Fibonacci Sequence
- Jung Hyeok Lee
- 10 hours ago
- 3 min read
The Fibonacci sequence is a series of numbers introduced in the book Liber Abaci in 1202 by Leonardo Fibonacci, a famous Italian mathematician, which follows the pattern of adding two previous numbers together. The first 4 numbers of the sequence are 0, 1, 1, and 2. As presented, 0 and 1 are added to create the next number, 1, and 1 and 1 are added to create the next number, 2. This sequence might seem random, but this, surprisingly, can be found on many occasions in nature. For example, the number of petals of the flowers is normally one of the numbers in the Fibonacci sequence, and the golden ratio, found in human ears, seashells, and galaxies, due to its most efficient way to grow, pack items, and optimize the resources, can be commonly found outside.
Alike Concepts
Similar to the Fibonacci sequence, there is the Lucas sequence, which also follows a recursive pattern just like Fibonacci’s, but instead starts with 2 and goes 1, 3, 4, 7, 11, and so on. The Lucas sequence, introduced in the 1870s and 1880s, is an extension of the Fibonacci sequence for mathematicians to dive deeper into the concept. The series appears in number theory, cryptography, and even in financials, while its ratios are still approaching the golden ratio. These diverse applications drive the foundation of the Lucas numbers. As a trivial fact, the Lucas numbers can always be represented by two Fibonacci numbers that are ±1 of the Lucas number’s n value, where n represents the nth term of it. Way back, around 200 BCE, ancient India introduced the Fibonacci sequence. It was done by Pingala, and he discovered this pattern while working to count syllable patterns, which also appeared in works of Bharata Muni, Virahanka, and Hemachandra for metrical analysis, exhibiting India’s deep understanding of math back then.
Additional Facts
There are several trivial yet fascinating facts about the Fibonacci sequence: the patterns in the squared numbers of Fibonacci and the golden ratio. Figure 1 shows the squared numbers of the sequence showing a particular pattern. The sum of the squared numbers can be represented by the product of the Fibonacci sequence. This is possible because squared numbers also indicate the area of a square, which allows people to visualize the image, like the image shown below, Figure 2.
The Golden Ratio
As mentioned before, the Fibonacci sequence appears in a diverse range of arts, called the golden ratio, which is found by dividing the larger number by a smaller one that makes up the sum of the squares of the series. 13/8 equals 1.625, 21/13 makes 1.615, 34/21 produces 1.619, and so on, gradually getting to a number of 1.61803398875, the golden ratio. This is used in drawings and paintings of art, not just because it is common in nature, but also because it is believed to create visually pleasing compositions, fostering harmony, balance, and natural beauty for the human eye. Due to this fact, photographers have frequently used it to flawlessly guide the viewer’s focus onto their main subject of the photo. Figure 3 shown below well-demonstrates this concept by placing the human figure in the place where the curve converges, naturally emphasizing the main subject.
References
Singh, Parmanand (1985), "The So-called Fibonacci numbers in ancient and medieval India", Historia Mathematica, 12 (3): 229–244, doi:10.1016/0315-0860(85)90021-7
Sigler, L. E. (2002), Fibonacci's Liber Abaci: A Translation into Modern English of Leonardo Pisano's Book of Calculation, Sources and Studies in the History of Mathematics and Physical Sciences, Springer, ISBN 978-0-387-95419-6
Applying Lucas Sequence For Optimal Stock Market Entry And (*Department of Mathematics, Chandigarh University, Gharuan, Punjab, India, 2014, #)
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