I have used the Euler Identity in a holistic mathematical fashion (Type 2 Mathematics) to demonstrate three stages of specialised contemplative development (preceding the full radial unfolding of stages).

This can however be given a more precise expression.

So we now define e^(k*2*i*pi) = 1^x (where k = 1, – 1 or 0 and x = 1, 0 and both 1 and 0 respectively.

So when k = 1,

1) e^(k*2*i*pi)= e^(2*i* pi) = 1^1

In holistic terms this corresponds to the rational linear (i.e. 1-dimensional) appreciation of (mystical) union.

Now with the 1st stage (of specialised contemplative development) a slight imbalance remains whereby appreciation remains unduly transcendent in nature.

In other words in emphasising the transcendent nature of spiritual reality as inherently empty and thereby beyond all notions of form, one to a degree still represses the corresponding immanent nature of such reality as prior and thereby inherent in all form.

This in effect leads to a subtle lingering rational attachment (of a necessarily linear nature) to the notion of unity (as phenomenal form) thereby preventing full realisation of complementary intuitive recognition (that is literally non-dimensional as empty of all form)

2) So when k = – 1,

e^(k*2*i*pi)= e^(- 2*i*pi) = 1^0

This is so as e^(– 2*i*pi) = 1/{e^(2*i*pi)} = (1^1)/(1^1) = 1^(1 – 1) = 1^0.

With the 2nd stage of such specialised development, this remaining lingering attachment to the notion of union (as a phenomenal point) is gradually negated.

Therefore as remaining involuntary rational attachment is eroded, the complementary pure intuitive realisation of the nature of union can unfold. Strictly this does not mean that the rational aspect of understanding now ceases with union but rather that it can interact with intuition in a very refined - and thereby transparent - fashion (due to the erosion of an excess element of involuntary attachment).

So whereas in the 1st stage we emphasised the refined rational linear appreciation of form (with respect to pure spiritual awareness), here in complementary fashion we are emphasising the corresponding intuitive (non-dimensional i.e. 0-dimensional) aspect of such awareness.

Put another way the emphasis here is on the immanent - as opposed to the transcendent - aspect of spiritual awareness.

3) when k = 0

e^(k*2*i*pi)= e^0 = (1^1)*(1^0)

This is so as e^0 = e^(2*i*pi)* e^(- 2*i*pi) i.e. e^(2*i*pi - 2*i*pi).

and as we have seen,

e^(2*pi*i)* e^(– 2*pi*i) = (1^1)*(1^0)

Now in Type 1 Mathematics, we would simply add the powers 1 and 0. However in Type 2 Mathematics these remain of a qualitatively distinct nature.

So the significance of the third formulation is that it represents the most specialised stage of contemplative awareness, where form and emptiness are successfully united in experience. Put another way it represents the full integration of both the transcendent and immanent aspects of spiritual understanding (which in turn implies the most refined interaction possible as between both the rational and intuitive modes of understanding).

So here we have pure emptiness serving as the potential for the entire world of (actual) created phenomenal form.

In mathematical terms this represents the most complete appreciation of both the linear rational (1-dimensional) nature of dimension that serves as the basis for quantitative appreciation of Type 1 Mathematics and the purely circular intuitive (0-dimensional) appreciation that serves as the basis for corresponding qualitative appreciation of Type 2 Mathematics.

This in turn with the radial unfolding of stages allows for the growing interpenetration of both Type 1 and Type 2 (in what constitutes Type 3 Mathematics).

We can perhaps see here the truly remarkable nature of e.

In conventional Type 1 terms any number raised to the power of 0 = 1. This represents therefore a merely reduced linear interpretation of 0 (as dimension). and of course when e is raised to 0 (in this reduced quantitative sense) its value is likewise 0.

However what is unique about e is that when it raised to 0 in Type 2 terms (reflecting the circular nature of dimension) its value is likewise 0.

Now we can appreciate why this is so with reference to the fact that by its very nature e fully reconciles corresponding notions of both differentiation (linear) and integration (circular).

So in Type 1 terms both the differentiation and integration of e^x are identical.

Likewise in Type 2 terms at the the most developed level of contemplative awareness differentiated (discrete) and integrated (continuous) elements are seamless so that phenomena of form no longer appear to arise in experience.

So e uniquely combines in its inherent nature both quantitative (discrete) and qualitative (continuous) aspects.

Now I have long identified the inherent nature of a prime number in that it too combines both extreme quantitative and qualitative aspects in its nature. Thus from a discrete independent nature prime numbers seemingly display no pattern. However from a continuous holistic perspective they display (en bloc) a truly remarkable regularity.

Not surprisingly therefore e has a vital role with respect to understanding the nature of primes.

Quite simply if we want to find the average space between primes (in any region of the number system) we simply find the dimensional power to which e must be raised to attain that number.

So for example to obtain 1,000,000, one must raise e to 13.8155... (i.e. the natural log of 1,000,000).

That means in the region of 1,000,000 we would expect the average space between primes to be just less than 14!

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