PagesHinduism & Quantum Physics
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Nature Uses Maths
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Nature Uses Maths (2)
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 havent 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.
3. The thirteen chakras
The same method for determining the location of the head
The physical location of each chakra is important in terms of
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.
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.
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 QBasics 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.
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
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
>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 ones 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.
Nature Uses Maths (2)
Rectangles and Shell Spirals
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).
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.
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987
Can you see the two sets of spirals?
(the lines are drawn connecting the centres of
Here is another pine cone. It is not only smaller,
Shows only the pinecone
Show the other set of spirals
How many spirals are there in each direction?
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!
Look at your own hand:
2 hands each of which has .
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?
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.
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.
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:
where t/n means each leaf is t/n of a turn after the
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
The time span of
planetary orbits in our solar system (i.e., the
Specifically, given that the Earth takes one
year to orbit the
Explanations in greater details
In October 1982 there was a 10-planet
alignment at which
In August 1987, with the "Harmonic
Convergence" and a
(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
In June 1992 there was a 5-planet alignment with a solar eclipse
The last major peak in the stock market occurred at Dow 4000
(see "Rare Planet Alignment Bodes A Bust For Booming
At the current juncture, the stock market is
likely peaking above
Now this is where things get interesting.
If you take the time span between the peak in Jan. of 1994
(The Fibonacci golden ratio is based upon simple numerology.
Now, if you take the time span between the
If you keep doing this to the time periods between each of
Now, to understand why this is so, you might note that
Specifically, given that the Earth takes one year to orbit the
Again, the correlation is around 99 percent (excluding Pluto).
Given this property of the solar system, the "astroharmonics"
This is all quite remarkable given how R.N. Elliott's
The "greatest European mathematician
He called himself Fibonacci [pronounced fib-on-arch-ee or
So Leonardo grew up with a North African
education under the
D E Smith points out that another famous
Italian - St Francis of
[The portrait here is a link to the University of St Andrew's site
1 2 3 4 5 6 7 8 9 . and 0
His book on how to do
arithmetic in the decimal system, called
The book describes (in Latin) the rules we
all now learn at
The Numerals are letters
The method in use in Europe until
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
The Additive rule
For instance, 13 would be written as XIII
or perhaps IIIXX. This
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.
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