The Secret Enneagram of Gurdjieff modified by Universal Vortex Math

© 2014/2018 by Dylan Stephens

In geometry, an enneagram is a nine-pointed plane figure.

Its prefix, ennea- is Greek for 9. It can be created by three equilateral triangles shifted 40°

It is interesting that the Enneagram shown here:

has become the symbol of Gurdjieff, similar to the Jesus and the cross and yet Gurdjieff made no mention of it in his trilogy:
"All and Everything" ("Beelzebub's Tales to His Grandson", "Meetings with Remarkable Men", and the shell of the Third Series)!

The Enneagram comes originally from Ouspensky's "In Search of the Miraculous", Chapter 14 where Gurdjieff gave it in one of his lectures early on. Since then it has been expanded upon by millions of people as a tool of spiritual advancement. A huge industry has evolved from the study of its intersecting lines, specifically in the mapping of nine personality types (arbitrarily chosen) and their interactions. Many have become rich giving presentations, drawing diagrams, and acting as therapists in these studies that have become synonymous with the enneagram, called the Enneagram_of_Personality by Wikipedia, which was lifted from Gurdjieff by Oscar Ichazo and Helen Palmer. The Fourth Way Enneagram in Wikipedia plays second fiddle to it now.

This brings to mind the statement from Beelzebub's Tales, Chapter 41, The Bokharian Dervish Hadji-Asvatz-Troov,pg 871/2

"One day we started talking about what is called there the ‘ancient-Chinese-science’ named ‘Shat-Chai-Mernis.’
"This science is nothing but fragments of the above mentioned totality of true information concerning the sacred Heptaparaparshinokh cognized by the great Chinese twin brothers and by other genuine ancient scientists and then called by them the ‘totality of true information about the Law of Ninefoldness.’
"I have already told you that certain fragments of this knowledge remained intact and passed from generation to generation through a very limited number of initiated beings there."I must here say that if these fragments, which have by chance remained intact and which have passed and are still passing there from generation to generation through this very limited number of initiated beings, do not fall into the hands of contemporary ‘scientists’ there, then this will be a great stroke of luck for the future three-brained beings of your planet.
"And it will be a great stroke of luck because, if these surviving fragments of genuine knowledge were to fall into the hands of the contemporary ‘scientists’ there, then thanks to their inherency of wiseacring, they would without fail cook up all kinds of their ‘scientific porridge’ about the sense put into these fragments, and thereby the, without this, scarcely smoldering Reason of all the other three-brained beings would be entirely extinguished; and besides, these last remnants of the former great attainments of their ancestors would thereby be also completely wiped off the face of this ill-fated planet. "

Astrology has twelve types, but they are of the essence, whereas these nine types seek to map the personality. The personality is merely the artificial structure that we create to shield our essence from the world. Ultimately, this study of personalities is a complete waste of time in the building of a Soul, which is built upon essence. It is possible that this knowledge could be useful in the manipulation of others who do not have a clue that they have an essence, but clearly serves no purpose to one's spiritual growth.

The irony is that 'Ninefoldness' refers to something larger than any of the more 'scientific' studies of the Enneagram that improperly draw the directions of the six lines in the Enneagram, as I will show here. The hidden law of 'Ninefoldness' is the unique property of nine which is revealed in Vortex Math. For example, one of the properties of nine is that when it is added to any number, it gives that same base number:

  • 1+9=10 which reduces to 1+0 which equals(==) 1
  • 8+9==17=1+7=8
  • 12345 (1+2+3+4+5=15=6==12345+9=12354 which 1+2+3+5+4=15=6
  • ..

Nine has many other amazing features as I will show. Disappointingly, "Beezebub's Tales to His Grandson" Chapter 6 Perpetual Motion does not reveal it, except to describe the ship that they are traveling in as an analogy of being-Partkdolg duty. In the excerpt from Chapter 41, that was presented previously, Gurdjieff as Beelzebub mentions the Law of 'Ninefoldness', but then the subject is lost in a tangential story that is clearly about his failure to prevent his wife's death from cancer. It is interesting that he uses "Ninefoldness" to describe Heptaparaparshinokh (Sevenfoldness) as this indicates that he is talking about the Enneagram's relationship to the notes that surround the circle.

If Gurdjieff had had access to the concepts of Vortex Math as presented by Marko Rodin (1911 - 2007), he could have elaborated further on the Enneagram and even shown the mathematical derivation of "The Nirioonossian-World-Sound" (being just that sound the vibrations of which have been taken from ancient times - and even at the present time there, is still taken, it is true, for a very small number of your favorites, of course, of this same China - for their sound-producing instruments as the 'absolute vibrations' of the note 'do'.) (The Law of Heptaparaparshinokh, Ch 40)

The mathematically elegant derivation of the Gurdjieff Enneagram lines is as follows:

Mark nine points on a circle numbered 9,1,2,3,4,5,6,7,8.

Calculate the decimal value of one divided by seven for the Law of Sevenfoldness:
1/7, 2/7, 3/7, 4/7, 5/7, 6/7 giving .142857..., .285714..., .428571...,571428..., .714285..., .857142 ...
(Note that these results contain a repeating decimal.)

Next draw a line connecting these numbers. The missing numbers 3, 6, and 9 can then be connected to form an inner triangle with it apex at 'do' and the higher 'do'. Thus the famous symbol appears! (We will see that Vortex Math shows a relationship between 3 and 6, where they swap places.)

All of these repeating decimal positions can be assigned to the eight positions of the octave as follows: 're' to position 1, 'mi' to position 3, 'fa' to position 4, 'sol' to position 5, 'la' to position 7 and 'ti' to position 8. The apex of the triangle at 9 is then 'do' to represent lower 'do' and its completion at the higher 'do'. (More on this later.)

Here is the Enhanced Enneagram with all possible lines
where the Gurdjieff Enneagram is shown within it in yellow and green:

In my first lecture: "The Key to Soul Is Within The Missing Stopinder in the Food Octave", I prove that a substance is missing at 'la' in the Food Octave in Chapter 39 of "Beelzebub's Tales to His Grandson" and conjecture what this substance is.

One author, Richard J. Defouw, in his book: "The Enneagram in the Writings of Gurdjieff" (Amazon), being one of the few that acknowledged the missing substance 'la' sets out to show that it was intentionally left out in order to demonstrate the action of the lines in the Enneagram. He presents a well-thought out process of two sets of three functions that create the energy necessary to pass through the missing 'la' position. (His proof relies primarily on proving that Beezebub's six "Voyages to the Earth" follow the Enneagram order - this is logical as there are not seven, but six - but I will prove that accepted Enneagram order is wrong.)

On page 78, Richard shows a map of dancers performing a "par de deux". I was intrigued by this and built the following animation which follows the spirit of Richard's diagram of six systems. (Please be advised to check the author's book for the correct one.) One dancer in the middle is a pivot point for the other dancer who is moved from the point on the right to a point on the left with the left dancer becoming the pivot point for the next set. For the dancers I used the angel and the devil from the cover design of the prospectus for the Gurdjieff Institute (1919). Technically, the angel is supposed to represent the passive force and the devil the active force, in my figure they must swap natures alternately. I have shown the dancer at 'la' having to move to position 'ti' to allow for the problem of the missing position in the Food Octave. Other than that the diagrams match the author's.

Freeze frames:

Richard's comment about the above dancers that I created: "It is pretty, but misleading. To see why it is misleading, recall that the angel and devil represent good and evil, that is, the passive and active forces. The dancer who is being swung ahead is playing the passive role and therefore should always be represented by the angel. But it looks like you have the angel and devil taking turns being swung ahead. Actually, it would be better not to use the angel and devil to represent the dancers at all, since that just confuses the two dualities (one duality consists of the hexagon cycle and the triangle cycle: the two dancers; the other duality consists of the passive and active forces: the angel and the devil). He agrees that 'la' certainly exists, but it is just not the result of Harnelmiatznel. 'La' is Piandjoehary that has moved to the place of Exioehary. Only after this move does another process of Harnelmiatznel take place, converting the moved Piandjoehary into Exioehary. See the end of section 5.5 (quote 5.6 and the paragraph that follows it). What is cool is that this two-step process (Piandjoehary first moves and then is converted into Exioehary by Harnelmiatznel) occurs at exactly that phase of the six-phase cycle which seems to pass over an extra note. That is because each phase of the cycle corresponds to a process of Harnelmiatznel, so 'la' doesn't get its own enneagram."

The Marko Rodin Vortex claims to describe all energy processes and its derivation is similar to the Gurdjieff Enneagram, so it makes sense to compare it. In the Gurdjieff Enneagram, the order of the repeating numbers has been used to determine the direction of action (the arrows)
1 to 4, 4 to 2, 2 to 8, 8 to 5, 5 to 7 and 7 back to 1, but this assumption may be flawed when compared to the more universal Marko Rodin Vortex.

The Marko Rodin Vortex begins like the Enneagram with a circle with numbers 1-9. A series of numbers 1, 2, 4, 8, 16 (7), 32 (5), 64 from is drawn in a different way than above, it is built by doubling and reducing by adding the digits of the number over 8 thus 1, 2, 4, 8, 16 (7), 32 (5), 64 (10 = 1 thus repeating).

Starting with 1 doubled shows the line is drawn from 1 to 2;
2 doubled shows the line is drawn from 2 to 4;
4 doubled shows the line is drawn from 4 to 8;
8 doubled is 16 which is then reduced to 7 thus shows the line is drawn from 8 to 7;
16 reduces to 7 placing it at 7 and 16 doubled is 32 which reduces it to 5 thus shows the line drawn from 7 to 5;
32 reduces to 5 placing it at 5 and 32 doubled is 64 which reduces to 10 and then 1 thus shows a line drawn from 5 to 1.
The triangle points have a unique quality that when added point and multiplied connect with the other triangle points:
At point 3, 3+3 connects with 6 and 3*3 connects with 9.
At point 6, 6+6 = 12 reduced to 3 thus connects with 3 and 6*6=36 reduced to 9 connects with 9. The 9 at the top vertex or apex divides the circle into mirrored halves since 8+1=9; 7+2=9; 6+3=9; and 5+4=9

It is the same as the Gurdjieff Enneagram flipped:

but shows the true perpetual motion.

Firstly, when considering the similarity of these two diagrams especially in having a mirrored structures left and right, there is a difference. In the Marko Rodin Vortex where the direction is 'mi' to 'fa' and 'la' to 'sol', but in the accepted Gurdjieff Enneagram is they are reversed, therefore I propose that these lines in the Gurdjieff Enneagram be aligned to the Marko Rodin Vortex direction as shown in pink as indicated:

Incidentally, the Marko Rodin Vortex also works by halving. In this case the direction is reversed:

Note: The Vortex Math of Marko Rodin has been ridiculed as being the obvious result of the Number Ten mathematical system and point out that it will not work for other numbers. This is an absurd compliant as we have ten fingers and toes and we count by ten! Also, there have been experiments performed that appear to show that free energy or at least enhanced energy can be produced using a "Rodin Coil".

Nikola Tesla (1856-1943), the famous inventor, is known for the quote “If you only knew the magnificence of the 3, 6 and 9, then you would have the key to the universe.”

In his "Map to Multiplication - Composites and Primes as a self organized system" (shown here), his observation of

  • 2 is "2 and 10 act as doublers alternating between the doubling of prime positions next to them and across
  • 3 is "the number three multiplies itself through the system as a perfect square. It bounces from position 3, to 6, to 9, to 12.
    All multiples of 3 are found in these positions"
Vortex Math uses doubling and its inner triangle is 9-3-6.

Vortex Math also works for the Fibonacci Series as shown by Maff Math:
(Note that the Fibonacci sequence is in the inner circle with its digits over 9 vortex reduced
and realizing that it repeats after each 24 numbers (giving 24 hours in a day!).
The vortex addition is then applied above and below to give the same vortex pattern: 1,5,7,8,4,2,1 with 9 being at the top and bottom)

yielding forward and backwards spirals:

Further, it is necessary to review the Marko Rodin Vortex number system in greater detail to reveal other possible relationships with it and the enneagram. In the table below, we will find that there is polarity on each side of the enneagram which is evidenced by repeating reverse numbers.

Enneagram vortex of all numbers

Next, let us study the relationships of the notes using vortex math.

This table calculated below derives the hertz values of the Circle of Fifths
using the basis numbers that created the vortex diagram above.
These are located in the first column of the table below:
multiplied by 2:
1,2,4,8,7(16: 6+1)),5(32: 3+2),
1(64: 6+4=10=1+0=1),2(128: 1+2+8=11=1+1=2), 4 (256: (2+5+6=13=1+3=4), 512: (4=5+1+2=8) ...
divided by 2:
5(1/2=.5),7(.5/2=.25=2+5=7),8(.125=1+2+5=8),4(.0625=6+2+5=13=1+3=4) ...
then each column value in the next column on the right is created by multiplying the adjacent column on the left by 3:
(Yellow colored values are the three possible universal values in the key of C for the Circle of Fifths:
C,G,D,A,E,B,F#/Gb,C#/Db,G#/Ab,D#Eb,A#Bb,F in the heading of each column.
Note that A (in green) is 432; not the artificial standard of 440.)
(Note: All the digits except C & G add to 9; G alternates between 6 & 3 and C contains the repeating vortex numbers of 1,2,4,8,7,5)
(See YouTube by Roman Blouny))

Listen to their values on a keyboard

(adapted from Keith William Horwood's program that plays standard notes.)

Circle of Fifths
derived from the table above (showing exact digits)

By moving the notes to their normalized respective places and giving a direction
based on the order of derivation in the Vortex Music Table (Circle of Fifths order) with
blue lines representing positive energies (full notes) initiated by do(C)-sol(G) and
red lines representing negative energies (half notes) ending in fa(F)-do(C),
we have the following figure:

All these triads are suspended chords, but their relationships reveal other influences:
Positive energies:
'do' (9) influences 'sol' (6&3); which influences 're', which influences 'la'; which influences 'mi'; which influences 'ti'
Negative energies:
'ti' influences the space between 'sol' and 'fa'; which influences the space between 'do' and 're'; which influences the space between 'sol' and 'la'; which influences the space between 're' and 'mi'; which influences the space between 'la' and 'ti'.
Note that 'ti' is the joining point of the positive and negative energies on it return back through 'fa' on it's return to 'do' (9).
Here are normalized note postitions vector lines with the Enneagram overlayed:
There are two lines unaccounted for: A ('la') to E ('mi') and G ('sol') to D ('re'), which deserve further study. Of interest is the line D ('re') to A ('la') in this Twelve Note diagram which is opposite to the traditional direction of the Gurdjieff Enneagram, which suggests the change as shown below:
Comparing these functions to the Food Octave. Then there are two symmetrical actions 're'-'fa'-'mi' (food and air directed to invigorating the automatic system of the cerebellum) and 'ti'-'sol'-'la' (sexual energy and hormonal energy directed to the cerebrum) that are joined by the lines 'mi'-'ti' and 'la'-'re' indicated by the crossed laurels having a similar function to those in the prospectus for the Gurdjieff Institute (following my figure). The crossing point of the two lines could be considered as the seventh point of the six enneagram points, just below God at 'do'. This is similar to the energy point of the Marko Rodin Energy crossing point that is the completed energy point. (both shown as light blue.) The laurel of the devil sends energy to 'ti' Sexual Center and that of the angel pulls energy to 'la', the Endocrine System.
return to main page