The Fibonacci series is a sequence of numbers that starts with 0 and 1, and each subsequent number is the sum of the previous two. For example: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, and so on. The Fibonacci series is named after the Italian mathematician Leonardo Fibonacci, who introduced it to the Western world in his book Liber Abaci in 1202.
The Fibonacci series has many fascinating properties and applications in mathematics, science, art, and nature. Here are some of them:
The Golden Ratio
The ratio of two consecutive Fibonacci numbers tends to a constant value as the series progresses. This value is approximately equal to 1.61803, and is known as the golden ratio or the divine proportion. The golden ratio has been considered as an ideal aesthetic standard for beauty and harmony since ancient times. It can be found in many geometric shapes, such as pentagons, pentagrams, and decagons. It can also be seen in many architectural and artistic works, such as the Parthenon, the Mona Lisa, and the Taj Mahal.
The Fibonacci Spiral
The Fibonacci spiral is a curve that approximates the golden spiral, which is a logarithmic spiral whose growth factor is equal to the golden ratio. The Fibonacci spiral can be constructed by drawing arcs connecting the opposite corners of squares in a Fibonacci tiling, which is a pattern of squares whose side lengths are successive Fibonacci numbers. The Fibonacci spiral can be observed in many natural phenomena, such as the arrangement of seeds in a sunflower, the shape of a nautilus shell, or the pattern of a hurricane.
The Binet Formula
The Binet formula is an algebraic expression that can be used to calculate any Fibonacci number without having to know the previous ones. It was derived by the French mathematician Jacques Philippe Marie Binet in 1843. The Binet formula is given by:
where Fn is the nth Fibonacci number, ϕ is the golden ratio, ψ is its negative reciprocal (approximately equal to -0.61803), and n is any positive integer.
The Fibonacci code is a variable-length code that can be used to encode positive integers using only two symbols: 0 and 1. It was invented by the American mathematician Robert W. Floyd in 1963. The Fibonacci code is based on the Zeckendorf’s theorem, which states that every positive integer can be uniquely represented as a sum of non-consecutive Fibonacci numbers. The Fibonacci code assigns a binary string to each Fibonacci number as follows:
F1=1 is encoded as 11
F2=1 is encoded as 01
F3=2 is encoded as 001
F4=3 is encoded as 101
F5=5 is encoded as 0001
F6=8 is encoded as 1001
F7=13 is encoded as 01001
and so on.
To encode any positive integer using the Fibonacci code, one simply adds the binary strings corresponding to the Fibonacci numbers that make up its Zeckendorf’s representation. For example:
To encode 4, which is equal to F4, one uses 101
To encode 6, which is equal to F4+F2, one uses 10101
To encode 9, which is equal to F6+F1, one uses 100111
To encode 19, which is equal to F7+F5+F2, one uses 01001001
The Fibonacci code has some advantages over other variable-length codes, such as the unary code or the Huffman code. For instance, it does not require any prefix or suffix symbols to indicate the end of a codeword, since every codeword ends with 11. It also has a simple decoding algorithm that does not require any lookup tables or dictionaries.