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Fibonacci Sequence - Golden Ratios

Maybe you’ve heard about Fibonacci Sequence, Fibonacci Ratios, Fibonacci Retracements, Fibonacci Extensions, Fibonacci Time Zones, Fibonacci Series, Fibonacci Spirals, Fibonacci Fans, Fib Nodes and so on, or maybe not. How about, The Golden Mean also referred to as Golden Ratio, Golden Section, Golden Number, Phi, Φ, Divine Proportion and what not, or Golden Spiral, Golden Triangle and Golden Rectangle.

All this terminology comes from a simple number sequence of the form: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987 and so on to infinity, where each number is the sum of the previous two numbers. Leonardo of Pisa (c1170 - 1250), known as Fibonacci, "the son of Bonacci," a brilliant thirteenth century mathematician developed this number sequence, hence the name Fibonacci Sequence. He also introduced the Arabic numbers into Western Culture.

The Fibonacci Series is a sequence of numbers so deceptively simple, but its ramifications and applications are nearly limitless. It has fascinated and perplexed mathematicians for over 700 years, and nearly everyone who has worked with it has added a new piece to the Fibonacci puzzle, a new tidbit of information about this sequence and the way it works.

If we study the beauty of nature, art, or architecture we will discover a common principle running throughout. This common principle is the universal recognition of pleasant proportion. This magical connecting thread of proportion, known since antiquity, is none other than the Golden Proportion, a phenomenon related to beauty.

This article illustrates just a few aspects of the mystery and magic of this Golden Ratio. The understanding of Fibonacci Series will enable the reader to take the first step on a journey of discovery into an unexpected dimension of our life that when applied to any field or endeavor produces success. Ergo figuratively speaking – “Mining Gold with The Golden Mean”.

It may surprise some of you to learn that the universe, the constellations, the galaxy, flowers, oceans, plant life, man, natural science, music, architecture and the financial markets have one thing in common: the FIBONACCI SEQUENCE.

Fibonacci Sequence

You can observe a curious evolution of Fibonacci expansions played out in musical progressions, crystal formations, and even in the growth of rabbit populations. Whether in the human DNA spiral, the pre-programmed construction of a beehive honeycomb, or the inspirational great pyramid of Giza, Fibonacci relationships abound. The human body itself is a study in Fibonacci relationships.

This sequence of numbers has some very important properties.

For example:

  • The ratio of any number to the next number in the Fibonacci Sequence is 0.618 to 1 and to the next lower number is 1.618.
  • Between alternate numbers in the Fibonacci Series, the ratio is 2.618 or its inverse 0.382.

These numbers have some special relationship of their own such as

  • 2.618 - 1.618 = 1  
  • 1 - 0.618 = 0.382
  • 2.618 x 0.382 = 1
  • 0.618 x 0.618 = 0.382
  • 1.618 x 1.618 = 2.618

Additional phenomena relating to the Fibonacci sequence includes:

  • No two consecutive numbers in the sequence have any common factors.
  • The sum of any ten numbers in the sequence is divisible by 11.
  • The sum of all Fibonacci numbers in the sequence +1 equals the Fibonacci number two steps ahead.
  • The square of a Fibonacci number minus the square of the second number below it in the sequence is always a Fibonacci Number.

There are numerous relationships within this series, but the most important is 1.618 or 0.618. It is known as the Golden Ratio or Golden Mean and governs nature´s growth patterns.

The Golden Mean - Φ

The Golden Mean (or Golden Section), represented by the Greek letter phi, is one of those mysterious natural numbers, like e or pi, that seem to arise out of the basic structure of our cosmos. Unlike those abstract numbers, however, phi appears clearly and regularly in the realm of things that grow and unfold in steps, and that includes living things.

The decimal representation of phi is 1.6180339887499...

Fibonacci correlations - more than coincidence

We know that people unconsciously identify beauty by facial geometries and physical proportions that possess Fibonacci Ratios or Fibonacci Numbers and symmetric qualities. People also identify the significance of opportunities and threats in a similar fashion - probably unconsciously - certain patterns and ratios are attractive or threatening and evoke an emotional response. People start looking for a gas station when their tank is 3/8 full. Think about it.

A man’s body, if you divide the body at the navel - from the navel to the top of the head is about a 0.618 ratio of the lower part of the body (from the navel to the feet).

The ratio 0.618 to 1 is the mathematical basis for the Parthenon (a temple on the Athenian Acropolis, Greece, dedicated to the maiden goddess Athena), sunflowers, snail shells, and spiral galaxies of outer space or the human DNA spiral. Spirals on shells, when examined more closely, reveal to have arcs whose lengths are Fibonacci Ratios. Their diameters equal to 1.618, and the larger radius relates to the smaller radius by 1.618. This is known as the Golden Spiral. The Greeks based much of their art and architecture on this proportion.

Financial markets have the same mathematical basis as natural laws. This is because the markets are not only numbers or economic factors but most important, they reflect the human nature: emotions in motion.

Fibonacci Ratios are used (in Technical Analysis) by traders and investors in making entry and exit decisions for each trade. The first ratio 38.2% is used as an entry point in a trending market. The ratio 0% is used as the exit point. The question that you may ask is what the reason markets react to these levels is. You should not forget currency markets are just investors and speculators buying and selling financial instruments.

Practical Application of Fibonacci Sequence


The Golden Mean - Fibonacci Ratio Phi

The Fibonacci number series has more interesting aspects to it than most of us can imagine or would wish to. If you want to pursue the subject, there are many good books written discussing the more esoteric aspects. They certainly do a better job than I do, describing the mathematics involved. While we might get dizzy considering the possibilities, let me put things in a different perspective here. Fibonaci Sequences, Fibonacci Ratios and Fibonacci Spirals are all around us in nature and inside us too and that’s a fact, whether you know it or you don’t. My intention is to make you consciously aware of this fact and how you can make use of Fibonacci Sequence and mine “your gold” with The Golden Mean.

  • Plants and flowers are using the Golden Mean to “mine their gold” by positioning their leaves and petals optimally to collect sunlight.
  • Bees are using the Golden Mean to “mine their gold” when constructing a beehive honeycomb.
  • A healthy human heart beats in rhythm compliant to the Golden Mean.
  • Plastic surgeons, dentists, architects are using the Golden Mean to “mine their gold”.
  • Traders and investors are using the Golden Mean to “mine their gold” The financial markets exhibit Fibonacci proportions in a number of ways; particularly they are powerful tools for calculating price targets and placing stops.
  • Musical therapy revolves around the Fibonacci Sequence and gets positive results.
  • Corporations design their products and even packaging using the Golden Mean to “mine their gold”.

Are You Living Within The “ Golden Mean”? What traits do you already possess? What are the ones you need to work on?

The Golden Mean refers to the ideas of philosopher Aristotle, of four traits that would lead to the fulfilled human life. Too much or too little of one trait is considered an imbalance of the mean, and is something that needs to be improved upon.

Here is the list of the characteristics of someone who is living the fullest life should posses.


Fibonacci Ratio - Golden Mean - Balance

Well, I can go on and on, but you get the idea. Next time you are looking for a solution to whatever matter you are dealing with, you are far better off by starting to look for a way to implement something that involves the Golden Mean, a Fibonacci Number or a Golden Spiral to “mine your gold”.

Now don’t knock yourself out and go overboard with “Fibbing” ratios and entangling spirals all the time. Have fun with it and help yourself and others with this knowledge. Make sure you smile just about 61.8% of the time and you’ll mine whatever “gold” you are after with ease. Never mind the word count of this article is not a Fibonacci Sequence Number, I sincerely hope you will find it beneficial anyway.

Good luck!


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