Anahata Nada
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       =======  Understanding Hinduism  =======

Nature Uses Maths

Click on underscored words to open paragraph

Nature Uses Maths   (1)
Anahata  Nada -Uncreated Sound
1 Natural Number Series defined
2 Chakras of the Head
3. The thirteen chakras
4 Determination of the pitches for each chakra note
5 Colours
6 Smell, Taste and Sensation
7 Brain functions
8 Practical applications

Nature Uses Maths   (2)
The Fibonacci Rectangles and Shell Spirals
Pine cones
Vegetables and Fruit
Seed heads
Fibonacci Fingers?
Arrangements of the leaves
Planetary orbits in our solar system
Stock Exchange Price Cycles
Who was Fibonacci?
Introducing the Decimal Number system into Europe

An Essay

Anahata  Nada -Uncreated Sound
By Mr. Roger Gould-King
              AUM (The most sacred symbol in Hinduism)

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Mr. Roger Gould-King
Total silence is perfection. Silence is necessary for sound to exist, it is the aural canvas for melody’s brush in our three dimensional world. The sounds we know are produced by at least two elements – the waves on a shore, wind in the trees, the blades of a bagpipe reed vibrating together, two lips, drum and drumstick, fingering a guitar string and so on.

One of the koans of Zen asks, "What is the sound of one hand clapping?". This sound is known in the Sanskrit tradition as "Anahata Nada," the "Unmade Sound". This means a sound not made in the way we know of – it is the "sound" of the universe, the primal sound of energy itself.

Ancient tradition says that the audible sound which most resembles this unmade sound is the sound of "AUM" (OM). ( Brahma Randhra: Brahma-aperture; opening in the crown of the head; "the tiniest of apertures, in which is the silent, primordial sound, which gives you the impression that you are, but you really are not" (Nisargadatta)). According to the Vedas, AUM is the most sacred of all words, out of which emanated the universe. The symbol of both the personal God and the Brahman or Absolute. AUM is regarded by Hindus as the greatest mantra being of incalculable spiritual potency.

Aum is not, in itself, the un-struck sound, but leads one to it. The mantra is composed of four elements. Three are vocal sounds: A, U, and M, while the fourth element is the silence which begins and ends the audible sound, the silence which supports it. The objective of intoning AUM is not only the mantra itself, but the experience of perceiving the unmade sound that supports it. This is the same as seeking the space which "supports" the universe and its galaxies : the "emptiness" or nothingness of space is necessary for the existence of everything seen and unseen.

Everything seen and unseen, heard and unheard, smelt and un-smelt, felt and unfelt, tasted and un-tasted, are manifestations of pure energy. This energy is the "container" for all things, and it is the seemingly elusive Source people have given numerous names to – God, Self, Brahman , Godhead. The Absolute, the Supreme Reality, the Ultimate Reality, Truth or the Self of the Vedanta Philosophy are also used interchangeably for Brahman, and so on. Our interface with the material world is through our senses and the interpretation of these sensations with our minds – our thoughts. At a basic level, we tend to see things in their material form, not as energy manifested in many different ways. We categorise created forms with labels, naming this form a rock, that a rose, the other water, another a human being, and yet others as "animals". What we seek is LOVE, but in applying labels to created forms, we end up with:

A weed is a flower that has never been loved

There are about 50 definitions of the word "love" in an English dictionary, but not one equates love to "God". In defining love in these many different ways, we define our perception of everything, but if we become perfect Love ourselves, all apparent differences disappear and we see things as they really are. There is no such thing as "ugly" or "beautiful", for these are only descriptions we apply to describe our own perceptions of things we haven’t yet seen as they see themselves in their own created form; we seek to be ekanta vasa [ekaant vaas]: free from mental concepts; "dwelling in mental solitude".

While classification appears to be useful as a means of physical identification in a three-dimensional world created in the mind, it is the major barrier to identifying oneself with all created things that all came from the same Source, and this includes us. From this Source we are born to don the garment of our bodies in order to experience this world, and in the death of the body we "return" to the Source which we in fact, never left : we are always part of this energy and it is only the manifestation of our present life form coupled to the interference of our minds, which leads to the illusion that each one of us is a "separate" person.

This essay is not about suggesting some "path" for someone to follow, because a path implies distance, a distance separating oneself from the destination : Self, God, Brahman. To go on some religious path or other often turns out to be like someone on one of those exercise machine treadmills- one can walk forever and never get to the desired destination because one never left it in the first place.

Chakras and the Natural Number series

Natural Number Series defined

Many years ago I was enthralled by the sight of a certain genus of flowering plant in a remote mountainous area, being pollinated by bees called by the plants by their emission of a distinct humming sound. After recording the television documentary, I checked the frequency of the fundamental frequency generated by the plants and found it to be 432 Hertz, or cycles per second. This prompted me to place small battery powered sound generators in the flower beds on my farm where I kept bee hives, and to discovering a whole new world of plant and bee intelligence.

All life used to live in synergy, and still strives to, despite the decimation of the environment by humankind. This interdependence establishes harmony on the physical and spiritual planes, each life form being dependent on the other. The eradication of a species creates an irreplaceable void which nature tries to balance as best it can, but usually, this leads often to further extinction of other species because a fundamental link in the material life form and spiritual chain has been destroyed.

The Natural Number series is formed by adding a succeeding number to the previous root number, and is as follows :

0,1,1,2,3,5,8,13,21,34,55,89,144,233,377 … and so on. In other words, 5+8=13, 8+13=21, 21+13=34 etc. If one divides one number by another, one gets a ratio, 89/144= 0.618, or, 144/89 = 1.618, and, 144/233=0.618.

This ratio, known variously as the golden section, golden mean or divine proportion, can be found everywhere in nature. It is represented by a rectangle in which the width compared to the length, is in the same proportion as the length compared to the sum of the width and length; i.e. "the smaller is to the larger, as the larger is to the whole". This sequence will be found in genetics and geometry, snail shells and in the growth formations of plants and other life forms, including the proportions of the human body.

2 Chakras of the Head

If one assigns the value of one to the distance between the chin and the crown of the human head, then 0.618 of this distance will be found to correspond exactly to the various locations of the head chakras.

2.1 Location of Head Chakras

In his exhaustive definitive work on the chakras according to the traditional Indian understanding, Layayoga - an Advanced Method of Concentration, Shyam Sundar Goswami, citing many references, describes thirteen chakras altogether; the seven standard chakras and six minor ones. The following lists the chakras according to the 13-chakra model.

The location of these chakras is shown in the drawing.

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3. The thirteen chakras
Chakra Location Petals Association Colour Tone
Sahasrara above head 1000 transcendent gold 486
Guru above head 12 transcendent silver  
Nirvana crown 100 mind origin white  
Indu forehead 16 buddhi    
Manas forehead 6 chitta    
Ajna brow 2 manas violet 432
Talu/Lalana roof of mouth 12 or 64      
Vishuddha throat 16 space blue 405
Anahata heart 12 air green 370
Hrit below heart 8      
Manipura sternum base 10 fire yellow 324
Svadhistana below navel 6 water orange 288
Muladhara base of spine 4 earth red 270


The same method for determining the location of the head
chakras, is used to locate those for the rest of the torso.
This is illustrated in the diagram.


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The physical location of each chakra is important in terms of
its response to vibrations, i.e. tones, colours and so on.

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4 Determination of the pitches for each chakra note

The standard Western musical scale is based on an equally tempered (also called the 12-tone chromatic) scale. In this scale, an octave consists of 12 equally spaced frequencies, each note being 2^1/12 (twelfth root of 2) from the previous note, as shown in the following table for the A440 musical scale. Therefore, the twelfth note is double the frequency of the first note.


Musical Note Frequency N Natural tube
Equation Hz


A 440 440.00 432.00 432 397mm do
A# 440 x 2^1/12 466.16        
B 440 x 2^2/12 493.88 484.90 486 353mm re
C 440 x 2^3/12 523.25 513.75      
C# 440 x 2^4/12 554.37 544.29 540 318mm mi
D 440 x 2^5/12 587.33 576.65 576 298mm fa
D# 440 x 2^6/12 622.25        
E 440 x 2^7/12 659.26 647.27 648 265mm sol
F 440 x 2^8/12 698.46        
F# 440 x 2^9/12 739.99 726.53 740 232mm la
G 440 x 2^10/12 783.99        
G# 440 x 2^11/12 830.61 815.51 810 212mm ti
A2 440 x 2^12/12 880.00 864.00     do


The natural scale is the scale used by me for the chakras, and not the Western diatonic scale per se. A long time ago the Highland bagpipe and other traditional instruments had a fundamental "A" of 432Hz, but this was increased to 440Hz with advent of the "standard" of 440, followed later by further increases in pitch to get pipe bands to perform with brass bands and orchestras. This Westernisation of folk instruments destroyed their inherent relationships with natural phenomena, and consequent loss of spiritual synergy with nature.

The natural scale proposed is calculated on the harmonization of any note with the upper harmonics of the fundamental A=432. These "natural" frequencies will in turn harmonize with the relevant chakra centres in the body – as heard by the ears, and interpreted by the brain as sound.


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To hear what these notes actually sound like, the easiest thing to make is a set of pitch pipes as illustrated. Any thin-walled tubing will do, such as bamboo or reed, or even metal curtain rod. Closing one end of such a pipe and blowing across the open end will produce a note of a pitch determined by the length of the pipe. The lengths of these pipes is given in the preceding table, calculated for sea level at standard temperature.

To hear the sounds using your PC, use QBasic’s SOUND function to write a simple program to play the tones via your PC speaker – e.g. SOUND 432,18.2 will sound A for 1 second.

5 Colours

Like sound, colours vibrate at specific frequencies, and therefore have their corresponding sound equivalents. Blue for example, vibrates at about 749.999 billion Hertz or cycles per second, and corresponds to the throat chakra and sound frequency of G. Red vibrates at half that of blue, and corresponds to Muladhara and C. Therefore sound and/or colours can be used in conjunction with specific chakras.

6 Smell, Taste and Sensation
These three functions of the body all respond to vibrations. In meditation it is important to try to harmonise these three as well. Sitting in a very hot and humid room, surrounded by foul smells and having a bitter taste in the mouth, are not conducive to meditation.

7 Brain functions

The electroencephalograph (AKA. EEG) is a machine that monitors brainwave activity. Laboratories have conducted many studies and experiments using these tools to understand the four main brainwave patterns: BETA, ALPHA, THETA and DELTA.

Each frequency has a characteristic blueprint, and produces a distinctive state of consciousness. BETA waves (14 cycles per second and above) dominate the normal waking state of consciousness when attention is directed towards the outside world. ALPHA waves (8-13 cycles-per second) are present during dreaming and light meditation when the eyes are closed. THETA waves (4-7 cycles per second) occur in sleep and are dominant in the higher state of mediation.

In deep meditation and deep sleep, DELTA waves (0.5 to 3 cycles per second) are experienced. Each of these brainwave frequencies serves an important function. The optimum level for deep no-thought is in the realm of THETA. When in THETA, the senses are withdrawn from the external world and focused on the inner one. DELTA waves enable a total disassociation from three-dimensional existence and provide the most profound feelings of peace – and it is in this state that all life is "connected" directly to Self –"God".

In direct correlation, we see similar effects brought on by the constant and rhythmic drone of Tibetan Buddhist chants, which transport the monks and even other listeners into realms of meditation. These chants are at 432Hz in frequency, above or below this fundamental by one octave – 432 = 4.5 x 8 x 3 x 2 x 2.

Like sound waves, the brain has its own set of vibrations it uses to communicate with itself and the rest of the body. EEG equipment distinguishes these waves by measuring the speed with which neurons fire in-cycles per second.

Beta waves range between 13-40 HZ. The beta state is associated with peak concentration, heightened alertness and visual acuity. Nobel Prize Winner, Sir Francis Crick and other scientists believe that the 40HZ beta frequency may be key to the act of cognition.

Alpha waves range between 7-12 HZ. This is a place of deep relaxation, but not quite meditation. In Alpha, we begin to access the wealth of creativity that lies just below our conscious awareness - it is the gateway, the entry point that leads into deeper states of consciousness.

Theta waves range between 4-7 HZ. Theta is one of the more elusive and extraordinary realms one can explore. It is also known as the twilight state which we normally only experience briefly as we rise from delta upon waking, or drifting off to sleep. In theta we are in a waking dream, vivid imagery flashes before the mind's eye and we are receptive to information beyond our normal conscious awareness.

Delta waves range between 0-4 HZ. Delta is
associated with deep sleep.

8 Practical applications

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>The foregoing sections detail briefly the physical aspects of intoning AUM as a mantra to experience God, Self, Brahman , Godhead. The Absolute, the Supreme Reality, the Ultimate Reality, Truth.

It is at the moment when the end of the mmm is reached and terminated, that the "Silence" which is God, Self, Brahman , Godhead. The Absolute, the Supreme Reality, the Ultimate Reality, Truth, is experienced. Once this experience is attained, what one seeks has been identified as it were, and the destination of one’s path has been reached.

Please note that the intonation of any mantra can be done mentally, that is, it is not necessary to do so audibly.

From this point on, the objective is to attain a constant state of one-ness with Self, perfect bliss. It is then that one knows (as opposed to beliefs) that all life and every creature has a soul, that all things are sacred.

I offer these thoughts to you in humility and pray that you will find them of interest, and perhaps of use to you. I humbly offer this essay in the spirit of kshama-prarthana and ask your forgiveness for the shortcomings and mistakes it may contain.

Roger Gould-King
South Africa
January 2002

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Nature Uses Maths (2)

The Fibonacci Rectangles and Shell Spirals






The Fibonacci Rectangles and Shell Spirals
By Ron Knott

We can make another picture showing the Fibonacci numbers 1,1,2,3,5,8,13,21,.. if we start with two small squares of size 1 next to each other. On top of both of these draw a square of size 2 (=1+1).

We can now draw a new square - touching both a unit square and the latest square of side 2 - so having sides 3 units long; and then another touching both the 2-square and the 3-square (which has sides of 5 units). We can continue adding squares around the picture, each new square having a side which is as long as the sum of the latest two square's sides. This set of rectangles whose sides are two successive Fibonacci numbers in length and which are composed of squares with sides which are Fibonacci numbers, we will call the Fibonacci Rectangles



The next diagram shows that we can draw a spiral by putting together quarter circles, one in each new square. This is a spiral (the Fibonacci Spiral). A similar curve to this occurs in nature as the shape of a snail shell or some sea shells. Whereas the Fibonacci Rectangles spiral increases in size by a factor of Phi (1.618..) in a quarter of a turn (i.e. a point a further quarter of a turn round the curve is 1.618... times as far from the centre, and this applies to all points on the curve), the Nautilus spiral curve takes a whole turn before points move a factor of 1.618... from the centre.
These spiral shapes are called Equiangular or Logarithmic spirals.

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987


Pine cones

Can you see the two sets of spirals?
How many are there in each set?



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(the lines are drawn connecting the centres of
each segment of the pinecone):

Pine cones show the Fibonacci Spirals clearly.

Pine cones


Here is another pine cone. It is not only smaller,
but has a different spiral arrangement.
Use the buttons to help count the number
of spirals in each direction] on this pinecone.

Shows only the pinecone


Shows the segment edges


Show the outline only


Show one set of spiral



Show the other set of spirals



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Vegetables and Fruit


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Here is a picture of an ordinary cauliflower. Note how it is almost a pentagon in outline. Looking carefully, you can see a centre point, where the florets are smallest. Look again, and you will see the florets are organized in spirals around this centre in both directions.

How many spirals are there in each direction?
(lines are drawn between the florets):


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How many spirals are there in each direction?

Seed heads

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Fibonacci numbers can also be seen in the arrangement of seeds on flower heads. The picture here is Tim Stone's beautiful photograph of a Coneflower, used here by kind permission of Tim. The part of the flower in the picture is about 2 cm across. It is a member of the daisy family with the scientific name Echinacea purpura  and native to the Illinois prairie where he lives.

You can see that the orange "petals" seem to form spirals curving both to the left and to the right. At the edge of the picture, if you count those spiralling to the right as you go outwards, there are 55 spirals. A little further towards the centre and you can count 34 spirals. How many spirals go the other way at these places? You will see that the pair of numbers (counting spirals in curving left and curving right) are neighbours in the Fibonacci series.





The same happens in many seed and flower heads in nature. The reason seems to be that this arrangement forms an optimal packing of the seeds so that, no matter how large the seed head, they are uniformly packed at any stage, all the seeds being the same size, no crowding in the centre and not too sparse at the edges.

The spirals are patterns that the eye sees, "curvier" spirals appearing near the centre, flatter spirals (and more of them) appearing the farther out we go.

So the number of spirals we see, in either direction, is different for larger flower heads than for small. On a large flower head, we see more spirals further out than we do near the centre. The numbers of spirals in each direction are (almost always) neighbouring Fibonacci numbers!

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Fibonacci Fingers?

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Look at your own hand:

You have ...

2 hands each of which has .
5 fingers, each of which has ...
3 parts separated by ...
2 knuckles

Is this just a coincidence or not?????

However, if you measure the lengths of the bones in your finger (best seen by slightly bending the finger) does it look as if the ratio of the longest bone in a finger to the middle bone is Phi?
What about the ratio of the middle bone to the shortest bone (at the end of the finger) - Phi again?

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Arrangements of the leaves

Also, many plants show the Fibonacci numbers in the arrangements of the leaves around their stems. If we look down on a plant, the leaves are often arranged so that leaves above do not hide leaves below. This means that each gets a good share of the sunlight and catches the most rain to channel down to the roots as it runs down the leaf to the stem.
The computer generated ray-traced picture here is created by my brother, Brian.

Leaves per turn


The Fibonacci numbers occur when counting both the number of times we go around the stem, going from leaf to leaf, as well as counting the leaves we meet until we encounter a leaf directly above the starting one.

If we count in the other direction, we get a different number of turns for the same number of leaves.

The number of turns in each direction and the number of leaves met are three consecutive Fibonacci numbers!

For example, in the top plant in the picture above, we have 3 clockwise rotations before we meet a leaf directly above the first, passing 5 leaves on the way. If we go anti-clockwise, we need only 2 turns. Notice that 2, 3 and 5 are consecutive Fibonacci numbers.
For the lower plant in the picture, we have 5 clockwise rotations passing 8 leaves, or just 3 rotations in the anti-clockwise direction. This time 3, 5 and 8 are consecutive numbers in the Fibonacci sequence.

We can write this as, for the top plant, 3/5 clockwise rotations per leaf ( or 2/5 for the anticlockwise direction). For the second plant it is 5/8 of a turn per leaf (or 3/8).

Leaf arrangements of some common plants

The above are computer-generated "plants", but you can see the same thing on real plants. One estimate is that 90 percent of all plants exhibit this pattern of leaves involving the Fibonacci numbers.

Some common trees with their Fibonacci leaf arrangement numbers are:

1/2 elm, linden, lime, grasses
1/3 beech, hazel, grasses, blackberry
2/5 oak, cherry, apple, holly, plum, common groundsel
3/8 poplar, rose, pear, willow
5/13 pussy willow, almond


where t/n means each leaf is t/n of a turn after the
last leaf or that there is there are t turns for n leaves.

Cactus's spines often show the same spirals as we have already seen on pine cones, petals and leaf arrangements, but they are much more clearly visible. Charles Dills has noted that the Fibonacci numbers occur in Bromeliads.

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From other sources

Planetary orbits in our solar system

The time span of planetary orbits in our solar system (i.e., the
time it takes for each planet to make one complete revolution
around the sun), are related by Fibonacci relationships.

Specifically, given that the Earth takes one year to orbit the
sun, the time that it takes Venus to orbit the sun is
1.618 to the -1 years. The time it takes Mercury to orbit
the sun is approximately 1.618 to the -3 years. Mars is roughly
1.618 years, the asteroid belt is roughly 1.618 to the 3rd years,
Jupiter is approximately 1.618 to the 5th years, so on and so forth
(note that Pluto does not fit).

Explanations in greater details

Stock Exchange Price Cycles

In October 1982 there was a 10-planet alignment at which
point stock prices bottomed at the lower channel line which
is drawn across the 1942 and 1974 alignment/lows and then
the Dow exploded through the "Magic 1000" barrier and the
great bull market began that many believe continues to this day.

In August 1987, with the "Harmonic Convergence" and a
five-planet alignment, stock prices peaked along the upper
trendline running through the 1937 and 1962 peaks, then

(see "The End Of The World" in 8/17 issue of Newsweek.)

In January 1991 stock prices hit a major low with a 5-planet
alignment and solar eclipse. This marked the deadline for Iraq
to pull-out of Kuwait and the following day the Gulf War began.
The Allies success triggered an sharp rally in stock prices that
carried the DJIA through the 3000 barrier and since that time
stock prices have not gone lower.

In June 1992 there was a 5-planet alignment with a solar eclipse
at which point stock prices peaked and dropped off into October.
Notably, this alignment coincided with the beginning of Western
intervention in the Yugoslavian civil war.

The last major peak in the stock market occurred at Dow 4000
the upper trendline across the 1937, 1962 and 1987 peaks
in January of 1994 with a tight 7-planet alignment.
The market corrected for about a year after which the rally
began which carried stock prices through Dow 5000 last

(see "Rare Planet Alignment Bodes A Bust For Booming
Stock Market" in 1/12/94 issue of the Wall St. Journal, P.B1.)

At the current juncture, the stock market is likely peaking above
a Supercycle upper trendline running through the 1937 peak
and a 1966 peak at Dow 1000 that coincided with a six-planet alignment.This peak comes with a five-planet alignment with
the new moon a day prior to the March 20th Spring Equinox

Given how the current possible peak is connected by long-run trendlines,  planetary alignment turning points and, as is explained below, Fibonacci mathematical relationships, to the Elliott Wave Supercycle upswing in stock prices that started following the Great Depression of the 1930's, clearly the ingredients are in place for this to be a Grand Supercycle or even Millenium Cycle turning point.

Now this is where things get interesting.

If you take the time span between the peak in Jan. of 1994
and November/December of 1995 (when the Dow broke 5000
in association with a major planetary alignment in Sagitarrius)
and multiply it by the Fibonacci "Golden Ratio"
of 1.6180339887498949, then you get the approximate
time span between the 1992 alignment/peak and the
current alignment/possible peak.

(The Fibonacci golden ratio is based upon simple numerology.
Take any two numbers, e.g., 0 & 1, and add them together.
Next, take the sum and add it to the larger of the two numbers,
i.e., 1 + 1 = 2. Now add the new sum to the old sum, i.e.,
2 + 1 = 3. As you continue doing this you'll get a numerical
series like the following: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34...
The further out you go in the series, the closer the ratio of
two consecutive numbers comes to 1.6180339887498949
(or, if you take the ratio of a number to the next higher number,
On a final note, if you create a new series based upon the
exponential values of the golden ratio, i.e., 1.618034 to
the X power, you'll get a new Fibonacci series in which
each number is the sum of the two previous numbers, e.g.,
1.618 to the -1 plus 1.618 to the 0 equals
0.618 + 1.00 = 1.618,
and 1.618 is 1.618 to the 1st power.)

Now, if you take the time span between the June 1992
alignment/peak and the current possible alignment/peak and
multiply it by 1.618... (which is also 2.618, or 1.618 to the 2nd
power, times the length of time between the January 1994
alignment/peak and the current possible alignment peak),
then you get the time span from the 1991 alignment/price-low
and today.

If you keep doing this to the time periods between each of
the alignment/Elliott Wave stock market turning points above,
all the way to multiplying the time span between now and Jan.
1994 by 1.618 to the 6th power to find the time span from the
1962 peak to the present, then you uncover the code of Elliott
Waves and historical cycles. The correlation found by
comparing the lengths of time between the key planetary
alignment/stock market turning points noted above with exact
Fibonacci relationships is over 99.9 percent.

Now, to understand why this is so, you might note that
the time span of plantery orbits in our solar system (i.e., the
time it takes for each planet to make one complete revolution
around the sun), are related by Fibonacci relationships.

Specifically, given that the Earth takes one year to orbit the
sun, the time that it takes Venus to orbit the sun is
1.618 to the -1 years. The time it takes Mercury to orbit
the sun is approximately 1.618 to the -3 years. Mars is roughly
1.618 years, the asteroid belt is roughly 1.618 to the 3rd years,
Jupiter is approximately 1.618 to the 5th years, so on and
so forth (note that Pluto does not fit).

Again, the correlation is around 99 percent (excluding Pluto).

Given this property of the solar system, the "astroharmonics"
of when alignments will occur will be based upon Fibonacci
relationships. Thus, the time spans between planetary
alignments, which apparently affect turning points in
mass mood and world history, are related by Fibonacci

This is all quite remarkable given how R.N. Elliott's
theoretical Elliot Wave fractal-based wave pattern
is characterized by Fibonacci relationships. Yet, he
didn't realize the connection between Elliott
Wave patterns and astroharmonics! 

Who was Fibonacci?

Fibonacci's portrait






The "greatest European mathematician
of the middle ages", his full name was Leonardo of Pisa,
or Leonardo Pisano in Italian since he was born in Pisa (Italy),
the city with the famous Leaning Tower, about 1175 AD.

He called himself Fibonacci [pronounced fib-on-arch-ee or
fee-bur-narch-ee] short for filius Bonacci which means son of
Bonacci. Since Fibonacci in Latin is "filius Bonacci" and
means "the son of Bonacci",

So Leonardo grew up with a North African education under the
Moors and later travelled extensively around the Mediterranean
coast. He would have met with many merchants and learned of
their systems of doing arithmetic. He soon realised the many
advantages of the "Hindu-Arabic" system over all the others.

D E Smith points out that another famous Italian - St Francis of
Assisi (a nearby Italian town) - was also alive at the same time
as Fibonacci: St Francis was born about 1182 (after Fibonacci's
around 1175) and died in 1226 (before Fibonacci's death
commonly assumed to be around 1250).

[The portrait here is a link to the University of St Andrew's site
which has more on Fibonacci himself, his life and works.]

Introducing the Decimal Number
system into Europe

He was one of the first people to introduce the Hindu-Arabic
number system into Europe - the positional system we use
today - based on ten digits with its decimal point and a symbol
for zero:

    1  2  3  4  5  6  7  8  9  .  and 0                                                  

His book on how to do arithmetic in the decimal system, called
Liber abbaci (meaning Book of the Abacus or Book of
Calculating)  completed in 1202 persuaded many European mathematicians of his day to use this "new" system.

The book describes (in Latin) the rules we all now learn at
elementary school for adding numbers, subtracting, multiplying
and dividing, together with many problems to illustrate the methods:

1 7 4 +                   1 7 4 -                     1 7 4 x     
   2 8                         2 8                           2 8          
-----                          -----                        -------
2 0 2                     1 4 6                      3 4 8 0 +
-----                          -----                     1 3 9 2
                                                           4 8 7 2

1 7 4 28
6 remainder 6

Let's first of all look at the Roman number system still in use
in Europe at that time (1200) and see how awkward it was for

 The Numerals are letters

The method in use in Europe until
then used the Roman numerals:

  I = 1, 
  V = 5, 
  X = 10, 
  L = 50, 
  C = 100, 
  D = 500 and 
  M = 1000                                                   

You can still see them used on foundation stones of old buildings
and on some clocks.

The Additive rule

For instance, 13 would be written as XIII or perhaps IIIXX. This
is reflected in the Roman language of Latin where 23 is spoken
as tres et viginti which translates as three and twenty.

Arithmetic was not easy in the Roman system:

    CLXXIIII added to XXVIII  is CCII
    CLXXIIII less     XXVIII  is CXXXXVI                                                  

The Decimal Positional System

The system that Fibonacci introduced into Europe came from India and Arabia and used the Arabic symbols 1, 2, 3, 4, 5, 6, 7, 8, 9 with, most importantly, a symbol for zero 0.
With Roman numbers, 2003 could be written as MMIII or, just as clearly, it could be written as IIIMM - the order does not matter since the values of the letters are added to make the number in the original (unabbreviated) system. With the abbreviated system of IX meaning 9, then the order did matter but it seems this sytem was not often used in Roman times.
In the "new system", the order does matter always since 23 is quite a different number to 32. Also, since the position of each digit is important, then we may need a zero to get the digits into their correct places (columns) eg 2003 which has no tens and no hundreds. (The Roman system would have just omitted the values not used so had no need of "zero".)

This decimal positional system, as we call it, uses the ten symbols of Arabic origin and the "methods" used by Indian Hindu mathematicians many years before they were imported into Europe. It has been commented that in India, the concept of nothing is important in its early religion and philosophy and so it was much more natural to have a symbol for it than for the Latin (Roman) and Greek systems.
Related articles

Vedic Mathematics

History of Mathematics in India
Amazing Science (Part 7)

Amazing Science
That Thou Art

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