The Golden Ratio

A Mathematically Metallic Mean

In mathematics, the Fibonacci numbers form a sequence, such that each number is the sum of the two preceding numbers, starting from 0 and 1:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55,…

These are named after the Italian mathematician Leonardo of Pisa, later known as Fibonacci, but they were really discovered much earlier by the ancient Indian mathematician Acharya Pingala. The set of numbers appears to have first arisen circa 200 BCE in a work regarding the enumeration of possible patterns of poetry formed from syllables of two lengths. Thus, the numbers were originally part of Eastern Asian mathematics. Then, more than a millennium later, Fibonacci introduced the sequence to Western European mathematics, in his 1202 book Liber Abaci. So, the so-called Fibonacci sequence should really be known as the Pingala sequence, but I will use the label that everyone is familiar with to avoid confusion.

In line with this, a specific ratio can be derived from the Fibonacci sequence. This results in what is known as the Golden Ratio:

1:1.618

This can also be obtained by cutting a line segment into unequal pieces of length denoted “a” and “b”, such that the ratio of a to b is the same as the ratio of a+b to a. In other words, the large segment over the medium segment equals the medium segment over the small segment:

This gives rise to the most irrational of all irrational numbers if such a thing can be said to exist. Simply put, an irrational number is one that cannot be expressed by a fraction of integers, or whole numbers. So, although the actual ratio is an irrational number that never resolves, it can be functionally reduced to the specific growth rate of 1 to 1.618. Mathematically, this is known as phi, and it is represented by the symbol Ф. This is best illustrated by the Golden Rectangle whose longer edge is 1.618 times longer than the shorter edge. Therefore, if a rectangle has an aspect ratio of Ф then it can be continuously subdivided into a square and a Golden Rectangle. Then, a Golden Spiral which necessarily has a growth factor of Ф…