Fibonacci Spiral

 

The Fibonacci spiral also known as golden spiral has an association with the golden mean, and it is based on the Fibonacci sequence. Fibonacci spiral is also reefed to as golden spiral.  In logarithm, it means a logarithmic spiral which gets wider by a factor of ɸ after making a quarter turn.  A Fibonacci spiral having an initial radius of 1 has a polar equation similar to that of other logarithmic spirals

 

The Fibonacci Spiral

Derivation of Fibonacci sequence

Fibonacci spiral is also considered as one of the approximates of the golden spiral.  Fibonacci spiral is based on Fibonacci numbers, which are set of numbers arranged in integer sequences referred to as the Fibonacci sequence. These numbers are characterized in such a way that each of the numbers after the first two numbers represent the sum of two numbers before it.

The first two numbers in the sequence can either be 0 and 1 or 1 and 1.  The starting point notwithstanding, the subsequent number is always the sum of the two numbers before it. The sequence of these numbers can be defined by what is termed the Recurrence Relation.

Origin 

The Fibonacci spiral has its origin in Indian mathematics and also associated with Sanskrit prosody. According to Susantha Goonatilake, a part of the Fibonacci spiral is related to the Pingala (200 BC) and Virahanka (c. 700 AD), Hemachandra (c. 1150) and Gopala (c. 1135). Be that as it may, the clearest of all the expositions regarding the Fibonacci spiral and its sequences can be found in the findings conducted by Virahanka (c. 700 AD). Though his work is lost, it can be found in quotations from Gopala’s work in c. 1135. The sequence also got a mention from Hemachandra in c. 1150, who was a Jain scholar. The Fibonacci spiral equally has popularity outside India.

Applications 

Fibonacci spiral is based on the Fibonacci sequence and each quarter in the spiral is as big as the last two quarters. The Fibonacci spiral equally crates the 16:9 golden ratio, which is used for formatting purposes and applications by many smartphones and televisions.  Research has shown that the faces of many of the celebrities out there today have a strong match to the 16:9 ratio.

The golden phi or number is 1.69, and the Golden Ratio is also 1:1.69.  This simply means that each of the curves on the Fibonacci spiral will end up being 1.69 times bigger than the previous curve.

The golden ratio represents the ratio of all the sides that make up a rectangle.  If a square with sides represented by the smaller sides of the rectangle is removed from the rectangle, the rectangle remaining will still have sides having the same ratio.

A golden ratio is never a transcendental number, unlike popular numbers like the e and pi. As a result, a finite number of exponentiations, divisions, multiplications, subtractions, additions or integers cannot be used in representing it.  Also, the Fibonacci spiral can serve as the base for representing positive integers expressed in a non-repeating form, which is usually referred to as metallic series. The golden ratio is also depicted by European paper sizes, making it easy to scale a paper size up or down.