।।तरति शोकमात्मवित्।।

Functions of a Complex Variable;

Open and Closed Contours --

Pure and Applied Mathematics.

--

A Sequel to the last post :

"Real and Unreal"

--

The 7 Dimensions of the Universe --

Have been defined in the earlier post. 

Just like as on the Real Plane represented by (X, Y) there exists the 

Conjugate Imaginary Plane (X, iY).

The Functions of a Complex Variable could be expressed on either or both of the two planes. Here we are concerned with the two dimensional imaginary plane :

(X, iY) only,  which could be further extended in application to the :

Whole 7-Dimensional Universe.

Now, as an example, let us consider the standard equation of a Circle :

On the Real plane (X, Y), the equation  x squared + y squared = r squared 

represents a Circle with a

Closed Contour

With respect to Real Numbers.

If we consider the equation :

x squared - y squared = r squared,

The same holds true in the case of the

Complex numbers (x, iy)

on the 

Complex Plane (X, iY).

What significance holds for 'i' ?

Could this be related by :

"t" where "t" denotes "time"?

The premise :

"The  Observer is the Observed"

Points out to the undeniable, irrefutable existence of :

"an observer",

as the Essential Underlying Principle.

This is evident that :

The consciousness of the individual, the attention and the Awareness are but the synonyms of the "Observer".

"The Observed" could be either of the two :

An object as a sentient being, or some insentient thing or object.

The premise, notion or the hypothesis of "time" in terms of "past" "present" or "future" too applies at the level of the individual, the collective entity named as the objective existence or the "world" and the essential factor / truth of the "observer".

As has been already pointed out above, in the bold italics, this is the same as :

The individual consciousness,  Attention,  Awareness, 

The Inevitable Reality  and 

The  Potential Possibility.

Not to be found in any Philosophy or any Philosophical Thought.

Here lies the practical Vedanta truth of The Dharma: The Sanatana Dharma.

आत्मानं विद्धि.. 

तद्विद्धि प्रणिपातेन परिप्रश्नेन सेवया।।

उपदेक्ष्यन्ति ते ज्ञानं ज्ञानिनस्तत्वदर्शिनः।।३४।।

- श्रीमद्भगवद्गीता अध्याय ४,

Gita Chapter 4, verse 34.

***

 

Real and Unreal.

Mathematics

of 

The Real

and

The Unreal.

The 7 Dimensions :

The Seven Dimensions in Space Time could be explained by Mathematical approach as under.

We know the Cartesian Co-ordinates.

First we design a 3-D model with three axis, namely : X, Y and Z.

Then the Real Material Space is assumed to hold the Space, or the Space is held through and in these three dimensions.

Now let us turn to the the assumption of the imaginary plane, based upon the idea of an imaginary number i, generated when we try to find out the square root of the integer -1.

I don't know who might have 'invented' the idea of this number but first of all, I learnt of this concept in the Plain Trigonometry.

I remember how I started with The :

De Moiver's Theorem,  which states :

(cos x +i sin x) ^n = cos nx +i sin nx, for all real values of 'n'.

While proving this theorem however, the 'i' just gets eliminated.

The same. 

'i' = √-1, 

With the help of a real and an imaginery one helps in creating a Plane where the two axis are X and iY.

A point on this plane could be expressed in the form :

(x, y) or (x+iy) .

So,  there is the Real Plane XY where a point will be of the form (x, y), where x and y are real numbers. 

And likewise we devised this imaginary one, where a point will be of the form (x, iy).

Extending this rationale to the Space with the three dimensions X, Y, Z  we could get ix, iy  and iz.

So a number on any of the three imaginary planes would be of the any or the either of the form :

(x, iy), (x, iz), (y, ix), (y, iz), (z, ix) and;

(z, iy).

This gives the idea of three Real Planes, namely : {XY, XZ, and YZ}. 

And their counterparts as:

{iXY,  iXZ, and iYZ}.

Is this all hypothetical?

Obviously, the Real Space {XY, XZ, YZ} Is perceived through the senses or the sensory organs and is a material Reality, though may or may not be a substantial one.

Still there is an assumed Real Time is known or thought of to exist in terms of The Past, The Present and The Future.

Also at the level of experience the three are though intangible, but their existence could be ascertained by The Scientific Physical Experiment.

This is again the Dimensions of Time, with these three Dimensions referred to as above.

What about The Reality of these three kind of Time?

Even if The Past, The Present, and The Future couldn't be thought of to exist apart and independent of / from "Now", could be "proved" beyond doubt to exist in the realm of "The Mind".

The Three therefore could be conceived of as the "Mental Reality", corresponding to to "the Space {iXY, iXZ, iYZ}.

Even more depending upon the knowledge of the The Past, and The Present could be proved to exist and by thinking of, accepting the"Experience", "The Future" too could be anticipated / projected. And Though a particular "event" may be predicted to take place in this " the anticipated / projected Future", the same could not be forced to be brought in "The Now".

Still the "The History" is believed to have "happened" by everyone without any doubt and is individual and different from others, one hardly denies the "Collective History" of a society, culture, country.

In what "Time" does this "History" happen?

There could be another point of view :

Suppose, and let us consider a point {0, 0, 0}  in the two Two Spaces : namely the material physical space, and the mental imaginary of time, could the moment of "Now and Here" be thought of as The Timeless Now, though without a beginning or an end?

Apart from the imagined Past, and The "Thought" of The passing Present, and other than the anticipated / projected Future, is it not the very essence of Time itself? Again is it not the very essence of Space also?

Could this "Here and Now" / The Eternity beyond the above six Dimensions, could be accepted as "The Seventh Dimension?

--

Meanwhile :

Incidentally and interestingly looking for the Hebrew word for the English word "Future", I got a word in Hebrew that sounds like the Sanskrit / Hindi word : "अतीत".

This Sanskrit word when pronounced like this, means : "The Past"!

Two verses of yore come to mind :

अस्ति कश्चित् स्वयं नित्यं अहं-प्रत्ययलम्बनः।।

अवस्था-त्रय साक्षी सन् पञ्चकोषविलक्षणः।।१२५।।

(शङ्कराचार्यकृत विवेक चूडामणि)

यदा ते मोहकलिलं बुद्धिर्व्यतितरिष्यति।।

तदा गन्तासि निर्वेदं श्रोतव्यस्य श्रुतस्य च।।५२।।

(श्रीमद्भगवद्गीता अध्याय २) 

***