Fibonacci sequence
Introduction
The Fibonacci sequence is a series of numbers in which each number is the sum of the two preceding ones, usually starting with 0 and 1. Named after Italian mathematician Leonardo of Pisa, also known as Fibonacci, this sequence has been recognized in various forms in mathematics for centuries.
Historical Background
The Fibonacci sequence was first introduced to the west by Fibonacci in his 1202 book, Liber Abaci. However, the sequence had been previously described in Indian mathematics. Fibonacci's Liber Abaci introduced the sequence to Western mathematics, although the sequence had been previously described in Indian mathematics.
Mathematical Explanation
In mathematical terms, the Fibonacci sequence is defined recursively as:
F(0) = 0, F(1) = 1
and
F(n) = F(n-1) + F(n-2) for n > 1
This means that each number in the sequence is the sum of the two numbers that precede it.
Properties of the Fibonacci Sequence
The Fibonacci sequence has many interesting mathematical properties. For example, the ratio of consecutive Fibonacci numbers converges on the golden ratio, approximately 1.6180339887. This ratio is often found in nature and art, suggesting a universal aesthetic appeal.
Fibonacci Sequence in Nature
The Fibonacci sequence is often observed in nature, particularly in the arrangement of leaves on a stem, the branching of trees, the flowering of an artichoke, an uncurling fern, the arrangement of a pine cone, and the growth of a snail shell. This phenomenon is often attributed to the efficiency of packing, allowing for maximum exposure to sunlight and other factors.
Fibonacci Sequence in Art and Architecture
The Fibonacci sequence and the related golden ratio have been used in art and architecture throughout history. This is due to the aesthetically pleasing proportions produced by the ratio. Examples include the Parthenon in Athens, the pyramids of Egypt, and many works of art from the Renaissance period.
Fibonacci Sequence in Computer Science
In computer science, the Fibonacci sequence is used in various algorithms, including those for sorting and searching. The sequence is also used in the mathematical analysis of algorithms.