What Is So Special About The Number 1.61803?

5 min readMay 5, 2019

PHI(φ) is an irrational, non-terminating number as PI(π), but its significance is far more than PI(π) ;

Π = 3.14159265359…(pi)
Φ = 1.61803398874…(phi)

The Golden Ratio (phi = φ) is often called The Most Beautiful Number In The Universe.

The reason φ is so extraordinary is because it can be visualized almost everywhere, starting from geometry to the human body itself!

The Renaissance Artists called this “The Divine Proportion” or “The Golden Ratio”.

PHI(φ) can be seen appearing in the following ways:

1. Fibonacci Series

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144…

This series was developed by an Italian mathematician known as Leonardo Fibonacci. Other than the fact that each term is the sum of its two preceding consecutive terms, it can also be seen that if we divide a term greater than 2 by a term preceding it that the ratio always tends to 1.618…!

And if we continue this division after the 13th term we will always get a fixed number = 1.618

Example;

89/55 = 1.618

144/89 = 1.618

233/144 = 1.618

377/233 = 1.618

610/377 = 1.618

987/610 = 1.618

So on…!!!

2. The Human Body

For instance, if you divide the length from your head to toe by the length from your bellybutton to toe you will find the answer tending to φ.
Now, divide the length from your shoulder to the tip of the index finger by the length from your elbow to the wrist (of the same arm) and you’ll get φ again..!!
Divide the length from the top of the head to the shoulder by the length from your top of the head to your chin, φ again!
Top of your head to belly button by the length between you head and shoulder…..BANG….φ again!!!
Distance between your bellybutton and the knee, by the distance between knee and the bottom of the foot….φ again!
Now divide the length of your face to the width of the face……BAAM…φ again!!
Width of your two upper teeth to that of its height, and you’ll get φ again!
Lips to eyebrow divided by the length of the nose, φ again!

3. Plants

A sunflower grows in opposing spirals, the ratio of its rotation’s diameter to the next is 1.618…..i.e. φ again!
The ratio between the margin of a leaf to its veins(some plants) also gives φ.

4. DNA Of Organisms

DNA of the cell appears as a double-stranded helix referred to as B-DNA. This form of DNA has a two groove in its spirals, with a ratio of φ in the proportion of the major groove to the minor groove.
A cross-sectional view from the top of the DNA double helix forms a decagon. A decagon is actually two pentagons, with one rotated by 36 degrees from the other, so each spiral of the double helix must trace out the shape of a pentagon. The ratio of the diagonal of a pentagon to its side is φ to 1.

5. The Solar System

The average of the mean orbital distances of each successive planet in relation to the one before, tends to φ.
The Kepler’s Triangle(the triangle formed by utilizing the moon and the earth) is formed by a Pythagorean relation, in which the three sides of the right-angled triangle formed are always of this order:

Hypotenuse = φ
Perpendicular = √φ
Height = 1

If the rings of Saturn are closely looked at we will see that there is a ring that is quite denser than the other rings. Miraculously this inner ring exhibits the same golden section proportion as the brighter outer ring i.e. φ
Venus and the Earth are linked in an unusual relationship involving φ. If Mercury represents the basic unit of orbital distance and period in the solar system:

we find:

√Period of Venus * φ = Distance of the Earth
√2.5490 * 1.6180339 = 1.5966 * 1.6180339
= 2.5833 million kilometers

6. Art And Architecture

The Golden Ratio was probably most utilized by artists and architects while building their masterpieces. The following 5 pieces of work are specifically mentioned in the list as the golden ratio has been extensively used while creating them!

The Great Pyramid of Giza
Notre Dame
The Vitruvian Man
The Last Supper
The Parthenon

7. Music

If we divide an octave by a perfect fifth, (13/20) = φ

If we divide a perfect fifth by an octave, (8/13) = φ

If we divide a perfect fourth by a major sixth, (6/10) = φ

And if we divide a major third by a perfect fifth, (5/8) = φ

Therefore we can see that φ is indeed a mystical number which can be visualized all around us.

And if we observe closely we can find its traces going back before humanity was even inhabiting earth, for example, the skin folds of extinct dinosaurs, rare ancient insect segmentation, and much beyond that.

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Edit: I can see that some people are confused about the fact that how dividing 13 by 20 gives the answer as “phi”. So here’s the answer:

One of the very basic properties of “phi” is that its reciprocal is 1 less than itself(or o.618). You can either compare the numbers listed in the above quoted part of the article(about the octaves) to this value(0.618) or just flip the fraction to get the answer around 1.618.

Therefore you can use 1.618 and 0.618 interchangeably!