The Fibonacci series is a series formed by the Fibonacci numbers.The numbers in this series are given as 0, 1, 1, 2, 3, 5, 8, 13, 21, 38, . . . This series is named after a renowned mathematicianLeonardo Pisano Bogollo, later known as Fibonacci.

In a Fibonacci series, every term is the sum of the preceding two terms, starting from 0 and 1 as the first and second terms. In some cases ‘0’ is omitted.

Given the first term, F_{0} and second term, F_{1} as ‘0’ and ‘1’,

the third term is given as, F_{2 }= 0 + 1 = 1

Similarly,

F_{3} = 1 + 1 = 2

F_{4} = 2 + 1 = 3

Therefore, to represent any (n+1)^{th} term in this series, we can give the expression as, F_{n} = F_{n-1} + F_{n-2}.

The fibonacci series spiral is the representation of Fibonacci numbers in pattern form. The spiral starts in a plane in the shape of a rectangle whose dimensions (length × breadth) follow the principle of a “Golden Ratio” (≈1.618), and is referred to as the “Golden Rectangle”.

Fibonacci series and Pascal's triangle

Another method to find the numbers in the Fibonacci series is Pascal’s triangle.

Pascal’s triangle is a triangular array comprising the binomial coefficients. In a Fibonacci series, Fibonacci numbers can be derived by calculating the sum of elements on the rising diagonal lines in Pascal’s triangle.

From this triangle, let the first term be 0 and the following terms can be calculated by summing the diagonal elements as given above.

Application of Fibonacci series in nature

1. Seed Heads

The head of a flower is also subject to the Fibonaccian Pattern. Typically, seeds are produced at the centre, and then migrate towards the outside to fill all the space. Sunflowers provide a great example of these spiralling patterns.

2. Flower Petals

The number of petals in a flower follows the Fibonacci sequence. Famous examples include the lily, which has three petals, buttercups, which have five petals, the chicory’s petals are 21, the daisy’s is 34, and so on.

3. Pine Cones

The seed pods on a pinecone are arranged in a spiral pattern. Each cone consists of a pair of spirals, each one spiraling upwards in opposing directions. The number of steps will almost always match a pair of consecutive Fibonacci numbers. For example, a 3-5 cone is a cone which meets at the back after three steps along the left spiral, and five steps along the right.

4. Tree Branches

The main trunk will grow until it produces a branch, which creates two growth points. Then, one of the new stems branches into two, while the other one lies dormant. This pattern of branching is repeated for each of the new stems.

5. Shells

The shape of a shell, a rectangle in which the ratio of the sides is equal to the golden mean (phi), can result in a nesting process that can be repeated into infinity — and which takes on the form of a spiral. It’s called the logarithmic spiral.

6. Hurricanes and Spiral Galaxies

Hurricanes and spiral galaxies also follow the familiar Fibonacci pattern. Our galaxy, The Milky Way has several spiral arms, each of them a logarithmic spiral of about 12 degrees.

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