The Fingerprint Of God

The Fingerprint of God is a guide that explains how to incorporate Fibonacci ratios in trading strategies and analysis.

1. The Finger Print of God
By Jacques Joubert
3
rd March 2012



You’re in a moment - staring deeply into the imagination of an artist as you contemplate the
creativity of his works. What is it that makes this painting so beautiful you ask yourself? You
notice how calm you feel and the sound of jazz music in the background of the museum as
you shift your gaze to admire the architecture.

There comes a point where you ask yourself: what is it that defines beauty?

There is something that all beauty has in common, a ratio so unique that it can only be
described as the fingerprint of God, the Divine Proportion 1.618.

It might not surprise you that there is a direct relationship between what we find
aesthetically pleasing and mathematics. This is because “mans senses enjoy objects that are
properly proportioned: Thomas Aquinas”. This relationship is known to many as the golden
ratio, which can be explained using the Fibonacci Summation.

Who was Fibonacci?
Born in Italy 1175, was a mathematician by the name of Leonardo Pisano (Fibonacci) - many
a man would question the roots of his name and how it came to pass that he is referred to
as Fibonacci. For further reading: Autobiography of Leonardo Pisano. The only important
thing that you need to take out of that story is that Fibonacci is a kind of nickname. (Richard
Grimm 1973)

Note: Fibonacci is not to be confused with “Captain Awesome” Leonardo da Vinci who was
born many moons later.

Fibonacci became famous for being one of the first to introduce the decimal number system
(Hindu-Arabic number system) to Europe. He published a book named “Libar Abaci” that
quickly influenced many mathematicians in Europe as they were using the Stone Age Roman
numeral system (I II III IV V X). Have you ever tried doing arithmetic using roman
numerals? I didn’t think so.

2. In chapter 12 of his book “Libar Abaci” Fibonacci goes on to explain the solution of how to
calculate the birth rates of rabbits. It is this very calculation that gave birth to the Fibonacci
Summation series.

The Fibonacci Series
Prepare to be blown away!
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584...

The series is produced by adding two consecutive numbers to produce the next. I.e. Each
subsequent number is the sum of the previous 2. Example: 21 + 34 = 55.

The Marvel of this series comes into effect when we divide any number in the series by the
previous number. Example: 89 / 55 = 1.618. You will notice that we can do this with any pair
in the series and we will approach 1.618 asymptotically. The ratio is irrational, meaning that
it produces a never ending unpredictable combination of decimals after the initial 1.618.
(Phi and the Fibonacci series [SA])

1.618 is the Golden Ratio, the Divine Proportion and Golden Mean. It is a phenomenon that
has been studied for hundreds of years and was coined by Kepler “as one of the jewels of
geometry”. The golden ratio is better known to some mathematicians as Phi (1.618). An
easier to understand definition would be: The Golden Ratio is a term used to describe
aesthetically pleasing proportioning within a piece. However, it is not merely a term - it is an
actual ratio. (Golden Ratio [SA])

There are plenty of examples of the Golden Ratio occurring in nature and beauty, to name
but a few:
 Structure of the lungs
 The human body
 The arrangement of sunflower seeds
 Architecture
 Music
 Art
 Human thinking patterns
 Hearing and balance
 In DNA
 The manner in which certain plants grow
 Sea shells
 Financial markets

A great example of how awe striking the Golden Ratio is would be to investigate the role it
plays in the clock cycle of our brain waves. In 2003 Weiss, H and Weiss, V wrote a paper on
this topic (Neuroscience2 2008: 145-154.).

To quote an abstract from the paper: “The principle of information coding by the brain
seems to be based on the golden mean. Since decades psychologists have claimed memory
span to be the missing link between psychometric intelligence and cognition.”

3. A more popular example of the Golden Ratio in nature would be that of the Nautilus shell
and how it relates to a visual representation of the logarithmic spiral.












The Golden Ratio even plays its role in philosophy. Aristotle proposed that moral values
follow 2 extremes. One part excess and the other part deficiency, the desirable point would
be the Golden Mean between the two extremes. This philosophy was also expanded on by
St. Thomas Aquinas, a great catholic philosopher. (Kraut, R. 2001)

How does the Golden Ratio relate to Financial Markets?
Given the above, we can theorise that other interactions of man would follow that of a
fractal nature. The markets are filled with human wants and needs, mass psychology and
emotion, which creates and environment of chaos and non-linear phenomenon that we can
measure using fractal geometry, like a Fibonacci Retracement. (Williams, B [SA])

Fibonacci Ratios
So how do we calculate the points: 68.1%, 50%, 38.2%, and 23.6% and why are they more
important than say 165% or 40%? Going back to the Fibonacci sequence (0, 1, 1, 2, 3, 5, 8,
13, 21, 34, 55, 89, 144...) we can calculate the percentages.

By combining the Fibonacci series with the Golden Mean we can find mathematical
relationships and then move those relationships to our trading strategies.

There are two main sets of Fibonacci Ratios that I like to point out:
1. Pure Fibonacci Ratios
 61.8%, 38.2%, 23.6% etc..
 1.618%, 2.618%, 4.236% etc...

2. Assumed Ratios
 76.8, 78.6%, 85.1%, 88.6%, 90.8% etc...

Pure Fibonacci ratios
Carry more weight than the others as they are not based on any assumptions. The ratios
are: 61.8%, 38.2%, 23.6%, 1.618%, 2.618%, 4.236% etc...

4. Calculations for 23.6%, 38.2%, and 61.8%
 68.1 % Take any number in the Fibonacci series and divide it by the 1st number to
the right and you will get 68.1% (Example: 89/144 = 0.681)
 38.2%
Take any number in the series and divide it by the 2nd number to the right of the
sequence and you will get 38.2% (Example: 34/89 = 0.382)
 23.6%
Take any number in the series and divide it by the 3rd number to the right of the
sequence and you will get 23.6% (Example: 34/144 = 0.236)
(The Sequence and Ratios [SA])

Calculations for 1.618%, 2.618%, and 4.236%
 1.681 % Take any number in the Fibonacci series and divide it by the 1st number to
the left and you will get 1.681% (Example: 144/89 = 1.681)
 2.681 % Take any number in the series and divide it by the 2nd number to the left of
the sequence and you will get 2.681% (Example: 144/55 = 2.681)
 4.236 % Take any number in the series and divide it by the 3rd number to the left of
the sequence and you will get 4.236% (Example: 144/34 = 4.236)

Assumed Ratios
We assume that the same basic ratios that are found between Pure Fibonacci Ratios can be
used to calculate points that can’t be found using the Fibonacci sequence.

These ratios are also Fibonacci Ratios; however they are based on the assumption that this
ratio can be used over and over again to find others.

One such example is by squaring Phi or by calculating the square root of Phi and then
looking for mathematical relationships.

To the power of Phi (1.618)
1.618 to the power of 2 = 2.617
1.618 to the power of 3 = 4.235
Note: 2.617 and 4.235 are the Points used in Fibonacci Expansions

To the power of 61.8%
61.8% to the power of 2 = 38.1%
61.8% to the power of 3 = 23.6%
61.8% to the power of 4 = 14.5%

Now implement the root function of the Fibonacci ratios to find the assumed points as they
follow the same mathematical pattern but in the opposite direction.

5. Root Function:
Nth root of 61.8% = Assumed Fibonacci ratio
2nd root of 61.8% = 78.6%
3
rd
root of 61.8% = 85.1%
4
th
root of 61.8% = 88.6%
5
th
root of 61.8% = 90.8%

The inverse function:
1 – Pure Fibonacci ratio = inverse function
1 – 0.236 = 76.4%
1 – 0.381 = 61.9%
1 – 0.618 = 38.2%

Note:
There is a difference between the two assumed ratios 76.4% and 78.6%, which one should
we use in our calculations? Luckily both numbers are relatively close to each other and
therefore it won’t make a big difference as to which one you choose.

6. The Fibonacci Tool Box
There are 5 main Fibonacci studies / indicators we use to analyse the markets:
1. Fibonacci retracements
2. Fibonacci Expansions/ extensions
3. Fibonacci Fans
4. Fibonacci Arcs
5. Fibonacci Time Zones


Fibonacci Retracements
Financial markets also tend towards the golden ratio 1.618.

In technical analysis Fibonacci retracements are used as a tool to identify areas of support
and resistance. A line is drawn between a swing low and a swing high (or vice versa, in this
case it is an upward trend.) the bottom of the line will represent 100% and the top 0%.
Several more lines are then drawn at points 78.6%, 68.1%, 50%, 38.2%, and 23.6% as shown
in figure 1.2.

Each one of these lines represents a level of support and resistance based on the Swing
move. We can see that the 38.2 point acted as a temporary support before it broke and
retraced all the way to the 50% mark where it found support and then shot up to make a
new High. We could have increased our position size at the 50% and placed our stop losses
just below the 50% marker.

It is common for the market to retrace a percentage of the swing move before making a
new high as shown in Figure 1.1.The difficult part is trying to determine how far the market
will retrace. Later on we will explore Elliot wave theory which provides more information on
how to place a trade similar to this one.

Figure 1.1

7. Figure 1.2

When the market retraces to a certain percentage we need to make sure that we fully
understand how important that percentage is therefore I have listed a general guide line
below:

 23.6%
A very weak support line but during periods of high momentum the market will
often bounce off of this point.

 38.2%
This is the first important line of support. If the market breaks this point it is the first
indicator that the main trend is weakening.

 50%
This point is based on Dow Theory and basses its convenience on the fact that the
market often bounces from 50% of a retracement.

 68.1 %
The Golden Mean – this is the strongest of the support lines. Should price break this
point it would indicate that the previous swing move is no longer in effect and that
traders should keep an eye out for a reversal.

 78.6%
This point doesn’t carry as much weight as 61.8% and it is also an “assumed ratio”,
none the less it is a Fibonacci ratio and can be seen as a possible turning point. It can
be viewed to carry the same weight as 38.2%.

8.  100%
Look for double tops and bottoms to confirm trading ideas.
(Fibonacci Retracements [SA])

When fractal geometry is used to analyse the markets consideration must be given to
multiple theories. One such theory is Dow Theory that states: “The thirty-five year record of
the averages shows a fairly uniform recovery after every major primary action, and such
recoveries average around 50% of the ground lost on the decline”. (The Dow Theory [SA])

By adding this information to the Fibonacci retracements we can build a new and improved
model for identifying possible support and resistance levels; this is the reason behind adding
the 50%.

9. Fibonacci Expansions

What are Fibonacci Expansions?
It is an indicator used by traders and market analysts to indicate S/R zones but more
importantly to determine points at which to take profits.

Often a problem will arise where the market will hit an all time high and we no longer have
previous market data from which to draw S/R lines. This is where the Fibonacci Expansions
come into play. They are used for three main reasons:
1. To find possible S/R zones, should the current trend continue. Most useful when
there is no historical data to fall back on.
2. Fibonacci retracements are used to take a position in the market; Fibonacci
Expansions are used to indicate points at which to take profits (close a position).
3. To determine the distance of wave 3. (see Figure 2.5)





Figure 2.1

10. Figure 2.2

As you can see from figure 2.1 and figure 2.2 that there are 2 points that need to be
calculated on this instrument. The first line is drawn from the swing low to the swing high,
point A to B (in terms of a Bull trend) and the second from where the retracement begins to
where it ends, point B to C. Once these points have been determined we can calculate the
Expansions.

The maths behind the madness:
To better understand how the instrument works, a mathematical explanation is provided:

The first step in the expansion principle is to calculate the length from which to work. To do
this we must first determine the distance of the impulse move less the distance of the
retracement which will equal the new distance being measured.

By looking at figure 2.3 we can provide an example of the equation, A – B = C.

11. Figure 2.3

The Length of C is then used to calculate future points of support and resistance by relating
its length to Phi (1.618).

In figure 2.4 we can see that b is in the ratio 1.618 to a. Now imagine C from figure 2.3
represents b in figure 2.4. By using mathematics we can simply multiply the length of b by
one of the Fibonacci ratios to get the points 68%, 168%, 261%, and 423% this is what we do
when using the Fibonacci Expansions. (Trading Fibonacci Expansions [SA])


Figure 2.4

The 3 most popular points used in this indicator are: 61.8%, 100% and 168%. (261.8% and
423% are also Fibonacci zones but are not as popular in expansions)
 168% point represents the total distance before reaching point a from b
 100% does not form part of the Fibonacci sequence but is derived from Elliot Wave
Theory that states “The Fibonacci Expansion's 100% level is merely the technique
known as a ‘measured move’. A measured move takes advantage of the fact that
Wave C will often move the same amount that Wave A did, though perhaps at a
different angle.” (GrayGhost 01 May 2008)
 68% is a key Fibonacci level

It is important to note that the third wave (see figure 2.5) is considered to end near these
levels. (Fibonacci Expansion [SA])

12. Figure 2.5

How can I use this indicator in my trading?
Example: Traders will often use the Fibonacci retracements to indicate at which price to
take a position, there after they will use Fibonacci expansions to indicate possible points at
which to take profit and close the position. (Note: it is important to use Fibonacci indicators
in conjunction with others in order to increase your chances of a successful trade.) (Trading
Fibonacci Expansions [SA])

A nifty thing about the expansion indicator is that it forces traders to move with the trend.
This automatically helps to reduce risk and improve trading success. (Trading Fibonacci
Expansions [SA])

13. Fibonacci Fans

What are Fibonacci Fans (FFs)?
FFs are a technical indicator that helps traders to see possible S/R lines in a developing trend
by displaying these points in a price channel formation. (Fibonacci Fan 1 [SA])

They are created by drawing a line from a swing low to a swing high, point A to point B in
terms of a bullish move (see Figure 3.1 & Figure 3.2) or from a swing high to a swing low, in
terms of a bearish move. We then apply the Fibonacci ratios, the most common being 38.2%
and 61.8%. We can however add all the Fibonacci ratios. (Fibonacci Fun [SA])


Figure 3.1

Figure 3.2

Important: a lot of programmes will automatically input 50% into the Fibonacci Fan but this
is wrong! The 50% line is derived from Dow Theory and has no use in the Fibonacci Fan
model. (The Dow Theory [SA])


The maths behind the madness:
(Fibonacci Fun [SA]):

14. First a line is drawn between a swing low and swing high (We are talking about a bull move,
the opposite applies for a bearish move.). These two points represent extreme points.
Figure 3.2

Figure 3.2

Then we draw an invisible vertical line through the second extreme point and break the
invisible line up by dividing it proportionately using the Fibonacci sequences. Figure 3.3


Figure 3.3

Diagonal lines are then drawn from the first extreme point through the marked Fibonacci
points on the invisible line. Figure 3.4

Figure 3.4
We then only include numbers that are part of the Fibonacci ratios (23.6%, 38.2% and
61.8%).

How can I use this indicator in my trading?
 A way to find possible S/R zones. This will help us to minimize risk on positions as
well as possible entry and exit points depending on your trading strategy.
 Help to predict the range of a prevailing trend. (Fibonacci Fan 1 [SA])
 Fibonacci Fans are often used in conjunction with the Fibonacci Arc.

Note: Use in conjunction with other indicators

15. Fibonacci Arcs

What is a Fibonacci Arc (FA)?
A FA is an indicator that helps to spot possible S/R zones based on previous impulse moves.
It is drawn from a swing low to swing high, points A to B in Figure 4.1 & 4.2 (in terms of a
bullish move). It then generates curved lines that aid in pointing out key S/R zones and the
possible range of the prevailing trend. (Fibonacci Arcs1 [SA])

Figure 4.1

Figure4.2

16. The maths behind the madness
A line is drawn from a swing low to swing high or vice versa depending on the trend. The
line is then divided into proportions by using the Fibonacci ratios. These distances are then
used to create a radius around the line which is the indicator we term a Fibonacci Arc.
(Fibonacci Arcs2 [SA])

How can I use this indicator in my trading?

 FAs are simply another way to find S/R zones however they have the added
advantage of forecasting a prediction on how the market will range. This is useful for
traders when trying to make profits from a sideways / ranging market.
 A popular strategy is to combine Fibonacci Arcs with Fibonacci fans and the points
where the two indicators cross will indicate extremely strong areas of S/R. (Fibonacci
Arcs3 [SA])

Important: take note that the Fibonacci Arcs will differ in size depending on the settings of
your X and Y axis. A savvy trader will remember to experiment with the indicator in order to
find the best fit (Fibonacci Arcs3 [SA]). It is my opinion however that as soon as we start to
“experiment” we start to lose focus on the academic side of how to trade and we move into
a territory that is rife with incorrect assumptions.

A better indicator to use than the Fibonacci Arc would be a Fibonacci Spiral as it maintains
the integrity of the golden mean, 1.618. The Fibonacci spiral/ logarithmic spiral is not a
popular indicator and would probably take some work trying to find it or build it on your
own.

17. Fibonacci Time Zones

What is a Fibonacci Time zone?
With the previous indicators our focus has been the price of the asset but what about
timing? Surely we can determine timing by applying the Fibonacci ratios to the markets? If
human beings are a living expression of the Golden ratio then surely 1.618 will indicate
times of peak performance or increased interest in a buying or selling decision. The
Fibonacci Time Zones try to measure this phenomenon.

The main difference with this indicator is that it measures the chart using a vertical lines to
determine points of increased volatility and the times at which they will occur. (Fibonacci
time Zones2 [SA])

There are two main methods for plotting the Fibonacci Time Zones:

Method 1
Plot the time zones from swing low to swing high. (Figure 5.1) (Drawing Fibonacci time
Zones [SA])

Figure 5.1



Method 2
Plot the time zones form double tops or double bottoms. (Figure 5.2) (Fibonacci time
Zones1 [SA])

18. Figure 5.2

Both methods will work however it is important to note that we are looking to measure the
amount of time it took for significant events/ moves and then use that distance for a base
interval. (Fibonacci time Zones1 [SA])

The maths behind the madness:
The Fibonacci Time Zones follow the Fibonacci sequence very closely (1, 2, 3, 5, 8, 13, 21,
34...).

Calculation for the time zones:
The base interval is simply multiplied by 1.618 every time to get the next Fibonacci ratio.
(Fibonacci Time Zones1 [SA])



Note: Keep in mind that the ratio trends asymptotically towards 1.618, meaning that the
first few vertical lines don’t carry as much weight as the others.

How can I use this indicator in my trading?

Time zones have many applications, to mention a few:
 It could be used in break out trading to get into a position. For this we recommend
using it in conjunction with Bollinger bands.
 Other reasons could be to protect investments where you wouldn’t like to see a lot
of volatility.
 Overall this indicator is best suited for momentum traders looking for movement.

19. Again this is an indicator to be used in conjunction with other indicators in order to increase
the possibility of a successful trade.



Conclusion on the 4 main Fibonacci indicators used in the markets

It is my opinion that the three most reliable indicators are:
 Fibonacci Retracements
 Fibonacci Expansions

I back up this conclusion by having a look at the mathematics behind the indicators and
point out they follow sensible logic. Too many people don’t understand the maths behind
the indicators they are using and then they extrapolate the wrong ratios in other indicators.
Example: the 50% mark is not a Fibonacci Ratio, it stems from Dow Theory. Now it is
appropriate to use the 50% mark in a Fibonacci retracement as the two theories
complement each other, however it is incorrect to include the 50% mark in any of the other
Fibonacci indicators as it makes no logical sense as to what we are trying to measure.

The 2 most unreliable indicators in my opinion are:
 Fibonacci time Zones
 Fibonacci arcs

The reasons are as follows:
 Fibonacci time zones: although the idea behind the indicator has good merit, it is
hard to tell from what point to what point we should draw the indicator as well as
what outcome we should expect from the forecasted Time zones.
 Fibonacci Arcs: the arcs are subjective to the user’s opinion. This is because the Arcs
are all of different shapes and sizes depending on the size of your X and Y axis. To
quote an above text “as soon as we start to ‘experiment’ we start to lose focus on the
academic side of how to trade and we move into a territory that is rife with incorrect
assumptions”

Note: there are many other Fibonacci indicators that can be manipulated and developed for
trading, however in this chapter we only cover the 5 main indicators: Retracements,
Expansions, Fans, arcs and time zones.

20. Advanced Fibonacci Trading Techniques

Fibonacci Clusters

Because Fibonacci tools are implemented manually it creates an environment where S/R
zones are subjective to the individual investor – for this reason that we use Fibonacci
Clusters.

Fibonacci Clustering is a method used by traders in order to see which S/R zones have a
higher probability of a reversal than others.

In order to do this a trader will combine different Fibonacci tools and draw them from
various points, after which we will be able to see a visual representation of the S/R zones
that carry more weight. (Fibonacci clusters [SA])

The more Fibonacci lines there are grouped in a particular area the stronger the S/R will be
and the greater the probability of a reversal.

This can in fact be done with all the Fibonacci tools.




Figure 6.1

In Figure 6.1 we can see the area of grouped S/R lines around the blue cluster of Fibonacci
zones will act stronger than those zones with a smaller or widely spread cluster.

21. Gartley Patterns:

A Gartley Pattern is a complex price pattern that falls within the scope of Price Harmonics
and is covered in a later section.



Common mistakes when using Fibonacci tools
Biggest weakness is the fact that Fibonacci tools is subject to the traders discretion and
therefore it is easier to miss calculate S/R zones. Placing the indicators in the wrong place
leads to cumulative losses and poor entry and exit points

1. Forgetting to use multiple time frame analysis
Often a newbie will get so excited about the current time frame under analysis that
he will forget to check the others in order to support his decision.

It is fair to say that Fibonacci indicators carry the same weight on all time frames
because the markets follow a fractal pattern. We can even use the smallest of time
frames, so long as you remember to make use of multiple time frame analysis.

2. Only using Fibonacci indicators
Any indicator used on its own is weak but coupling it with another dramatically
increases the chances of a successful trading idea.

In such a situation I feel the need to recite the phrase “kiss” Keep it simple stupid.
When combining indicators it is important to not lose focus of the bigger picture. Try
to combine 2 or 3 indicators together. The more indicators you use, the more
conflicting trading ideas you will receive and the greater your odds of being a victim
of analysis paralysis.

3. Body to body and wick to wick
When drawing in Fibonacci support zone it is important to draw the indicator from
the beginning of the impulse move to the end, at the same time keeping consistency,
drawing from a candle body to another or from wick to wick. Not: body to wick or
vice versa.

4. Avoid outliers
If the wick of a particular candle is abnormally long then it should not be used.
Outliers in price data cause indicators to lose accuracy. In such a situation it would

22. be better to use discretion and look for a better point from which to extrapolate
information.



Keeping integrity
People tend to make trading too mathematical, so mathematical that the people
implementing the models don’t know what’s going on. That is exactly what is happening to
traders that use Fibonacci indicators and don’t understand what percentages should be in
what indicator.

These percentages have no Fibonacci significance what so ever:
 50% Derived from Dow Theory (can only be implemented in a Fibonacci
Retracement)
 100% derived from Elliot wave theory and used in Fibonacci Expansions. Make sure
to understand the theory behind this as it refers to the ABC correction.

Trading Best practices using Fibonacci tools
As a trader it is your job to let profits run and cut losses short. By this very logic it means
that we should use 23.6%, 38.2%, and 61.8% ratios in our tool box, however when we start
to implement ratios like 78.6% and those above it we start to base our trades on hope which
can lead to a very big losing streak and revenge trades.

No Holy Grail in trading!
The truth is that there is no Holy Grail indicator in trading. There is no trading system that
will have 100% accuracy and there is no point in breaking your back trying to find this magic
indicator.

It’s time to whip out the old legal pad and draw a pros and cons list:
There are two types of trading strategies:
1. Trend following
2. Trend breakout

23. Each one of these techniques has strengths and weaknesses. The best way to explain why
there is no point in mixing indicators to find a Holy Grail would be to discuss the study of
genetic algorithms. To sum it up a genetic algorithm runs all the possible combinations of
trading indicators and time periods as well as other variables using computers to come up
with the most accurate trading system.

It is because of this technology that we can be certain that with using the current
mathematical models we can’t develop a perfect trading system. This study then further
bridges out into fuzzy logic.

I’d like to conclude that Fibonacci indicators and the Golden Ratio all build up on the field of
Fractal Geometry and how it can be used within financial markets. You will notice that when
drawing the main support and resistance levels manually they will line up with the Fibonacci
zones

The question stands as to whether or not price behaviour is attributable to a natural law or
if the Fibonacci retracements are a self fulfilling prophecy. This is a question for the
philosophers to sit and figure out. From a practical point of view I can stand testament to
the accuracy of basing trade ideas on the Fibonacci sequence.

24. Source List:
The Title: “the Fingerprint of God” A common phrase used for the Fibonacci pattern/ Golden
ratio amongst evangelists.

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Golden Ratio. [SA].[O]. Available:
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Accessed 03 March 2012

Harald Weiss and Volkmar Weiss. 2003. The golden mean as clock cycle of brain waves [O].
Available:
http://www.v-weiss.de/chaos.html
Accessed 03 March 2012

Kraut, R. 2001. Stanford encyclopaedia of philosophy.[O]. Available:
http://plato.stanford.edu/entries/aristotle-ethics/#DocMea
Accessed 03 March 2012

Phi and the Fibonacci series.[SA].[O]. Available:
http://www.goldennumber.net/fibonser.htm
Accessed 03 March 2012

Richard E. Grimm. 1973. Autobiography of Leonardo Pisano [O]. Available:
http://www.fq.math.ca/Scanned/11-1/grimm.pdf
Accessed 03 March 2012

Roopun, Anita K. et al. (2008). Temporal interactions between cortical rhythms. Frontiers in
Neuroscience 2, 145-154.

The Dow Theory.[SA].[O]. Available:
http://www.marketthoughts.com/dow_theory.html
Accessed 03 March 2012

The Sequence and Ratios.[SA].[O]. Available:
http://stockcharts.com/school/doku.php?id=chart_school:chart_analysis:fibonacci_retrace
men
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Williams, B.[SA].[O]. Available:
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25. http://fxcodebase.com/wiki/index.php/Fibonacci_Time_Zones
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Fibonacci Expansions.[SA].[O]. Available:
http://ta.mql4.com/linestudies/fibonacci_expansion
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Fibonacci time Zones2.[SA].[O]. Available:
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GrayGhost. 01 May 2008. Fibonacci Expansion. [O]. Available:
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Fibonacci Fan1.[SA].[O]. Available:
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Fibonacci Fun.[SA].[O]. Available:
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Images:
http://www.flickr.com/photos/mbell1975/2370055374/Michelangelo Sistine chapel
http://www.creativeautomaton.com/designforbusiness/the-golden-ratio-a-brief-on-design/
http://browse.deviantart.com/photography/?q=Nautilus#/d2uphuh
http://futurepredictions.com/2011/08/the-great-philosophers/

Golden mean
http://www.absoluteastronomy.com/topics/Golden_ratio