Monday, October 7, 2013

Where Science and Art Coincide (7)

Another way of looking at the dimensional number system is as a spectrum entailing the dynamic interaction of both quantitative and qualitative aspects (in relative terms).

From this perspective the only number that remains undefined on this spectrum is 1, as it is the only number where both quantitative and qualitative characteristics are absolutely similar.

Again, as the 1st root of 1 is 1, this is the only number where the corresponding roots of 1 are identical with the original dimensional number (i.e. power) of the number!

This would further imply that the correct dynamic interpretation of the Riemann Zeta Functions is that it in fact represents the number system (considered as a spectrum) with the only number undefined in this system where s (the dimensional number) = 1.

This (Type 2) dimensional number system also has intimate connections with Euler's Identity.

In fact Euler's Identity - or at least the slightly modified form I employ - provides the ready means of calculating all the various root values of 1 (which in this new interpretation have both quantitative and qualitative interpretations which are complementary).

Now Euler's Identity can be expressed as,

e = – 1

However the more fundamental version is obtained from squaring both sides so that,

e2iπ   1

More correctly this can be expressed as

e2iπ   11

I have explained the profound holistic importance of these number symbols before.

e is unique in the sense that both its differential and integral are the same.

Therefore in experiential terms e plays the role of the symbol where both (analytic) differentiation and (holistic) integration are ultimately identical.

Now with respect to the unit circle, 2π represents the circumference. However in analytic terms this will still have positive extension.

However if we try and envisage a circle whose radius simultaneously has the same positive and negative unit value, in dynamic terms this will shrink to a point where the (linear) diameter and (circular) circumference approach identity.

Therefore if we are to properly interpret the Euler Identity we must combine both Type 1 (analytic) and Type 2 (holistic) interpretations.

In spiritual contemplative terms the very means of achieving ultimate unity (which of course can only be approximated in dynamic terms) entails seeking this ultimate identity with respect to both finite (analytic) and infinite (holistic) understanding.

And as number - properly understood - represents the very basis of such experience, the very notion of 1 comprehensively understanding 1 (as numerical unity) entails the same process.

Now the very reason why this is not apparent in customary terms is precisely because Conventional Mathematics employs a merely reduced (i.e. absolute) analytic notion of number.

However the journey to understanding this notion of 1 in a comprehensive dynamic manner (where both analytic and holistic appreciation interact) is inseparable from the spiritual contemplative process of achieving unity of all experience.

So again properly understood the fundamental Euler Identity opens up a remarkable window of appreciation to both the Type 1 and Type 2 aspects of the number system.

So for example any number (k) in the Type 1 system can be expressed as

ke2iπ

Therefore for example 2 in the Type 1 system is expressed as

2e2iπ  = 2 i.e. 21

Then k in the Type 2 system is expressed as

e(2iπ)k

Therefore 2 in the Type 2 system is expressed as

e(2iπ)2 = 2 i.e. 12

Now of course roots will be expressed as the reciprocals of whole numbers

Now De Moivre's formula can be easily obtained from Euler's Identity.

Therefore to obtain the corresponding roots of 1 (using the fundamental Euler Identity)

e2iπ = cos(2π) + i sin( ) where 2π (measured in radians) = 360 degrees

Therefore to get the k roots of 1 we obtain,

e2iπ/k  = cos(2πk) + i sin(2πk )  where k = 1/k, 2/k, ....k/k

So for example to obtain the 3 roots of 1, we let k = 1/3, 2/3 and 3/3 respectively.

Therefore the 1st root for k = 1/3 is

e2iπ/3 = cos(2π/3) + i sin(2π/3 )  = cos 120  + i sin 120

= – 1/2 + .866 i

The 2nd root for k = 2/3 is

e4iπ/3  = cos(4π/3) + i sin(4π/2) = cos 240 + i sin 240

= – 1/2  .866 i

The 3rd root for k = 3/3 is

e6iπ/3  = cos(6π/3) + i sin(6π/2) = cos 360 + i sin 360

= 1

There is an intimate connection here with the (finite) Zeta 2 Function

The Zeta 2 Function represents all the non-trivial solutions for roots i.e. all those except 1)

Therefore the first 2 solutions above for k = 1/3 and 2/3 represent the solutions for the Zeta 2 Function (where t = 3)

ζ2(s) =  1 + s+ s+ s+….. + st – 1 (with t prime) = 0, where t = 3

i.e. 1 + s+ s2   = 0