Sunday, January 10, 2010

The Number 24

24 in its own right is a fascinating number.

Firstly it represents all the permutations of 4 (containing 4 elements) that can be made from 4 which is 4 * 3 * 2 * 1.

However there is another very interesting property that if we add up the squares of the consecutive numbers from 1 to 24 (inclusive) that the result 4900 will be the square of an exact whole number i.e (70).
This is the only case known where the sum of squares of successive natural numbers is equal to the square of another whole number!

Interestingly the sum of 1 + 2 + 3 +....+ 24 = 300, while the sum of the prime numbers between 1 and 24 = 100!

24 - as we shall see - plays a key role in Ramanujan functions, which in turn provides a direct link to the number of dimensions in one of the earlier string theories.

24 also plays a crucial role in the search for the Monster Group (the largest known symmetrical object) which again provides a direct connection with string theory.


As we know, if the proper divisors of a perfect number are summed, the total is the same number. So for example 1, 2 and 3 are the proper divisors of 6 and the sume of 1 + 2 + 3 = 6. So the ratio here of proper divisors to the number = 1

Now 24 is not a perfect number. However interestingly the ratio of its proper divisors to the number = 1.5.


Another highly interesting relationship to 24 that equally results in 1.5 is worth commenting on in greater length.

I mentioned in an earlier contribution that I had found a mathematical justification as to why the 1, 2, 4 and 8 dimensions play an especially important role in an overall integral approach.

Once again the holistic qualitative notion of dimension is directly linked to the quantitative structure of its corresponding root (of unity).
So 1 qualitatively is related directly to the 1st root (of unity) which is identical. Thus in effect, no clear distinction exists as between qualitative and quantitative interpretation with respect to the 1st dimension.

To obtain the nth root of any number (which in qualitative terms corresponds with its nth dimension) we simply obtain cos (360/n) + i sin (360/n)

The value of both cos and sin will range in absolute terms as between 0 and 1.

Then when we add both terms and square the result, the absolute value will range between 1 and 2.

Now when the root and dimension is 1 the value is 1 + i (0) = 1.

So here we have a maximum in terms of the real part representing the absolute dominance of rational understanding according to one dimensional i.e. linear interpretation. Equally we have a minimum in terms of the square of sum of sin and cos values = 1.


When root and dimension is 2 the value is cos 180 + i sin 180 = - 1 + i (0) = - 1.
Now we have a maximum in terms of the negative real part. This reflects in turn the dominance of negative linear (i.e. intuitive) understanding.
Equally again the square of the absolute sum of sin and cos values = 1 (which again is a minimum in terms of what can be achieved).


When root and dimension is 4 the value is cos 90 + i sin 90 = 0 + i (1) = i.
Now we have a maximum in terms of the positive imaginary part. This reflects in turn the dominance of imaginary linear (i.e. the indirect rational expression of intuitive) understanding.
Equally again the square of the absolute sum of cos and sin values = 1 (which again is a minimum in terms of what can be achieved).


Finally when root and dimension is 8, the value is cos 45 + i sin 45 = 1/(sq. root of 2) + i/(sq. root of 2).
Now we have a balanced equality in terms of real and imaginary parts. This reflects in turn the harmonisation of rational (conscious) and intuitive (unconscious) understanding.
Equally again the square of the absolute sum of cos and sin values = 2 (which now is a maximum in terms of what can be achieved, representing the limit in terms of pure contemplative integration of experience).


So what we see is that 1, 2, 4 and 8 dimensional interpretation represent specialised extremes in terms of rational (real), intuitive, rational (imaginary) and empty understanding (as the equality of conscious and unconscious) respectively.

One could perhaps suggest therefore that the ideal radial balance (in terms of overall understanding) should come at the midpoint between the two extreme values i.e. 1 and 2 (for the square of absolute sum of cos and sin values).

Now this in fact happens when n = 24.

Therefore from this perspective, 24-dimensional understanding represents the ideal in terms of overall balanced understanding.

Coming back to the previous post on Personality Types, this would suggest that for truly balanced radial understanding of reality, one would need to successfully combine attributes of all 24 Personality Types.

So when we look at development from a radial perspective , the key goal is to successfully differentiate (to a degree) traits associated with each Personality Type before then integrating all in a simultaneous manner.

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