I have highlighted the importance of the holistic mathematical interpretation of 2, 4 and 8 dimensions respectively, for all development processes (psychological and physical).
Basically these relate to the manner in which the principle polarities, that govern all phenomena, dynamically interact.
Jung in his writings drew frequent attention to the importance of mandalas as pictorial images with a strong integrative value for the psyche.
The most common mandalas closely resemble symmetrical 4-fold, or alternatively 8-fold, circular geometrical patterns.
In fact though often presented in an extremely artistic ornate fashion, these pictures fundamentally represent the same structures that can be used to represent both the 4 roots and 8 roots of 1 respectively.
So the deeper explanation as to importance of these mandalas lies in the holistic mathematical nature of all development processes. Again these can most fruitfully be explained in terms of either two (internal/external and whole/part) or three (internal/external, whole/part and form/emptiness) polarity sets respectively.
One might further ask why such importance attaches precisely to these polarities!
Again this can be explained with respect to the nature of number.
If we obtain the 1st root of a number, no change takes place and we remain in the world of real positive numbers.
So the first root of 1 = + 1.
However when we now obtain the two roots of a number, we require both the notion of real positive and negative numbers. So the 2 roots of 1 are + 1 and – 1.
Then we obtain the four roots of a number, we additionally require the notion of both imaginary and real numbers (in positive and negative terms). So the 4 roots of 1 are + 1, – 1, + i and –i respectively.
Now universally, all numbers must be either positive or negative in real or imaginary terms.
So in moving from just 1 through 2 to 4 dimensions, we open up vital new notions with respect to all relationships in a holistic mathematical manner.
Now with 8 dimensions we additionally open up four new complex numbers (that have a unique limiting significance).
So the four additional roots (of the 8 roots of 1) are k(1 + i ), k(1 – i ), k(– 1 + i ) and
k(– 1 – i) respectively, where k = 1/√2.
The special importance of these roots is that geometrically, they can be represented as null lines.
Thus the ultimate equality of form and emptiness (which can only be experientially approximated in a dynamic relative manner), is provided in a holistic mathematical manner through these additional 4 polarities.
For many years, I attached considerable importance to 2, 4 and 8 dimensions as the basis for integration in experience.
In particular I looked at the three levels of the "higher" band (Band 3) as representing progressively higher levels of integration in a spiritual contemplative manner.
So I referred to Level 1 (where external and internal polarities are interactively fused) as Integral 1: then I referred to Level 2 (where whole and part polarities are interactively fused) as Integral 2 and the Level 3 (where now form and emptiness polarities are fused) as Integral 3.
Each level here in fact literally related to a dimensional number as a power of 2.
So Level 1 = 21; Level 2 = 22; Level 3 = 23.
One might raise the possible significance then of higher powers of 2! Would these not imply in turn higher levels of integration?
The answer I would give is yes! However this would not require the discovery of new polarity sets, with which to structure phenomena, but rather increasing refinement with respect to the existing sets.
Indeed in a certain valid sense, only the first two sets of polarities (internal/external and whole/part) are required to structure experience, for as we have seen the 3rd set simply represents a special limiting case with respect to the combined use of the first two sets!
In fact it is remarkably like directions on a compass.
In basic terms we can start with just two polar sets of directions i.e. N and S together with E and W which we can represent as 4 equal equidistant points on a circle.
Then we can introduce 4 new directions (that in a sense divide up the existing directions in an equal manner). So we can have NE and NW and also SE and SW.
And of course we needed end here. We could then divide up the 8 directions obtained in an equal manner to now obtain 16 more precise directions.
In fact, no matter how we divide up the circle, we can always attempt to represent directions obtained as some combination of the starting 4 coordinates.
In this way we can give a more universal significance to the holistic mathematical notion of dimension.
For example, 100 dimensional reality would be indirectly expressed in terms of the 100 roots of 1, which would give a much greater range of varying combinations in which the fundamental "directions" of external/internal and whole/part are configured.
Put another way, to interpret in 100 dimensions would require the ability to experience reality in terms of 100 relatively distinct perspectives, that yet all can all be seamlessly combined together in an integrated nondual fashion.
However despite all this progress with respect to the dynamic understanding of the holistic mathematical nature of the dimensions of experience, I only gradually come to the clear realisation of a crucial imbalance with respect to my understanding.
Putting it simply, I had concentrated largely on the even integral dimensions and then on the simplest form (that represented multiples of 2).
It only slowly dawned on me that the odd dimensions, relating to differentiation, would be equally important in experience (though not directly accessible in the same manner).
Now in fairness to myself, I had earlier in fact given considerable attention to the nature of 3-dimensional experience.
Basically I came to understand it like this!
The culmination of Level 1 (Band 3) represents a new form of temporary equilibrium (where both external and internal polarities are united in a nondual fashion).
This by its nature represents a passive contemplative state of spiritual intuitive awareness.
However in moving to the "higher" contemplative state of Level 2, a breakdown of the previous passive state is required where one gets actively involved in reality (through this advancing spiritual perspective).
In many ways this represents an asymmetrical state, where one's existing spiritual vision, initially tends to clash with one's day to day experiences. Understandably this leads to considerable conflict.
Then all going well, when one eventually reaches 4-dimensional experience, a new state of passive contemplative integration is attained.
So experience therefore correctly should keep alternating as between even dimensions (representing an existing passive state of spiritual integration) and odd dimensions (representing a new stage of differentiation with respect to one's active involvements). And ideally, one should keep alternating as between contemplation and activity in both a balanced and - temporarily - unbalanced manner.
However I came to realise in my own case that the emphasis was very much more on the passive even dimensions rather than active odd dimensions.
Therefore though I had indeed made considerable progress in spiritual contemplative terms, it had been far too narrowly based with insufficient exposure to every day activities and concerns.
Though in a sense, from a transcendent perspective, I was now approaching the summit of the spiritual mountain, there was still a vast amount of adjustment required, before I could hope to place my feet firmly again on the plains below,
So in many ways, this proved a very disillusioning time that quickly led to unexpected health difficulties.
Basically these relate to the manner in which the principle polarities, that govern all phenomena, dynamically interact.
Jung in his writings drew frequent attention to the importance of mandalas as pictorial images with a strong integrative value for the psyche.
The most common mandalas closely resemble symmetrical 4-fold, or alternatively 8-fold, circular geometrical patterns.
In fact though often presented in an extremely artistic ornate fashion, these pictures fundamentally represent the same structures that can be used to represent both the 4 roots and 8 roots of 1 respectively.
So the deeper explanation as to importance of these mandalas lies in the holistic mathematical nature of all development processes. Again these can most fruitfully be explained in terms of either two (internal/external and whole/part) or three (internal/external, whole/part and form/emptiness) polarity sets respectively.
One might further ask why such importance attaches precisely to these polarities!
Again this can be explained with respect to the nature of number.
If we obtain the 1st root of a number, no change takes place and we remain in the world of real positive numbers.
So the first root of 1 = + 1.
However when we now obtain the two roots of a number, we require both the notion of real positive and negative numbers. So the 2 roots of 1 are + 1 and – 1.
Then we obtain the four roots of a number, we additionally require the notion of both imaginary and real numbers (in positive and negative terms). So the 4 roots of 1 are + 1, – 1, + i and –i respectively.
Now universally, all numbers must be either positive or negative in real or imaginary terms.
So in moving from just 1 through 2 to 4 dimensions, we open up vital new notions with respect to all relationships in a holistic mathematical manner.
Now with 8 dimensions we additionally open up four new complex numbers (that have a unique limiting significance).
So the four additional roots (of the 8 roots of 1) are k(1 + i ), k(1 – i ), k(– 1 + i ) and
k(– 1 – i) respectively, where k = 1/√2.
The special importance of these roots is that geometrically, they can be represented as null lines.
Thus the ultimate equality of form and emptiness (which can only be experientially approximated in a dynamic relative manner), is provided in a holistic mathematical manner through these additional 4 polarities.
For many years, I attached considerable importance to 2, 4 and 8 dimensions as the basis for integration in experience.
In particular I looked at the three levels of the "higher" band (Band 3) as representing progressively higher levels of integration in a spiritual contemplative manner.
So I referred to Level 1 (where external and internal polarities are interactively fused) as Integral 1: then I referred to Level 2 (where whole and part polarities are interactively fused) as Integral 2 and the Level 3 (where now form and emptiness polarities are fused) as Integral 3.
Each level here in fact literally related to a dimensional number as a power of 2.
So Level 1 = 21; Level 2 = 22; Level 3 = 23.
One might raise the possible significance then of higher powers of 2! Would these not imply in turn higher levels of integration?
The answer I would give is yes! However this would not require the discovery of new polarity sets, with which to structure phenomena, but rather increasing refinement with respect to the existing sets.
Indeed in a certain valid sense, only the first two sets of polarities (internal/external and whole/part) are required to structure experience, for as we have seen the 3rd set simply represents a special limiting case with respect to the combined use of the first two sets!
In fact it is remarkably like directions on a compass.
In basic terms we can start with just two polar sets of directions i.e. N and S together with E and W which we can represent as 4 equal equidistant points on a circle.
Then we can introduce 4 new directions (that in a sense divide up the existing directions in an equal manner). So we can have NE and NW and also SE and SW.
And of course we needed end here. We could then divide up the 8 directions obtained in an equal manner to now obtain 16 more precise directions.
In fact, no matter how we divide up the circle, we can always attempt to represent directions obtained as some combination of the starting 4 coordinates.
In this way we can give a more universal significance to the holistic mathematical notion of dimension.
For example, 100 dimensional reality would be indirectly expressed in terms of the 100 roots of 1, which would give a much greater range of varying combinations in which the fundamental "directions" of external/internal and whole/part are configured.
Put another way, to interpret in 100 dimensions would require the ability to experience reality in terms of 100 relatively distinct perspectives, that yet all can all be seamlessly combined together in an integrated nondual fashion.
However despite all this progress with respect to the dynamic understanding of the holistic mathematical nature of the dimensions of experience, I only gradually come to the clear realisation of a crucial imbalance with respect to my understanding.
Putting it simply, I had concentrated largely on the even integral dimensions and then on the simplest form (that represented multiples of 2).
It only slowly dawned on me that the odd dimensions, relating to differentiation, would be equally important in experience (though not directly accessible in the same manner).
Now in fairness to myself, I had earlier in fact given considerable attention to the nature of 3-dimensional experience.
Basically I came to understand it like this!
The culmination of Level 1 (Band 3) represents a new form of temporary equilibrium (where both external and internal polarities are united in a nondual fashion).
This by its nature represents a passive contemplative state of spiritual intuitive awareness.
However in moving to the "higher" contemplative state of Level 2, a breakdown of the previous passive state is required where one gets actively involved in reality (through this advancing spiritual perspective).
In many ways this represents an asymmetrical state, where one's existing spiritual vision, initially tends to clash with one's day to day experiences. Understandably this leads to considerable conflict.
Then all going well, when one eventually reaches 4-dimensional experience, a new state of passive contemplative integration is attained.
So experience therefore correctly should keep alternating as between even dimensions (representing an existing passive state of spiritual integration) and odd dimensions (representing a new stage of differentiation with respect to one's active involvements). And ideally, one should keep alternating as between contemplation and activity in both a balanced and - temporarily - unbalanced manner.
However I came to realise in my own case that the emphasis was very much more on the passive even dimensions rather than active odd dimensions.
Therefore though I had indeed made considerable progress in spiritual contemplative terms, it had been far too narrowly based with insufficient exposure to every day activities and concerns.
Though in a sense, from a transcendent perspective, I was now approaching the summit of the spiritual mountain, there was still a vast amount of adjustment required, before I could hope to place my feet firmly again on the plains below,
So in many ways, this proved a very disillusioning time that quickly led to unexpected health difficulties.
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