Fractals And Julia Sets In The Quran

If you have been following this blog, there is no denying the depth of the arithmetic underpinnings of the Quran.

As a brief survey this post considers taking the Chapter and Verse numbers in consideration as the  input values Re(c)= Chapter number and Im(c)=Corresponding verse.

A pattern emerges as to which chapters correspond to connected or disconnected Julia sets that are or are not part of the Mandelbrot set.

As a further study it is correlated which of these Chapters were revealed in Mecca or Medina versus their fractal nature and a brief survey of the corresponding Chapters with Muqati'at are given as a subset of the study.

Initially only the numeric portion of Chapter and verse were given as the Julia set values, and a further refinement was to keep the significant figures as the order of the Julia set values. For instance, Chapter 1 with 7 versus was investigated both as Re(c)=1 and .001 and Im(c)= 7 and .007 since both the chapters and versus range to 3 significant figures from 1 to 114, and 3 to 286, respectively...


Heyyy, Divide By Zero Is No-Fair!

We don't need a new math to prove this is just a matter of Opinion.

For the most part :

Number/0 == Undefined

But if the system settings are such that they default to the purist definition, then there are cases wherein dividing by '0' yields valid results.

The first case is should the zero be expressed by exponentials as any number n:

n/n^n where n= 0,

is a sensible answer since any number raised to the 0 'power' is 1, making the denominator in the above example 1 and the result is no longer undefined, s.t.

0/0^0 ==0/1==0

The second case is that wherein Zero by definition is the 'valid sum of all numbers'.

In a complete set of positive and negative numbers, every positive number added to its negative dual results in a 0 sum.

In the event that the numbers comprising the set of 'all numbers' is 'all known numbers', this, too, would yield a valid result.

This case is simply an extension of 'normalization' and the elements in the set may not all have an exact negative dual so that the zero sum results in an actual quantity:

Here,  A  comprises  'all known numbers', and by definition Zero= 'valid sum of all known numbers' For A={-1/23, -1/2, -1/4, 1, -1, 1/4, 1/2}
The valid sum == -1/23 which can serve as the '0' for this function, and is a valid divisor throughout the system defined by the set.

Deferring to the purist definitions does not only serve to make sense of systems generally rendered undefined, it may also be a clever deduction that renders some pursuits 'non-sensible.'

Take for example the hunt for the 'biggest' or 'greatest' prime number.

By definition, and pure understanding of the definition,

'A prime number is only divisible by itself and 1'

Results in the only 'greatest' or 'largest' prime number that fits this definition exactly as the number 2.

This is True, since every other number greater than 2, prime or non-prime, can still be divided by 2 and 1.

1/2^0 = 1
2/2^0 = 2

3/[2^1+ 2^0]= 1

5/[2^2+2^0]= 1
7/[2^2 + 2^1 + 2^0] = 1
8/[2^3]= 1
9/[2^3 + 2^0] = 1
10/[2^3 + 2^1] = 1...

And so on...

And the definition of what is and what is not a Prime number, can and may also be extended by the manner in which such a number is expressed.

For example, given that 1 and 2 are really the only 2 prime numbers based purely on definition, a subcategory of prime numbers emerges based on how a number can be expressed in terms of these 'primes'.

For example 8 in terms of 2^3, is now considered a 'Prime Eight', but in systems where the 8 is expressed as 4 + 4 or 4 x 2, or any combination of non-exact-primes, 7 + 1, can be considered an 'Even' or 'Odd' Eight simply by tracing its 'origin'.

By simply asking, "Did this Integer, or Natural number, start out as '8', '4 x 2', '7 + 1' or' 2^3'?

While this may seem like quibbling to merely get a banal point across, when considering the overall system, the means by which a number 'got there' or was 'arrived at' is pivotal when accountability and traceability are at the forefront of the system level considerations.

In due time, the intention here is that this measure be given due consideration.

The mechanism of establishing just means in determining system parameters, including weights and measures, and essentially calibrating whatever mechanism is in place to make a disinterested or unbiased judgement or value determinations is alluded to in Surah 55 Ar-Rahman.

Surah 55 Ar-Rahman (The Beneficent, The Merciful)

وَالسَّمَاء رَفَعَهَا وَوَضَعَ الْمِيزَانَ (55:7

55:7) And He Raised (Erected) The Heavens (The Sky), and Established (Manifested) the Balance (Just Measure, Libra)

أَلَّا تَطْغَوْا فِي الْمِيزَانِ (55:8

55:8 Such that you never transgress the measure [cheat the balance of Justice, weights-and-measures]
وَأَقِيمُوا الْوَزْنَ بِالْقِسْطِ وَلَا تُخْسِرُوا الْمِيزَانَ (55:9
55:9 And establish weights (the Scales...of Justice, Measures, etc.) honestly and do not cheat the measure (weights, scales, etc).

Again, while there is a brief allusion to one of the Zodiac signs (Libra), the inference is that how we carry out our daily business in determining weights and measures is parametrically to be in as honest a method as the reliability with which the Heavens themselves are established (Libra, the Scales, being the only sign in the Zodiac that is an instrument, an not associated with a being of some sort, be it human, animal or chimeric).