Real Life Applications of Fibonacci Sequence

Fibonacci Sequence, a series where each number is the sum of the two preceding ones, finds applications in nature, mathematics, and technology. The article explores the significance and applications of the Fibonacci Sequence in various fields, including nature, mathematics, technology, finance, cryptography, and poetry, offering insights and practical examples.

What is the Fibonacci Sequence?

Fibonacci sequence, also known as Fibonacci numbers, is defined as the sequence of numbers in which each number in the sequence is equal to the sum of two numbers before it. Fibonacci Sequence is given as:

Fibonacci Sequence = 0, 1, 1, 2, 3, 5, 8, 13, 21, …

Here, third term “1” is obtained by adding the first and second terms. (i.e., 0+1 = 1)

Similarly, “2” is obtained by adding the second and third terms (1+1 = 2)

“3” is obtained by adding the third and fourth terms (1+2) and so on.

For example, the next term after 21 can be found by adding 13 and 21. Therefore, the next term in the sequence is 34.

Applications of Fibonacci Sequence

Various applications of fibonacci sequence are:

In Flower Petals

Number of petals in a flower consistently follows the Fibonacci sequence. Famous examples include the lily, which has three petals, buttercups, which have five (pictured at left), the chicory's 21, the daisy's 34, and so on. Phi appears in petals on account of the ideal packing arrangement as selected by Darwinian processes; each petal is placed at 0.618034 per turn (out of a 360° circle) allowing for the best possible exposure to sunlight and other factors.

In Mathematics

Fibonacci sequence is used in number theory, algebra, and geometry. It has applications in the analysis of financial markets and computer algorithms.

In Biology

Fibonacci sequence appears in biological settings, such as the branching of trees, the arrangement of leaves on a stem, the flowering of artichokes, and the spiral arrangement of seeds in a sunflower.

In Computer Science

Fibonacci sequence is used in algorithms for tasks such as searching and sorting.

In Art and Design

Fibonacci sequence is used in art, architecture, and design to create aesthetically pleasing proportions and compositions.

In Finance

Fibonacci sequence is sometimes used in technical analysis of financial markets to identify potential levels of support and resistance.

In Fibonacci Series and Poetry (FIB)

Fib is explained as experimental Western poetry, similar to haiku, but based on the Fibonacci series. The typical Fib and another version of the modern Western haiku follow a strict structure. It is a copy of how characters were explained in ancient Sanskrit prosodies. A typical Fib is a six-line, 20-syllable poetry with a syllables count by lines of 1/1/2/3/5/8 – with many syllables as required.

Ancient form of contemporary haiku uses three or fewer lines and no more than 17 syllables. The only condition on a Fib is that the syllable count follows the Fibonacci Sequence.

In Application to Trading

One of the main applications of Fibonacci numbers outside of the realm of mathematics is in the area of stock market analysis. Many investors use what is called the Fibonacci Retracement Technique to estimate the action that the price of a particular stock will take, based on certain ratios found within the Fibonacci numbers.

The retracement uses lines across the 0, 23.6, 38.2, 50, 61.8, and 100 percentiles of the chosen high and low values. A trader would then use these estimates to buy stock when the value decreases to one of these percentages and sell stock as it peaks at another of the percentages.

In Fibonacci Sequence In Nature

Fibonacci can be found in nature not only in the famous rabbit experiment but also in beautiful flowers (Internet access, 12). On the head of a sunflower, the seeds are packed in a certain way so that they follow the pattern of the Fibonacci sequence. This spiral prevents the seed of the sunflower from crowding themselves out, thus helping them with survival. The petals of flowers and other plants may also be related to the Fibonacci sequence in the way that they create new petals

In Fibonacci in Coding

Recently Fibonacci sequence and golden ratio have been of great interest to researchers in many fields of science including high energy physics, quantum mechanics, Cryptography, and Coding. Raghu and Ravishankar(2015) developed a paper on the application of classical encryption techniques for securing data. (Raphael and Sundaram,2012) showed that communication may be secured by the use of Fibonacci numbers.

A similar application of Fibonacci in Cryptography is described here by a Simple Illustration. Suppose that the Original Message” CODE” is to be Encrypted. It is sent through an unsecured channel. The security key is chosen based on the Fibonacci number. Any one character may be chosen as the first security key to generate cipher text and then the Fibonacci sequence can be used.

Examples of Fibonacci Sequence

Example 1: Find the sum of the first 15 Fibonacci numbers.

Solution:

As we know,

Sum of Fibonacci Sequence:

⅀ F_i = F_{(n + 2)} - F₂

Thus,

Sum of first 15 Fibonacci Numbers = (15+2)^th term – 2^nd term

Sum of first 15 Fibonacci Numbers = 987 – 1 = 986

Example 2: Find the 5th Fibonacci number.

Solution:

As we know,

nth Fibonacci number is

F(x_n) = F(x_n-1) + F(x_n-2), for n>2

Then 5th Fibonacci number is,

F(x₅) = F(x_5-1) + F(x_5-2), for n=5

F(x₅) = F(x₄) + F(x₃)

F(x₅) = 2 + 1 = 3

Example 3: Find the next number when F14 = 377.

Solution:

Here,

F₁₅ = F₁₄ × Golden ratio = 377 × 1.618034 (up to 4 decimals)

F₁₅ = 609.9988 (up to 4 decimals), which is approximately 610

Hence, F₁₅ = 610

Example 4: Calculate the value of F(-6).

Solution:

As we know, F(-n) = (-1)n + 1.Fn

Here,

F(-6) = (-1)6 + 1.F₆

F(-6) = (-1) × 5 = -5

Practice Problems on Fibonacci Sequence

1. Describe how the Fibonacci sequence can be applied to model the growth of a rabbit population over several generations.
2. How can Fibonacci retracement levels be used to identify potential points of resistance or support in a stock’s price?
3. Write an algorithm that uses the Fibonacci sequence to implement a search method in a sorted array. Discuss its time complexity.
4. Develop a simple encryption method using Fibonacci numbers. Explain how the sequence helps in enhancing the security of the algorithm.
5. Identify the connection between the Fibonacci sequence and the golden ratio in a famous piece of architecture or art. Discuss how the proportions were derived.
6. Create a model using Fibonacci numbers to simulate the population growth of a species under ideal conditions. What assumptions does this model make?
7. Compose a short piece of music using rhythms or intervals based on the Fibonacci sequence. How does this influence the structure of the composition?
8. Implement a Fibonacci heap and compare its performance with a binary heap for priority queue operations. When is a Fibonacci heap more efficient?
9. Examine the arrangement of seeds in a sunflower and explain how the Fibonacci sequence appears in this pattern.
10. Solve an optimization problem using dynamic programming where Fibonacci numbers can be applied, such as calculating the minimum cost of a staircase.

Conclusion

In conclusion, the Fibonacci sequence, with its unique pattern of each number being the sum of the two preceding ones, holds significance across various fields. From nature's intricate designs to cryptography and trading strategies, its applications are diverse and profound. The sequence's connection to the golden ratio, its presence in natural phenomena, and its usefulness in algorithm design make it an essential tool in both theoretical and practical contexts.