The universe exhibits a mysterious underlying order. It is
interconnected at levels that are hidden from the most powerful
scientific minds in existence today. Nothing is as it first
appears. One of the prime examples of this is the Fibonacci
numbers – a sequence that is built into all levels of creation.
They crop up in many places and circumstances.
The Fibonacci numbers were discovered by the 13th century Italian
mathematician Leonardo Fibonacci and written about in his great
work Liber Abaci. It is more accurate to say that he
re-discovered them, because they were definitely known to the
Ancient Greeks and Egyptians. The Greeks based most of their Art
and Architecture upon the Fibonacci ratio, and Egyptians
incorporated this ratio into the Pyramids.
Fibonacci derived the number series to solve a mathematical
problem related to the breeding speed of rabbits. The series
itself is very easy to generate. These are the first few terms:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233,…..
Basically, the next term in the series is generated by adding the
two preceding terms, e.g. 144 = 55+89.
The Fibonacci ratio itself is also easily derived. Take any two
adjacent terms of the series and divide the earlier term by the
later. For example 2/3 = 0.666, 3/5 = 0.600, 5/8, 0.625, 8/13 =
0.615, 13/21 = 0.619, 21/34 = 0.617, 34/55 = 0.6182, 55/89 =
You will find that these ratios eventually converge to the
“magic” Fibonacci ratio of 0.618. This is also known as the
Golden Ratio. It is prevalent throughout Nature, as well as
in man-made creations. For example, the width of all standard
picture frames are 0.618 of their length. Notes in music are
related to each other by Fibonacci ratios.
The ratio itself has some weird mathematical properties. If you
square 0.618, i.e. multiply it by itself, you get 0.382:
0.618*0.618 = 0.382.
However, this is the SAME number you get by subtracting the
Fibonacci ratio 0.618 from unity, i.e. 1 – 0.618 = 0.382
If you take any term in the Fibonacci series and divide it by
term two places later, you again converge to 0.382 as you proceed
up the series. For instance, 21/55 = 0.3818, 34/89 = 0.3820,
55/144 = 0.3819, 89/144 = 0.382. Conversely, divide the same two
terms the other way round and you get 55/21 = 2.619, 89/34 =
2.6180, 144/55 = 2.6181, 233/89 = 2.6180. In other words, you get
2 + 0.618. The same Fibonacci ratio of 0.618 crops up yet again!
In truth, there are even more relationships than the ones pointed
out so far. However, these examples should demonstrate to even
the non-mathematical reader that this is a VERY strange number
series indeed! Suffice it to say that there is a Fibonacci
Association, complete with Fibonacci Quarterly Journal, dedicated
to analyzing all the strange relationships of these intriguing
The numbers themselves crop up throughout Nature. For instance,
it is said that the number of seeds and petals on a sunflower are
both Fibonacci numbers. Many other examples of the discrete
numbers abound in Nature. However, a particularly common and
interesting manifestation is the Golden Spiral.
The Fibonacci Spiral – Its Presence Throughout Nature.
As we have said, this ratio of 0.618 occurs widely throughout
Nature. The most famous example is the Golden Spiral, which is
the basis of all spirals witnessed in Nature; from the spirals on
the shells of marine creatures right out to the vast shapes of
spiral galaxies. Again, the Golden Spiral is very simply derived
Here are some examples:
(a) The classic example of the Golden Spiral is the shells of
shellfish like the Nautilus.
(b) The spiral can also be found in the vastness of galactic
space, in the form of some spiral galaxies a hundred thousand
lights years or more in diameter.
(c) On more earthly levels, skilled surfers moving through the
eye of a wave are covered by a partial Fibonacci spiral.
(d) The epeira spider spins its web in a Fibonacci spiral.
(e) Bacteria grow at an accelerating rate that can be plotted
along a Fibonacci spiral.
In fact, virtually everywhere that you see spirals in nature,
they are described by Fibonacci numbers.
The Fibonacci Ratio In the Financial Markets
Examples of Fibonacci phenomena abound in the stock, commodity,
bond and currency markets too. In fact, this is so much the case
that Fibonacci calculation methods are included as standard in
all software programs used to plot price charts. Currency
dealers, despite being a hard-boiled lot, do know their Fibonacci
levels extremely well for purposes of trading the markets.
The common appearance of Fibonacci is in the relationships
between highs and lows on a price chart. For example, if you
calculate the price difference between a market top and its
subsequent bottom, you will very often find that the next peak or
trough is formed at a Fibonacci relationship to this distance. In
other words, the market may retrace exactly 38.2% or 61.8% of the
previous move. This happens often enough, and exactly enough, for
it to be a law of the markets, and not merely wishful thinking.
The higher ratios work too. If a market made a top, followed by a
bottom, and then trades ùp to make even higher highs, you will
often find that a new top forms at a 1.382 or 1.618 multiple of
the distance between the previous top and bottom.
These relationships work to some extent in time too. Thus, a
third top or bottom may be found to have formed at a time
multiple of 1.382 or 1.618 of the time interval between two
previous two peaks. Also, the actual Fibonacci numbers themselves
work quite well in time, so analysts look for tops 7, 13,21,
34,55… weeks, months, or even years, after previous major tops
It should also be mentioned that these Fibonacci relationships
exist at all time scales. Thus, you can observe them on a price
chart where the minimum time increment is only 5 minutes, as well
as on very long-term charts where the minimum time increment is
Metaphysical Significance of Fibonacci Structures
The existence of Fibonacci number structures throughout the
universe, at every level, has tremendous implications. it is
important to understand THAT THERE IS NO SCIENTIFIC EXPLANATION
WHATSOEVER for their existence. That they do exist, no scientist
would ever dispute. However, WHY they exist is a complete and
Thus, with Fibonacci numbers, as well as other numerical
sequences that are present in nature, we have direct evidence for
an underlying structure to the universe; a shaping principle that
defies our rational intelligence to explain. If these numbers
were isolated in one instance, such as ONLY on the shells of
snails, we might think no more of it. It is a local phenomenon,
peculiar to the specific situation. However, the fact that they
occur almost everywhere you look, and in a breathtaking diversity
of forms, gives any open-minded observer serious pause for
Thus, there are certain engineering principles built into the
universe, and Fibonacci numbers are an outward evidence of this
secret order. Although we do not understand the mechanism, the
presence of this constant structure of numbers across a wide
range of nàtural phenomena is proof of a hidden order to
creation. At present, this hidden order remains within the realm
Fibonacci numbers also point to a powerful interconnection
between all things. For we have seen that living beings, like
snails and insects, inanimate objects like spiral galaxies, and
even abstract phenomena like stock price charts, all exhibit the
same underlying number sequence. This implies an
inter-connectedness between all things that is at present
unexplained. Science is coming to a realization of the extent to
which the universe exhibits this holistic structure. However, as
it has been for countless ages past, the mystery of Fibonacci
numbers, and the underlying unity of Creation with Consciousness
itself, remains within the realms of mystical exploration.
Copyright 2001, Asoka Selvarajah. All Rights Reserved.
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