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 can be placed over this:
In line with this, the Golden Ratio is one more than its reciprocal:
Ф=(1/Ф+1)
This is equivalent to the following quadratic equation:
Ф²-Ф-1=0
Of course, there are two conjugate solutions to this problem, with one being positive and the other negative:
(1+ √5)/2
(1- √5)/2
However, since phi is the ratio of two lengths, it has to be the positive solution. This must also mean that the conjugate of Ф is the negative reciprocal. To better understand how this all works, it’s important to know the difference between infinite irrational numbers and finite rational numbers. For instance, 23/16 is equal to 1 plus 1 over 2 plus 1 over 3 plus 1 over 2. This is an example of a finite continued fraction applied to a rational number, and this is very different from the endless elegance of the number Ф. Notice that if we zoom in on the first denominator it’s identical to the entire continued fraction:
Now consider cutting off the continuing fraction after one nesting level, and then two, and so on:
1/1, 2/1, 3/2, 5/3,…
These terms are the best rational approximations to Ф with that denominator or lower. Plus, those are precisely the ratios of successive Fibonacci terms. More importantly, this tends toward phi in the infinite limit. Consider a geometric sequence in which each successive term is exactly Ф times the previous term:
1, Ф, Ф², Ф³,…
Notice that the third term is exactly the sum of the previous two, remembering the quadratic equation that phi satisfies:
1+Ф=Ф²
Every term in the sequence is always the sum of the previous two. Thus, a geometric sequence whose term ratio is Ф will automatically satisfy the Fibonacci recurrence relation. In fact, any sequence satisfying this same recurrence relation will get closer and closer to being geometric with phi as the term ratio. The thing is that the Golden Ratio is about so much more than just math. The concept is used across a number of different disciplines. It’s incredibly useful for calculation, application, and even inspiration.
The Golden Ratio is commonly found in nature, among countless different animate and inanimate forms. Plus, when it’s used in design, as it often is, it fosters organic looking compositions that are very aesthetically pleasing. Our souls seem to yearn for the divine curve of the Golden Spiral. The way I see it, this is sacred geometry that evokes something very spiritual from deep within us. In addition to that, our eyes and brains are hard-wired to prefer objects that make use of the Golden Ratio. That’s why every great artist from Leonardo da Vinci to Salvador Dali has used the perfect proportion to create the most inherently pleasing shapes imaginable.
Ultimately, for all of these reasons and so many more, this specific metallic mean really does mean something special. As far as I can tell, Ф is the most profoundly purposeful number there is. The bottom line is that this particular string of digits is of fundamental importance:
1.61803398874989484820458…
