Application of the Fibonacci Sequence in Real-Life

Introduction

The Fibonacci Sequence is a unique and storied sequence of integers with diverse applications. Originally discovered in ancient India, the sequence has left its mark in history for over 2000 years. The first two elements of the sequence are defined explicitly as 1. Following these leading elements, the uniquem structure of the Fibonacci begins to take form. Starting with the third element, each element is defined as the sum of the two previous elements. Expressed mathematically, the Fibonacci Sequence is defined as a recurrence relation: FO = IFI =1Fn = Fn-l + Fn-2. Though simple and abstract in principle, the Fibonacci sequence features heavily in modern mathematics, and more unexpected areas of life. For example, the sequence is used in computer science to optimize data structures and algorithms. In the financial industry, Fibonacci numbers are used to predict market trends and analyze stock prices. The sequence has also been applied to the design of buildings, with architects using Fibonacci ratios to create visually appealing structures. In nature, the Fibonacci sequence is found in everything from the spirals of seashells to the arrangement of leaves on a stem. To discuss the application of Fibonacci sequence in real life, this essay would explore these and other fascinating real-world applications of this sequence.

Discovery

Though the Fibonacci Sequence was named after the Italian mathematician Leonardo Fibonacci (Leonardo Pisano), Fibonacci wasn’t actually the first to discover the sequence. The discovery of the Fibonacci Sequence dates all the way back to the 2nd or 3rd century B. C. E. through a long line of Indian mathematicians. The first of which was Pingala, who laid the groundwork for the sequence. Sometime in between A. D. 600 and 800, Virahanka expanded on Pingala’s work and created the Fibonacci Sequence that we know today. Virahanka’s work is now lost, but he is believed to be the first one to create the Fibonacci Sequence as his work is referenced in a journal written around 1135 A. D. by Gopala.

Fifteen years after that, in 1150 A. D. , Acharya Hemachandra popularized the sequence in his writings. It wasn’t until 1202 A. D. that Leonardo Fibonacci introduced the west to the Fibonacci sequence in his book Liber Abaci after picked up a copy of one of Hemanchandra’s books while on a trip through the Mediterranean world and North Africa, and simplifying the sequence by comparing it to an expanding rabbit population. As Fibonacci put it, “you can use the Fibonacci sequence in order to find the number of rabbits in a population for an unending number of months”. Even though Virhanka first discovered the sequence, Leonardo Fibonacci is given credit for its rediscovery by introducing it to the West where it has been used to model all sorts of structures and natural phenomenon.

Applications in Art

In art, the Fibonacci sequence is seen throughout history. When visualizing each number in the Fibonacci sequence as a series of interconnected squares, a spiral can be drawn through its corners to creates a logarithmic spiral commonly known as the “golden spiral”. This spiral’s approximate growth factor is the golden ratio: 1. 618. This golden spiral is used in artistic composition as “an expression of an aesthetically pleasing principle-- the rule of thirds”. The use of the spiral in art serves to increase the balance and flow of an artwork, making it more visually pleasing.

The golden spiral can be seen in paintings by Leonardo Da Vinci, Michelangelo, Raphael, Botticelli, and Salvador Dali, among others. It is often used to portray “beauty, balance, and harmony in art and design”, and the proportions of many famous art pieces utilize the golden ratio. For example, in Leonardo Da Vinci’s “The Last Supper”, the proportions of the background, the positioning of the subjects, and the composition of the piece as a whole fall in line with the golden ratio. Michelangelo’s paintings on the Sistine Chapel and “The Creation of Adam” all exhibit this composition. The golden ratio has also been used in architectural design. The proportions of many of the structures within the Parthenon in Ancient Greece utilize the “golden rectangle, ” a shape whose sides follow the golden ratio 1 : 1. 618.

The Fibonacci sequence also makes many appearances in nature such as in the structure of family trees, nautilus shells or even some galaxies. It is also extremely common in the assortment of plant structure (branches, leaves, petals, etc. ). Take the family tree of a honey bee. Honey bees are unique in that the “queen” bee is the only female that can reproduce, and these eggs produce offspring whether or not they are fertilized. An egg that is fertilized by a male will hatch into a female - referred to as a “worker” - or remain unfertilized and hatch into a male - referred to as a “drone”. This means that every drone has one female parent, while every worker has two parents (one male and one female). The family tree of honey bees of either sex follows the Fibonacci sequence. And the ratio of female to male honey bees at any given time mirrors the golden ratio: 1. 618.

Next, the nautilus shell follows the growth pattern of the Fibonacci Spiral. As an organism grows, its shell must grow in the same proportion and this self-similar geometric construction often results in the Fibonacci Spiral. On the other hand, it is sometimes gravity that causes this shape on a much larger scale. As with spiral galaxies, the closer objects are to the center, the faster they move, creating a spirillic appearance. However, it is still largely up for debate how these structures remain spiral and do not eventually wind out. For something more readily observable, this pattern can be observed in many different varieties of plants. Some plants branch with a Fibonacci number of growing points. This is because a plant shoot may take two months to be strong enough to support branching but will create new offshoots every month after.

The seeds of flower heads, form Fibonacci spirals left and right, forming the optimal packing arrangement for uniformity of seeds regardless of head size. Each seed is phi (0. 618) of a turn from the previous and there are Phi (1. 618) seeds per turn. Arrangement of leaves also follows this since their arrangement is often designed to not block sunlight from leaves below.

Applications in Computer Science

Yet another application for the Fibonacci sequence is for developing algorithms useful in computer science. This includes finding extrema, merging files, and providing a potentially useful method for searching sorted arrays. The latter application is called the Fibonacci search technique and has similarities to, but significant differences from, the binary search technique. A binary search divides the sorted array into two equal sizes, one of which is examined further by dividing that selected subarray into equal sizes once again, and so on. The Fibonacci sequence also implements a divide-and-conquer strategy, but instead splits the array using Fibonacci numbers. The algorithm is as follows:

Let m be smallest Fibonacci number greater than or equal to the size of the sorted array.

Use the (m-2)th number in the Fibonacci sequence as the comparison index.

If the target element is less than the element at the comparison index, recur for the subarray to the left by moving active Fibonacci variables m and (m-2) down by 2 numbers in the sequence each.

If the target element is greater than the element at the comparison index, recur for the subarray to the right by moving active Fibonacci variables m and (m-2) down by 1 number in the sequence each.

Repeat steps (3) and (4) until the target element is found or no elements remain to be searched.

It can be proven that both the binary search and the Fibonacci search have the same worst-case performance of O(log n). While the Fibonacci sequence does not improve complexity, it offers other potential advantages when used as a searching technique. The first is that the Fibonacci search only needs addition and subtraction to access array elements, whereas binary search requires division or multiplication, which can be costly for some CPUs. The second potential advantage is that the Fibonacci search examines relatively closer elements in subsequent steps than binary search, which in some cases reduces the average time to access a storage location, especially for large arrays that may not fit in CPU cache or RAM. The Fibonacci search technique is representative of how the Fibonacci sequence can be used in certain cases to improve common algorithms used in computer science.

Conclusion

The Fibonacci sequence has been a staple of mathematical theory for millennia due to its astonishing accuracy in modeling real-world phenomena and ability to provide elegant solutions to technical problems. This simplistic sequence, whose elements are derived by adding the previous two terms together, is seemingly unmatched in its relevancy to the patterns of the outside world in nature, art, technical fields, and more. All this originated from the desire to map rabbit population growth, giving it roots in biology. Notably, its numbers also form the golden ratio, which is widely used by artists and architects because of its visual appeal. The Fibonacci sequence is also an invaluable tool which can be used in algorithm development in computer science and provides inspiration and direction in solving and modeling many problems, all together making its scope so much greater than just a mathematical sequence of numbers.