“The Rational Irrational”
The Golden Ratio – The Divine Proportion
Phi is simply the most amazing thing I have ever come across in our worldly knowledge. In all my years studying Science and Mathematics at a tertiary level (and a very high one at that!), I have never heard of, nor have I been so intrigued and so drawn by its ubiquity in all things within this Universe.
An Introduction to Phi – Φ
Represented by the Greek letter Φ – Phi has a value of 1.618… where the decimals go on infinitely with no repetitions or any patterns. This is why it is considered an “irrational” number, yet I have never seen a more “rational” number than Phi. It is also known by many other names such as; “The Golden Ratio”, “The Golden Section”, and “The Divine Ratio”, to name a few.
Phi is not just a number or ratio that is expressed in digits. It is also inherent to all living things of which we are not fully aware of except on a sub-conscious level. It is the reason why we find things so pleasing and aesthetically beautiful. It’s the reason why the Human form looks appealing to us; the reason why anything that exhibits itself within this ratio is so alluring to us. It is not just confined to numbers but also found in Nature and also extends into the world of Art and Philosophy. Phi is found in many things like; Animals, Plants, DNA, Population Growth, Life, Acoustics, Architecture, Marketing, Music, Poetry, Stock Markets, Energy, “Sacred” Geometry, The Solar System, Quantum Time and The Universe itself. Even our heart beats in Phi ratios! All of this will be illustrated in my Phi pages.
When I first rediscovered this simple ratio for myself back in 2004, I immersed myself for months on end after I understood where the Phi ratio is derived from geometrically (as I mentioned in my main article – What followed after my discovery of OneEye). I found the answer thanks to a dream sequence over 3 nights… and this is what I got…
This is a more updated version of my original.
Back in 2004, it was extremely hard for me to find in-depth information on this phenomenal ratio as there was literally little or no information about it on the net, nor in the vast volumes of books in our libraries. I even asked my old Professor and he said I was “having fun with numbers”. Yeah, right! Now there seems to be much more awareness however, and much more information about it. But back in 2004, I felt all alone and lost as I went about trying to understand what this mystical ratio was! I was unraveling something that was real, perplexing, simple, but amazingly beautiful once you understand its connection with nature, the Universe, and ourselves; all of which exhibit this ratio in one way or another. To me it is a ratio that God uses to fashion and shape the way everything in this finely tuned Universe looks and behaves within. It is simply divine! I hope you can appreciate why?
While I was investigating the mathematical background, I noticed that ancient races like the Mayans, Greeks and Egyptians already knew this ratio in their time and they even used this ratio to construct their buildings like the Parthenon and the Pyramids, for example. Even during the Renaissance period, the great inventor/artist – Leonardo da Vinci – used PHI proportions in his works, with a few examples being; the “Vitruvian Man”, “Mona Lisa” and the “Last Supper”.
To quote Johannes Kepler – a German Mathematician, Scientist (1571-1630):
“Geometry has two great treasures: one is the Theorem of Pythagoras; the other, the division of a line into extreme and mean ratio. The first we may compare to a measure of gold; the second [Phi] we may name a precious jewel.”
The golden ratio has fascinated intellectuals of diverse interests for over 2,400 years:
“Some of the greatest mathematical minds of all ages, from Pythagoras and Euclid in ancient Greece, through the medieval Italian mathematician Leonardo of Pisa aka “Fibonacci” and the Renaissance astronomer Johannes Kepler, to present-day scientific figures such as Oxford physicist Roger Penrose, have spent endless hours over this simple ratio and its properties. But the fascination with the Golden Ratio is not confined just to mathematicians. Biologists, artists, musicians, historians, architects, psychologists, and even mystics have pondered and debated the basis of its ubiquity and appeal. In fact, it is probably fair to say that the Golden Ratio has inspired thinkers of all disciplines like no other number in the history of mathematics.” — Mario Livio, The Golden Ratio: The Story of Phi, The World’s Most Astonishing Number.
Trying to explain this amazing mathematical ratio (PHI) to someone new can make one feel almost like an alien coming down to Earth and offering new technology or knowledge. Whenever I talk about this wonderful number with people, I get all worked up and excited due to its immense influence on our lives, yet most people have no idea what it is! In fact, to this day, most of the schools in the academic world still do not teach it, and this is almost as equally intriguing! I frequently ask maths enthusiasts, scientists, engineers from all over the world and most of them have never heard of it! However, there are a few (and I do mean “few”) that have. And the one’s that do know, fall in love with it, as I have!
In schools all around the world, we are frequently taught about other irrational ratios like Pi – 3.1416 or e – 2.7183, but very rarely about Phi – 1.618. They teach us to work out the Pythagorean angles of PI though (15, 30, 45, 60, and 75 degrees), and in all other angles you can think of (as it should be), but you will probably not be taught about the angles of Phi (9, 18, 27, 36, 45, 54, 63, 72, and 81 degrees), nor the many expressions that comes with it.
What’s interesting about this wonderful ratio is that; not only has the world of academia withheld this information and excluded it in our syllabus’, but the one’s that DO know about it are holding us back. The thing is, the one’s that have mastered this knowledge don’t seem to want to share it, or at least with the same level of vigor shown when teaching other irrational ratios, which you may be familiar with already, such as Pi. And when they do teach us about this phenomenal ratio (Phi), it’s usually taught in a simplistic, and dare I say, diversionary manner. They just skim the surface and rarely talk about its fundamentals, particularly in Mathematics where Phi can be derived from a pentagram, or pentagon. You will only see this taught properly at 2nd and 3rd year tertiary level. You will rarely see it taught in high schools, when it should be!
Those of you familiar with the Phi ratio would most likely know of (or at least heard of) terms, such as “The Golden Ratio”, or the “Fibonacci Sequence”, and that’s about it! But I and many other wonderful sites, as well as authors of many books on Phi and the Golden Ratio are here to make you all aware of this, so that you too can learn and apply it in everyday life. How? This will also be covered further on.
Now let’s begin our little journey of knowledge into the world of Phi shall we? There’s a lot to cover, and should be of interest to many of you, as it relates to almost anything you can think of.
Here is a list of topics that I will post over time, so please take a moment to read them and enjoy, my phine friends! 😉
Phi in Mathematics
Phi in Science
Phi in Finance & Stock Markets
Phi in Marketing
Phi in Psychology
Phi in Art
Phi in Music
Phi in Beauty – Aesthetics
Phi in Architecture & Ancient Monuments
Phi in Philosophy
Phi in Theology
I’d also like to share with you all a fantastic video of Donald Duck In Mathemagic Land, which I think you should watch as it beautifully illustrates the Golden Ratio in a fun and simple way. It was also nominated for an Oscar in 1959.