of numbers? Does math fascinate you? If it does, you’re in the right place!

The Italian mathematician named ‘Fibonacci’ was the inspiration behind the naming of the series. If a series of numbers follow a specific integer sequence as given below, then the series or sequence is termed as Fibonacci series.

Fibonacci Series: 0, 1, 1, 2, 3, 5, 8, 13,

A Fibonacci series of numbers comprises of terms that are subsequently placed one after another; where every number is the su of the preceding two numbers. A Fibonacci sequence typically begins with a 0 and 1 or 1, 1 based on the choice of the starting point. Fibonacci sequence of numbers is often described to be a form of recurrence relation.

The origin of the Fibonacci sequence dates back to 1202. You can find the introductory beginnings to the series in a book authored by Fibonacci called “Liber Abaci”. This specific series is often said to have evolved with the evolution of Indian mathematics several years ago. You can find several close linking and connections between the Fibonacci series and the Golden Ratio. This series has close resemblance or is a lot related to Lucas numbers too.

The Golden Ratio:

Golden ratio is a ratio commonly used in mathematics. This ratio is defined of two terms which are said to be in a golden ration if and only if the ratio is equal to the ratio of the sum of the two quantities to the quantity larger of the two.

Symbolically: [(a + b)/a=a/b]; where a>b.

The golden ration is denoted or symbolized by the alphabet phi (φ). There are several other names that are also often used to describe the golden ration; some of which are described below.

  • Golden Mean
  • Golden Section
  • Extreme ratio
  • Mean Ratio

There is a very close relationship of the Fibonacci numbers with the golden ratio. It is observed that, the golden ratio of any two consecutive Fibonacci numbers is very close to the value of φ which is equivalent to 1.6180… This rule extends to a greater point to state that the value of the ratio becomes much closer to the approximated value with an increase in the value of the Fibonacci pair.

Clearly, in mathematics when there is one rule, it is often found to have an inverse application as well; much like in the aforementioned case where the ratio value obtained (Golden ratio) can be used to determine the pair of Fibonacci numbers used itself. The result of this produces a distinct whole number.

Formulae:

Xn = {φ – (1 – φ) n}/√5

The Golden Rectangle:

Have you heard of The Golden Rectangle? Have you tried calculating the ration of a rectangle’s length to its width? Don’t despair. An interesting theory behind the golden rectangle is that if you design a very beautiful rectangle and measure the ratio of its longer side to the shorter width, you’ll find it to be close to 1.6. However, the ratio should be very close to 1.6 and to do that one has to draw the most beautiful golden triangle of all. An accurately drawn golden triangle when divided into a square and a rectangle; you’ll discover the rectangle to be another golden rectangle while the sides of the rectangle being in sequence of Fibonacci series.

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