The Link in Accretions

The holy grail in one sense is to see visibly the link between physical and symbolic. This is what has been partially achieved in the cross addition of the numbers, their squares and the relation to the base of a 1, 3, 5 etc triangle.

But the accretive framework permits all manner of associations to become credible. For instance the original insight is that there exists a relation between the way we write the numbers 4 and 7 and their connection such that they synthesise easily like this:wpid-wp-1394206316034.jpeg.

Remember this is accretive content. Fourness attaches in the pneuma to the symbol but because the underlying nature of things contains a connection to sevenness (in this base and in physical systems) they manifest an easy ability to connect. This accretive logic is always at work in occultism.

The planets and astrological constellations are of course psychic accretions. The accretions form nexi of power which may be utilised in various ways. Do they belong with any necessity to the manifestation of substrate? No they do not. As soon as the substrate enters the monadic sphere, the pneuma attaches to give it a sense of some kind. What kind of sense is determined by other pneuminous accretions (hermeneutics).

There is a sense of truth to there being an accordance between the form the accretion takes and what is suggested by the substrate. Falsity can occur in this manner. However what the occultist knows is that any pneuma which accretes to form a phenomenon has power.

Middle Pillar

There is a pattern that follows the pattern. The pattern concerns the central pillar of the Chabbalach which comprises of Gimel, Kaph, and Qoph and relates to the morphology of them and sound.

Gimel looks like this:

Kaph looks like  this:

Qoph looks like this:

I see a morphological shift from Gimel to Qoph. It is not difficult to see. Thin Gimel retracts its tail and expands to Kaph, in Qoph, the structure seems to take flight towards the top right leaving behind a stick in the ground to the left (the tail of Gimel as was retracted in the first stage where it was originally on the right). The accretion could be speculatively said to move this stick/tail from right to left. And of course the sound of Kaph and Qoph is similar (I cannot link Gimel here), and the rough english equivalents G K Q are all hard sounds which can be morphed subtly. The path numbers are Gimel 3, Kaph 11(2), Qoph 19 (1), 3,2,1 a descent if ever there was one. But of course a descent in number is also an ascension in the Chabbalach. Do the symbol meanings show any morphic relation? Can we forge one? Camel-Palm of the hand-Back of the head. I cannot see it yet if there is something -but there is always a way. The tarot and astrological correspondences show a greater harmony: Priestess-Moon, Wheel of Fortune-Jupiter, Moon-Pisces. The higher moon is the Priestess, the lower physical moon rules the seas (Pisces). But what of Jupiter and its Wheel? The Wheel is obviously the oscillating 4 and 7 of abstract eternity. Fortune is related to Chesed’s status as fate. But why is there a manifestation of the moon above and below it? The lower moon can be cruel where as the higher one aids an exalted revelation of existence -the veil is peeled back. No good accretive answer has been formed here yet.

Cross Addition with the Professor (ii)


I don’t know if I quite get your operators is 10.x =10x (i.e. multiplication)?

[In one type of world you take numbers as the things that can be operated upon by ‘cross addition’. For example (21)+ = 3 (or the application of the operation of ‘cross addition’ on 21 results in 3). In another type of world you are taking numbers as the things that can be operated upon by ‘addition’. For example 2+3 = 5.

The two things that you operate upon are related. A number like 21 is the result of the addition of 20+1, or 10.2+1 (where the . stands for multiplication).

‘Cross adding’ a number like xy results in x+y, whil ‘adding’ xy results in 10.x+y. ]
The pattern you point out is familiar to me  though as you say not the issue here, indeed it would seem there is in a  sense no issue here.

[Mathematicians would see an issue, though. Some of the greatest work on number theory, starting from issues like yours, was done by Bernard Riemann. But I agree, it is not your issue.]

I will be interested to actually understand what you have done –and the math does not look outside of my range.

I suppose what I was really interested in is if there was anything that could be called a ‘reason’ why numbers which are arbitrarily related by a base like 13 and 4 or 16 and 7 appear to have a relation that is not contingent upon the base but rather appears to exist in a form that is not base contingent (the triangle).

[There are three elements in your argument. A. A system of numbers that that includes the operation of ‘cross addition’, like (xyz)+=x+y+z. B. A system of numbers that includes the operation of ‘addition’ as well as ‘multiplication’, like (10.x+y)2 (squared) = 100.x2+20xy+y2. C. A ‘physical’ system, the triangle.

If I understand you well, you conclude that the physical system C satisfies the conditions of the number system B. Then you ask whether it also satisfies the number system A. The answer you should expect ‘philosophically’ is no. The systems A and B have different structures, so searching for an overall agreement with C should lead to either A or B, but not both.]

So does the fact the pattern doesn’t repeat –it is clearly not possible in all instances to find a trick to reconcile the cross addition with base- mean there is nothing interesting going on there?

[If the structure C fits A, then it is possible that some elements in C also fit B and vice versa. This raises two questions.
I. Which elements in C fit both A and B? This is the question I tried to answer by suggesting to solve the equation.
II. Is there a trick or rule that helps to fit all elements in C to both A and B? I did not explore this, but referred to some of your rules (which were not general, e.g. ‘set 9 to zero (if there is a 9 for the 100s) and apply cross additon’. They are what is called ‘ad hoc’, i.e. different for different elements in C.]

Yet to relegate the fact that in the squares of 4 and 7  the relation looks concrete and real and defiant of cross additions sums contingent relation to the number themselves. But now I think vaguely philosophically and think what is the number itself out side of a base (a conceptual mediation), if I lay out 49 stones but have no base to count them in so there is just an infinite progression of symbols to delineate them, is that more real than through a base?

[The issue you raise here is related to my resistance to the word ‘mathematical’ in discussions of science – and my preference for saying that certain systems may be quantified, so operations like ‘additions’ are ‘allowed’ (or as may be the case ‘cross additions’).

Mathematicians have no difficulty studying systems of numbers that include ‘cross addition’ or, alternatively, ‘addition’. Scientists have to make a decision, however, as to whether a system C (the one they are studying: a triangle or any other physical object) fits the one (A) or the other (B). It may happen that the first 5 elements in C, or the first 50, or 500, or 5000, etc., all fit both A and B – but that differences appear later. The search for the first number that fits is well known in the context of linking the concept of probability (and its numerical implementation) to a system like C, i.e. throws of a die.

You are of course right in saying that B depends on the number 10. If we would take a number like 21 in base 5, we would get 20+1=5.2+1, and (21)+=3. Similarly we would get (21)2=(5.2+1)2=25.4+10.2+12 (1 squared)=121 and to (121)+=4 (in contrast: (21)2 in base 10 is equal to 441, with (441)+=9.]

Just a thought –the answer would seem to me to be no, though the appearance has the phantasy of the original to make it seem somehow more real.

[To a mathematician systems like A and B are ‘real’, whatever ‘thing’ (triangle or series of die throws) they relate to. A frequent problem in mathematics is whether A and B are equivalent. The proof may consist of finding a model (system C, possibly a number system) that fits both, or of mapping every element in A to one in B, etc. A scientist has no truck with this problem: he does not ‘find’ C given A and B, but ‘discovers’ C and searches for A or B.

Cross Addition with the Professor

Let me know whether I did understand your question.

[I add comments in red and between [ and ]]

[If I understand you well, you are interested in the relation between the number n, its square, the cross addition of the square and the base number. You have found a number of instances where the latter can be derived directly from the cross addition of the square. To explore the situation I have constructed an Excel demonstration. It is attached. The table shows an interesting pattern for the cross addition of n2 (in colour). But this is not what you appear to be after. Note that column F indicates the results of various ‘tricks’ to compute the base number from the cross additions of the square. Just exercises.

It seems useful to consider a general form of the problem. Let me start by considering the number n.

If we are in between 0 and 10, the number n can be written as x.

If we are in between 10 and 20, the number n can be written as xy, or 10.x + y.

If we are in between 20 and 29, the number n can be written as xy, or 210.x +y.

If we are in between 100 and 109, the number n can be written as xyz, or 100.x+10.y+z (where y = 0).


I think your problem is to solve the following equation (for numbers between 10 and 20):

(10.x+y)2+ = (100x2+20.x.y+y2)+ = 2.(10.x+y)-1.

It appears that there is no general solution. It is possible to find solutions by looking for combinations, for example.

If n=19, the equation is 100.1+20.9+81=361. The base number is 37, which one can derive by the trick of taking 30+(61)+=37.

If n=20, the equation is 100.4+20.0+02=400. The base number is 39, but 40+(00)+=40. Considering 40 separately is a trick that is not part of the mapping.

If n=21, the equation is 100.4+20.2.1+1=441. The base number is 41, but 40+4+1=45.


It would seem that the number of numbers for which the base number is the same as (10x+y)2+ is quite limited, and also not densely or equally distributed over the set of natural numbers.

Did I understand your problem?]

On 21/02/2014 09:59, Graham Freestone wrote:

Dear Gerard

Entirely aside from the forum I have a problem which you might be able to understand. These are sections taken from some online notes I have written but I think the issue is clear. Any thoughts appreciated.




The Numerical system and the reduction that squaring and cross addition performs upon it.

The foundation of the system is derived from the following squaring of base 10 coupled with the classic numerological process of cross addition formalized as n+ e.g.


The rule was once observed that ‘the highest incarnation of a number is itself multiplied by itself (ignore this kind of language in this instance)‘. The accretion of this rule was taken to heart and thus derives results even though the accretion itself is of course contingent and the term ‘highest incarnation’ subject to incoherentism. (Refer to the Tractatus for the understanding of these terms).

When these rules are applied the number system shows itself to be made up of these numbers only:


2=4=(16)+=7=(49)+=(13)+=4 and will repeat this endlessly thus we can write 4v7








Thus from this perspective, taking this rule as truth, the highest manifestation of the numbers is 1, and oscillation of 4v7 and 9.

The wary observer will be aware that all of this is contingent upon base 10 and can represent nothing greater than this one numerical perspective. What must be understood in the light of the Tractatus is that the accretion of base 10 is very powerful, and thus whilst contingent in one sense, in another is the only thing worth working with to generate further connections with this world.

[I understand the above to mean that every natural number can be mapped onto the set of single numbers 1, 4, 7 and 9. The mapping is the result of the operation of cross addition. As mentioned, there is a pattern: see the attachment, the coloured numbers.]

Cross addition and the problem itself.

Cross addition ((123)+=6) suggests the possibility of a relation, indeed says that under certain condition (a particular base) there is a relation between a number higher than the highest single integer and one of the single integers. The general impression would be that this is in a sense arbitrary or at least meaningless. What has 53 e.g. got to do with 8? Very little other than the cross addition relationship.

What is truly fascinating is the that the squares of 4 and 7 give numbers whose cross addition then has a demonstrable reality in a triangle comprised of units 1 at the top 3 on the next line down, then 5, 7 and so on.

Triangles of this kind of unit construction are remarkable as they also provide squares. The number that will be squared is the height of the triangle. So if I have a triangle of 2 height, the total number of units in the triangle will be 4, if 3 it will be nine and so on.

The relation to the issue of cross addition is as follows. If I have a triangle of a height of 4, necessarily it will be comprised of 16 units. The base however will be 7 units(1+6). There is one relation uncovered here for the base will always have the relation to the height 2n-1. If the height is 12 the base is 23 and so on.

The second relation that is more interesting to us here is the one concerning squares. Squares of 4s and 7s even of cross addition ones will always reduce to 7s or 4s respectively but the base seems to often (though not always) reveal a relation between the number itself, the square and the cross addition of the square.

These are the most concrete examples:

4 becomes 16 becomes 7 (the base units of 4)

7 becomes 49 becomes 13 (the base units of the 7 triangle) becomes 4

These require a tweak to make them work but are still quite convincing.

13 becomes 169 becomes (curiously by preserving the first two digits as a whole number) 16+9=25 is the base number.

16 becomes 256 becomes 31 (by the same logic above) which is the base number and also reduces to 4.

22 becomes 484 and a similar logic derives the base. This time we extract 40 and add 8+4=12=3, re-add them and we have 43, the base number.

31 gives us 961, if we cross out the 9 for 9=0 in base 10 cross addition we immediately have the base number again.

This one has no link though.

25*25=625=13=4, or course the 7/4 transformation is preserved but the base relation is not. The base would be 49 and 625 does not have a relation to it. And no doubt there are others.

The Crux

Given that the relation between 4 and 13 contingent entirely on the base we write it in, how is it that it seems that this relation is necessary even if only in some cases? I am not saying anything mystical here, the mystical can be bolted onto it, but that’s not the point? Isn’t it possible there is something numerically interesting here? ( I did not say mathematical ;-))


We are told the translation means door. It forms part of the circuit, indeed in a sense is the first path of the circuit insofar as there can be a first path. 2 to 3, Khaokmah to Binah; the attribution is Venus and the Empress, which suggests the female power emanates back against the current of the circuit. This resonates with the rising power of the star as contributing to this female nexus.

The path emanates out of Khaokmah’s blinding whiteness and is sucked into the fulligin of Binah. At the moment the path looks grey, it transforms from white into black. I thought I saw a serpent on it, which gives echos of Lust and Teth later and earlier. Let us not forget the phallus is not far behind (the Emporer/Hermit/Binary line of Khaokmah).

In the emanations of language bina-ry is cut short by Bin-ah. The Ree is the phallus now triangulated -Resh is the sun. The third point alters linearity into the possibility of space.