Golden Ratio

In mathematics, the Fibonacci numbers, commonly denoted $F n$ , form a sequence, the Fibonacci sequence, in which each number is the sum of the two preceding ones. The sequence commonly starts from 0 and 1, although some authors omit the initial terms and start the sequence from 1 and 1 or from 1 and 2. Starting from 0 and 1, the next few values in the sequence are:

0

1

2

3

5

8

13

21

34

The Fibonacci numbers were first described in Indian mathematics, as early as 200 BC in work by Pingala on enumerating possible patterns of Sanskrit poetry formed from syllables of two lengths. They are named after the Italian mathematician Leonardo of Pisa, later known as Fibonacci, who introduced the sequence to Western European mathematics in his 1202 book Liber Abaci.

Fibonacci numbers appear unexpectedly often in mathematics, so much so that there is an entire journal dedicated to their study, the Fibonacci Quarterly. Applications of Fibonacci numbers include computer algorithms such as the Fibonacci search technique and the Fibonacci heap data structure, and graphs called Fibonacci cubes used for interconnecting parallel and distributed systems. They also appear in biological settings, such as branching in trees, the arrangement of leaves on a stem, the fruit sprouts of a pineapple, the flowering of an artichoke, an uncurling fern, and the arrangement of a pine cone’s bracts.

The Fibonacci numbers may be defined by the recurrence relation

and

for $n > 1$ .

Fibonacci numbers are strongly related to the golden ratio: Binet’s formula expresses the $n$ th Fibonacci number in terms of $n$ and the golden ratio, and implies that the ratio of two consecutive Fibonacci numbers tends to the golden ratio as $n$ increases. Fibonacci numbers are also closely related to Lucas numbers, which obey the same recurrence relation and with the Fibonacci numbers form a complementary pair of Lucas sequences.

Fibonacci number – Wikipedia. [Accessed 7 January 2022].

Examples in Nature

The universe may be chaotic and unpredictable, but it’s also a highly organized physical realm bound by the laws of mathematics. One of the most fundamental (and strikingly beautiful) ways these laws manifest is through the golden ratio.

It’s not hard to find examples of this logarithmic phenomenon in nature — whether it’s a simple houseplant, a nautilus shell, an ocean wave or an expansive spiral galaxy, they all originate from the same mathematical concepts.