Filled to the brim with the enthusiasm accrued by having attended his lot of garden East of Nowhere, East of Eden, he sets out swooping as a falcon on the game, and incepts as a virtuoso sitting at the keyboards of a piano, outpouring the floodgates and animated by the intention of destroying the instrument, until all the creatures get saved, until all the beings get soaked, until the room too gets filled to the brim with the blare of melodies and the walls sparkle waves of sound, and eventually come tumbling down in a roaring clash.
«And the verse fall to the paper like dew to the pasture» [Pablo Neruda].
And since vile matter isn't but the most general metaphor of all, some calleth it manna, namely mystical droplets: droplets as lancets of rain. Thin rain. Thick thin rain.
And be mindful: fleshy angels only visit on rainy days: rain, and rain only, knows how to bring them along. No one ever saw a Messenger in the sun. Because «only rain has so small hands» [E. Cummings]
 
What is properly controversial is not what is controversial: what is controversial is what is obvious. As Confucius wrote:

«A common man marvels at uncommon things; a wise man marvels at the commonplace.»

Ponder before any "of course" more than before anything else.
And you have only one way things may appear or may even actually be easy to you: and it is when you've understood them not.
Wisdom means being puzzled like kids every day before the very same things, but every day in a new, different way. You never get an accomplished man, and as Voltaire even too rightly said "it is remarkably curious noting that never one single man of genius came out of an Academy": in fact academies are those places where academics study or teach what non academics taught them... from Voltaire to Einstein, all dilettanti & amateurs, and from pirate to Baron, at times it is only a matter of a plume on the head. Other times it is not, and my case fully belongs to these latter times, when it is not.
 
To me it's all a big mystery, and if you say to me Number 5, it is Number 5 to you only but not to me: to me it is a big mystery, for is it (4+1) or (3+2) or is it (5)? Is a pattern concealed behind the conglomerate it subsumes?
If you find the last statement lazy, oh well you've got two chances equally good: leave now or go on reading and see: it's gonna be all a big mystery to you soon.
But be warned, if you get this habit, one page in one book can engross you for weeks, and perhaps forever. Everything will be wearisome, and you will never make a Career.
«Inception of Philosophy is wonderment» Plato wrote.
But a career needs a stampede, and not the gaze of the comeback kid, forever bewildered before the World, every day washed away and renewed by a mystic Dew.
And none the less, we don't have a way out: for the mystery is there. Objectively.

What are we after here?
Here we're after reasonings that can reveal to us patterns through which a number can evolve by itself knowing nothing more but the elemental conditions it started with (ideally, just the self, just the number itself): in other words, from these conditions, I want to investigate whether it is possible for any number to beget recursively different offsprings moulded after logics/patterns entirely intrinsic to its self.
You should not read this like an orthodox mathematical essay: it is not, and doesn't mean to be such. You'd consider this an evidence that still in year 2003 men were busy making considerations on shapes and numbers that we thought lost in the times of the ancient greece.
The first half of the essay is widely speculative, but the last section eventually provides you with an interesting function.
 
Multiplication means to distribute a number N along as many Dimensions as a number D prescribes:
N*D
 
It should be

A Johannes Vermeer painting (1668)

noted that we speak of reproduction, and of a reproduction that follows a logics already somewhat bound to the self: the Number N begets copies of N itself (namely retrieves its logics from within) along the arrayed Dimensions D.
 
But if we would have meant a reproduction that genuinely follows a logic within the self, that would have been the sum, bud: because any number you start with is assumed as a mono-dimensional collection of N items, not of manifold Dimensions along which you distribute the items. As I said: is 5 actually 5 or (3,2) in two dimensions, or (1,1,1,2) in four dimensions? We cannot answer this question, for we cannot know from a result what went on in the process. We can know about the process only if and when we witness it from its onset, or trigger it: not when all we have aren't but its ultimate results. These whirlwinds leave no stream to track back.
 
A multiplication as a process, isn't but a sequence of sums in disguise: sum N to N (that is, to itself) for D rounds.
 
Now, this aspects brings in something else beyond the items: it spells Dimensions, which are given by the second number D brought in: N*D.
We are so used to cope with the eventual result, and take it as a whole, that we entirely forget that such whole isn't but the overall representation as a totality of what in detail is a scrupulous distribution of N Items along D Dimensions or Sets housing them. In other word, by focusing only on the eventual result, we implicitly assume that a new global Dimension has been created and that it now houses as many items as the result relinquished on the whole: but this is not the truth of what happens in the backstage of the algorithmic pattern and process, for behind the scenes a plurality of mansions are created to accommodate the items in different lots.
 
Our habit of dealing with multiplications considering only their output brought us to consider the terms of a multiplication as interchangeable: we assume, namely, that saying 2*3=6 is the very same than the inverse namely 3*2=6. Whereas we obviously do not concede this for divisions (2/3 yields a thoroughly different result than 3/2) we immediately grant it for multiplications, and this exactly because we're focused on the result only. But if we consider multiplications for what they do in the process, this interchangeability ceases.
 
It was a Pythagorean and euclidean attitude (a greek attitude, actually) to represent numbers by geometrical shapes; so it is not entirely without tradition if we consider that 3*2 means that number 3 must reproduce itself along 2 dimensions namely:
**
**
**
Read by columns, not by rows: each column is a dimension, each hosts 3 items. Conversely 2*3 is represented by:
***
***
three columns namely dimensions each hosting 2 items.
The shapes are different, although they're both rectangles of the same area: one is longer height-wise, the other longer in the breadth.
These shapes are unlike each other under a variety of approaches: if we even take into account processes like the shrinking of an object's length under Lorentz contractions [I'm referring to the 'ruler example' provided by Brian Greene in his book The Elegant Universe, Chapter 3: a ruler inside an accelerated system appears shorter in the front exposed to the acceleration], then by these two apparently identical objects we'd get two entirely different "contracted" objects; if they both expose their rightmost front to an acceleration, the effect would distribute evenly throughout the whole exposed surface, and if it slices off one dimension, then out of those shapes we'd get two entirely different conformations: a thinner rectangle and a square (the rendering on your browser may be not fully exact in showing the square):
*
*
*
which is a thin rectangle, and a square:
**
**
 
In the reproduction after the self performed by multiplications we imply, by bringing in a second number D, we know nothing as far as the distribution of these copies inside these Dimensions D could be arranged: we don't know whether these copies are meant to fill all the capacities of these Dimensions 5=(1,1,1,1,1), or if these Dimensions outrun such capacity or lack it and need subsequent redistributions 5=(2,1,2).
If we stick to the logics of the number, the Dimensions perfectly fit, without exceeding or lacking. We assume this is what is meant.
But uncover the assumption, in order not to be surprised if it would reveal itself wrong: the Number is distributed evenly only as far as the Number itself is regarded (it reproduces exact copies of its original amount of items per D Dimensions), but we don't know whether these items can also be allocated along those Dimensions with an homogeneous distribution: because these Dimensions are brought about by an intervening second number D; the physical Dimensions could even get saturated before the dispatchment is performed to the last. Mathematical operations are entirely theoretical insofar they assume this can never be the case: Distributions along Dimensions are always even, and these houses are perfect.
 
Dimensions seem to be brought about only starting with multiplication.
But as I noted Dimensions in multiplications are extraneous to the self of the Number: it needs a second number D (the number you multiplicate the former N by) to dictate how many these dimensions are to be.
Only in pow (as you will see) also Dimensions get deduced out of the structure of the self of the Number N, without introducing any necessity of a mate number D in order to deduce them. Indeed, we'd stop considering only eventual results as a whole if we want to grasp the logics of the patterns.
 
Division means once again to distribute a number N of items along as many Dimensions as number D prescribes:
N\D
Sounds quite like a multiplication, doesn't it?
 
But unlike multiplication, this shuffling (N/D is a sequence of subtractions in disguise: subtract D from N until 0 is achieved or surpassed, and return the rounds performed. Note by the way a missing feature: we do not have any official operation whose purpose is: subtract N from N for D rounds: why?) is not reproductive and replicative, but it is reductive: the number has to get fractioned in as many subunits as necessary to populate evenly each Dimension.
Can you sense a wither shade of weird here?
 
The operation has the same intention (symmetrical) in wording as a multiplication (distribute a number N items along as many Dimensions as number D prescribes), and none the less it carries along a handful of additional questions that multiplication didn't involve.
Now, let's examine closer what a division performs.
If I say
5*4=20
I mean that 5 items get distributed along 4 dimensions and give an overall 20:
items*dimensions=overall
So since a division coherent with the above multiplication would be:
20/5=4
or
20/4=5
A division means this:
overall/items=dimensions
or
overall/dimensions=items
 
I humbly believe that this implicit ambiguity, which I deem unescapable, is what is accountable for the conceptual difficulties some have envisioning divisions: they necessarily institute a plurality of reciprocally interfering connections that prevent the operation from being read in whatever univocal given direction, that forestall the division from being interpreted after whatever definitive play book. You have to agree upon some otherwise relative directionality, before you can handle divisions in an homogeneous fashion.
Now, although these inherently ambiguous operations appear consistent when we know the overall (20) and it is bigger than both the terms that got multiplied (5*4), these very same operations can lose such meaning if I don't know this overall: in fact if I say:
5/4=1.25, how can I make it fit with the schemes above?
If a divisions means either of the above, what kind of situation is the situation of 5/4 that predicates that I started with 4 items and my purpose was to distribute them along some dimensions being sure such distribution does not exceed 5 (5 is the overall)? It is clearly asymmetric. What type of dimensions I am envisioning to accommodate such a distribution?
We clearly have at least one dimension which fully accommodates 4 items out of 5, and one which is partial and houses one item only; or, in other words, we say that we have one dimension that certainly houses all the 5 items (one dimension for the result, which represents the dimensions, says: 1.25), and a remaining dimension (0.25) which hosts 0.25 of 5 namely 1 item:
*
*
*
**
 
As everybody sees, there is a missing clip to fill the scheme so that the coherence of the image can be recovered. Of course, its completion would have demanded 3 more items: the completion to 1 from 0.25 is 1-0.25=0.75, and 0.75*4=3
The Phytagoreans had a name for these "clips": they called them gnomon .

If we would have considered that in 5/4, number 4 is the number representing the dimensions instead than the amount of the starting items, and the result would have been the amount of items, namely if we would have considered the dimensions as given and fixed, our conclusions could be somewhat different. If I have to distribute an overall 5 through 4 dimensions, I could have placed 2 items in one dimensions, and one item only in all the remaining dimensions.
****
*
 
Still, the shape is asymmetric and longs for a completion. We certainly have one dimension (1.25) of the four which is complete, and a remaining 75% (the completion to the next 1 dimension from 0.25 is necessarily 0.75 for 1-0.25=0.75, namely 75%) of these four dimensions namely 3 dimensions that are lacking items: 4*0.75=3.
 
Either way we watch it, we have a missing clip: in both cases this clip implied 3 items had to be added; in the first case in order to complete one last dimension already present but nearly empty; in the second case to complete 3 dimensions already present but nearly empty that were in want of items: how many? each lacks, invariably, one item.
Consequently, in both cases the division seems to hint and to be furnishing me with data about what would be needed to restore some ideal symmetry.
 
Now, if this hinting to a further completion is in action for 5/4, couldn't it be there an analogous hint pointing to some other more "perfect" completion is in action also in the case of 20/5=4?
 
Let's try.
20/5=4.0
likewise in 5/4=1.25 and 1-0.25=0.75, now in 20/5=4 the fractional part is 4.0 so:
1-0=1
We can now multiply this by 5 and get still 5: do not forget that whereas the platitude of this outcome may make appear this operation meaningless and frigid because we're in the field of the evenly divided operators, none the less we have derived it from a situation where we needed a completion since the operators did not surrender an even result; which means that the ideal even a non floating result division hints at, is either to add one full dimension or accommodate 5 more items (the two perspectives are reversible). In other words, the ideal an even division hints at is increasing upward by lots equal to the divisor as its natural next completion. So 20/5=4 also implies that the division wants to produce 25 as its next completion because 20+5=25.
On the whole if:
x/y=n.f
whereas n is a integer and f the fractional part (if none: zero) of whatever division x/y, then the operation to get the "clips" is:
(1-f)*y
This is the missing elements. The sum of this with x gives the new whole of items the division was hinting at.
 
We can make likewise reasonings in the case of 4/5=0.8
It means that out of an overall 4 I have to distribute this 4 along 5 dimensions:
* * * * x
(1-f)*y=(1-0.8)*5=1
This means that I need to add 1 element to saturate the first row of the dimensions. This type of "ideal" perfection hinted at here is still evenness in the items distribution.
 
Now, what about 4/0.8=5?
It can mean that I have 0.8 items and I have 4 dimensions, namely in each dimension I have 0.8 items. This means that in the case of a less then zero dividend, the situation I'm referring to is no longer of lacking items, but of lack of dimensions. So it means I have to add one dimension to the four I have, thus yielding 5, in order to host 100% of each item in each dimension: therefore the gnomon is still 1, but the operation this time on
x/y.f=n
is
(1-f)*n
or (1-0.8)*5=1
We can convert this situation into 4/5 by setting:
y.f=1/(1-f)
namely y.f=1(1-0.8): this converts this gnomonic procedure 4/0.8 into the older 4/5, and all we have to keep in mind is that now the output of the gnomonic computations does not imply we have to add items (y>0) but that we have to add dimensions (y<0).
Or if I consider I have 0.8 items and 5 dimensions, it means they're filled up to the 80% so I need add one item to the four I already have. What the gnomon has to fill, namely either dimensions or items, entirely depends on how I read the implicitly ambiguous meanings of a division.
Or if I consider 0.8 as the amount of elements and 5 as the dimensions, it implies that these 5 dimensions are full for the 80% only.
This hints to the fact that 4/0.8 should be handled like an instance of 4/5, consequently I assume 0.8/4 should be converted into an instance of 5/4, which can be achieved by turning 0.8 into 5 by:
0.f/y=x:
0.f = 1/(0.f/y)
Namely applying the so called reciprocal number (1/x) which we're discussing more in depth in the next section.
 
If you apply this gnomonic procedure to number 1, you'll get a sequence of "ideal" completions of 1, whereas by "ideal" I mean a completion whose logics entirely derive from within the given operators' logics, which imply each subsequent completion of 1/1 isn't but the linear computation of the numbers:
1/1=1.0
1-0.0=1 (gnomon)
1+1=2 (completed. To next completion)
recursion:
2/1=2.0
1-0.0=1 (gnomon)
2+1=3 (completed. To next completion)
recursion:
3/1=3 (...)
We're therefore generating a linear output as a natural completion entirely drawn from a logic inside what the given operation seemed to suggest. The momentum of directionality the operation gains operates like a self generated propulsion. This can be a way to conceive the geometrical production of perpendicularity.

The way we usually adopt to consider divisions, is not gnomonic.
In the mainstream version of the current conception of division, the items get distributed not by replicating their identity, but by fractionating it, namely by potentially altering it. Consequently a division completely loses its implicit reference to an elsewhere. The standard vision of a division, is that it fragments objects, so what cannot be accommodated in the dimensions, gets conveniently grind.
Now, copies of a Number are arguably identical to the Number. But fractions of a Number are likely something else than the Number: 1 part of a car (division) is not a whole car any more; but 1 car * 2 positively is two comely & wholesome cars indeed.
 
So the first thing we sense here is this: all Numbers in multiplication imply that we are considering every Number as a whole whose identity gets respected while distributed along the Dimensions.
 
If consistency with this outlook has to be retained in division too, we have to imagine that each fraction, in order to withhold the identity it sprang from like multiplication positively allows for, is a full reproduction of the whole as well, or namely each subsequent step of a distribution performed by a division implies that at each subtractive round a full miniaturization of the whole gets performed, so that it maintains its original identity as a compressed powerhouse of the original version; otherwise, the very same logics of division would be incoherent with its conceptual match: multiplication, which conceptually isn't but the exact intentional copy of division: distributing N copies along D Dimensions.
We cannot predicate a communality in the intentions only to disavow it in the actions.
 
And, again, these are implications that jump to our eyes only when we cease considering the output of a mathematical operation as a whole: if we don't, then also 5*2=10 yields 10 and therefore we might be induced into thinking this 10 is a different object than 5: but it is not, for what the operation meant was a distribution of 2 identical copies of 5; how we consider the whole can be arbitrary, as (10) or as (5,5), but no arbitrariness whatsoever can be contended from within the nature of the process while it is ongoing and we analyze its recursive steps: there belongs no equivocation, their nature is crystal clear, exactly because all identities are respected throughout all and each of the steps.
 
Numbers always assume, therefore, that what you're getting when distributing is a sequence of whole and effectual copies, no matter whether you multiplied or divided. You don't get splinters. Mathematical operations are entirely theoretical.
 
None the less, this logics implied in the numbers has a, say, visual flaw: if we're ready to acknowledge, after these reasonings, that the product of 5*2=10 can be both a defaced 5 entirely transfigured into a 10 or a set of fully respected copies of fives, why we are not equally ready to acknowledge this for fractions? In fact, if we say 5/2=2.5 we're immediately prompt to acknowledge one thing and only one thing: 2.5 cannot be but a defiled identity of 5, which consequently has lost its profile (has been split) during the distributive process.
 
Calculations already have a procedure meant to consider a number as a whole, so it is not clear why this implication is so ingrained in whatever division as well: and such procedure and calculations where each Number always and most certainly represents a whole is called Reciprocal Numbers: a division of 1 by a Number:
1\D
Such expression means: how many times you have to fraction something considering it as a whole in order to get from it as many even parts to dispatch them into each of the D dimensions you divided (and consequently want to allocate) such fractions in.
 
1 is a very peculiar Number.
Even more than zero, number One's arci-enemy, said very much as Johnatan Swift wrote of "3 and its two arci-enemies: the 7 and the 9".
 
Peculiar, for it is fit to represent the whole and the bit as well. The gigantic, the leviathanic whole and the minimal, the quantic bit.
 
Therefore by the agency of 1 in regards to whichever Number (like, say, 5), Number 1 may represent the way we're currently considering 5: as the bigger unity which collects and gathers under its capable wing an array, or the minimal subunit we can find in within 5: a wider system which embraces it and is therefore throughout and thoroughly composed of... multiplications of it.
Number 1 is therefore the great synthesizer of the Scales and of the Hosts. This is why a cartesian cross should spring from 1, not from zero, and rightward proceed by multiplications by 1, leftward by divisions by 1.
 
--------------------------------- 1 ---------------------------------
/////////////////////////////////// 1 ******************************
...(1/5) (1/4) (1/3) (1/2) 1 (1*2) (1*3) (1*4) (1*5)...
 
You can see that in this way the perseverance of the identity is retained: for instance 4 has two embodiments: 1*4=4 and 1/4=0.25 which is its miniaturized scale.
 
If I say:
1\D
I'm actually fractioning, therefore, an anonymous whole entity: consequently the value I get is a value meant to be automatically universal: it is a result fit to be fit for whatever ulterior entity I meant as being embodied and represented as a whole by Number 1: therefore 1/4 isn't but a global formula meant to mean: 4/N.
Also, note that a great significance of introducing reciprocal numbers relies on the fact all integral numbers we consider are actually a disguised fraction: 3 is not 3 but actually indeed it is 3/1 (which still tantamount to 3): this is no surprise, really all numbers have to be considered as divisions, or more precisely ratios, and this is why they're called rational numbers (a pretty good essay on rational numbers can be found here - click). If we're forthwith ready to imagine them in their rational suit, namely in the shape of N/1, why aren't we as expeditious to consider them similarly gibing in their reciprocal outfit, namely in the shape of 1/N?

So:
1/5=0.2
yields 0.2 which, given what I just said, is going to be good for whatever whole I preferred or meant embodied by 1, as soon as and provided I mean to divide it by (distribute it along) 5 Dimensions.
If for instance by that 1 I meant, say, concealing number 50 as a whole, what I have to do to demonstrate what I said is true, is to multiply (and on why I multiply it you will know soon) 50*0.2: I will get 10: which is precisely the same result as 50\5. And now, here, I was multiplying.
 
1\N provides me with a parametric constant which is good to be used for whatever magnitude I'm pleased to hide inside Number 1, as soon as I mean to distribute it along N dimensions.
 
The reason I have to multiply the outcome of
1\5 namely 0.2
by a specific number to get its equivalent division as if it would have been not 1 to be divided by 5 but the specific number I meant as embodied in 1, is the following: insomuch as 1 represents an universal synthesis of whatever else number considered as a whole and fractioned (distributed along) by 5 (Dimensions), and namely it was an ideographic implosion of it, the only way to regain from such synthesis and implosion the specific value for a specific incarnation meant as concealed in number 1, is to analyze it and make it explode back.
Therefore if ratios (divisions) are, as we said earlier, the invariant that can be spotted within apparently different shapes namely regardless of the scale, reciprocal numbers represent the invariant of the invariants, the invariants of all ratios.
 
Namely I have to redistribute such outcome along the Directions number 1 included as imploded and synthesized inside its shell: and namely I have to multiply the outcome by the implied number to make its components explode back: 50 out of its disguise 1.
And so 1\5=0.2;    50*0.2=10 replicates along these dimensions what was originally stored in 1.
 
We

From Paolo Zellini, Gnomon: a decrease of dimensions can still be the equivalent of an increase: adding more squared items/fractions to the same bigger square, both fractions it making it smaller and produces more units thus increasing the squares.

have seen that a reciprocal number is the universal of a division: for instance 1/25 is the universal of whatever division by 25.
What is the reciprocal of a reciprocal? Namely, what is the reciprocal of 1/25? It is certainly 1/(1/25): what does this yield?
1/(1/25)
1/0.04=25
So, the reciprocal of a reciprocal reproduces the integer (25) on the multiplication axis stretching at the right of a cartesian cross springing from 1, as we previously envisioned. In other terms, from a sequence of recursive multiplications you cannot get the result of a division, but from a sequence of recursive divisions you do can get both the universal of whatever division by that number and the universal of whatever multiplication by that number. And that thing by which you can deduct more things, is also the thing which is more likely to have fathered all things.
This is the division.
So we can say that an axis whose origin is number 1, can be envisioned like proceeding in both directions by recursive divisions:
 
...(1/5) (1/4) (1/3) (1/2) 1 (1/(1/2)) (1/(1/3)) (1/(1/4)) (1/(1/5))...
 
We have previously seen that there is a procedure by which we can assume that once given number one, its "ideal" gnomonic completion would be to proceed to number 2, then to 3 and so on.
We are therefore on a threshold where we could device a system that can auto evolve itself both in the direction of the immensely big and in the direction of the immensely minute(because 1 is the converter of the immensely vast into the immensely tiny and viceversa), retrieving its logics for such a process generating such wide a multiplicity entirely within the self... itself!

When you consider an operation like powing, for instance 5 pow 2 which can be written 5^2, what the number does is to determine entirely from within itself how many Dimensions it has to create (exactly as many as items are in it: 5 on the whole, namely including the already exiting one, the first) and with how many items to populate each (exactly each with as many items are in the beginning: 5), namely 5^2 is:
 
* * * * *
* * * * *
* * * * *
* * * * *
* * * * *
 
25 items.
If you say 5 pow 3, it means that this procedure once attained 5^2 gets repeated: but how? It gest repeated considering now the last attained status as the new whole of items, generating 5 new dimensions, and allocating this time the new whole of 25 items per each new dimension:
25
25
25
25
25
125 items. Saying 5^4 would mean considering this last achieved 125 total as the amount of items to allocate in 5 new dimensions:
125
125
125
125
125
625 items.
 
You may have noticed that if powing is a way to develop a number through recursive steps deriving the inputs entirely from the self, this isn't but one among many other different self developing procedures that could be envisioned as well.
If you consider X as the amount of elements, Y as the amount of Dimensions, and R as the rounds you repeat the process, for each round you could have different ways to calculate the current X and Y:

1. You derive the next X (elements) from the current Y (dimensions) amount
   1. taking in only the first entry of the whole Y process
   2. taking in only the last produced entry of the whole Y process
   3. taking in the sum of all the previous entries of the whole Y process
   4. taking in only 1
2. You derive the next X (elements) from the current X (elements) amount
   1. taking in only the first entry of the whole X process
   2. taking in only the last produced entry of the whole X process
   3. taking in the sum of all the previous entries of the whole X process
   4. taking in only 1
3. You derive the next Y (dimensions) from the current Y (dimensions) amount
   1. taking in only the first entry of the whole Y process
   2. taking in only the last produced entry of the whole Y process
   3. taking in the sum of all the previous entries of the whole Y process
   4. taking in only 1
4. You derive the next Y (dimensions) from the current X (elements) amount
   1. taking in only the first entry of the whole X process
   2. taking in only the last produced entry of the whole X process
   3. taking in the sum of all the previous entries of the whole X process
   4. taking in only 1

You can therefore build sets of mathematical recursive progressions completely new and as much consistent within themselves as the process of powing was, considering that powing wasn't but one among the manifold combinations you can envision in the list above (and more precisely powing is, from the points above, combinations [2,1] for producing the elements X and [4,2] for producing the dimensions Y).
The aseroth algorithm tries to take into account all those combinations to produce different types of progressions using one single function and just changing its input arguments to produce these implicit troves.
 
The aseroth script includes a relatively ample validation, which gets executed only once upon starting the recursive process, and whose sizeable nature simply depends on the fact my personal approach to javaScript demands of it to be sturdy: a function which is a built in method or a C++ function may also afford the luxury to assume its users are invariably to be the most competent possible and they'd never pass to it the wrong arguments: but javaScript, the most widely used client side language, can't be so optimistic: it always has to take into account an extremely promiscuous environment of clients all busy making to the function any sort of nasty thing because of inexperience.
 
The script returns false in case of non numerical or bad formatted inputs, or if arguments are ok an Array whose each entry is another small Array of 2 entries only both representing numbers, entry[0] the Elements, entry[1] the dimensions for each round of the process.
As for the input arguments, here is an overview of them, what they are and what they do: