Fractal Pattern in a Quantum Material Confirmed for the First Time!

The word fractal has become increasingly popular, although the concept started more than two centuries ago in the 17th century with prominent and prolific mathematician and philosopher Gottfried Wilhelm Leibnitz. Leibnitz is believed to have addressed for the first time the notion of recursive self-similarity, and it wasn’t until 1960 that the concept was formally stabilized both theoretically and practically, through the mathematical development and computerized visualizations by Benoit Mandelbrot, who settled on the name “fractal”.

Fractals are defined mainly by three characteristics:

  1. Self-similarity: identical or very similar shapes and forms at all scales.
  2. Iteration: a recursive relationship limited only by computer capacity. With sufficiently high performance, the iterations could be infinite. This allows for very detailed shapes at every scale, that modify with respect to the first iteration, manifesting the original shape at some levels of iteration. Because of this, fractals may have emergent properties, which make them a suitable tool for complex systems.
  3. Fractal dimension, or fractional dimensions: describes the counter-intuitive notion that a measured length changes with the length of the measuring stick used; it quantifies how the number of scaled measuring sticks required to measure, for example, a coastline, changes with the scale applied to the stick.
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What Is So Special About The Number 1.61803?

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


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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