The Secret Enneagram of Gurdjieff and its relationship to Universal Vortex Math

© 2014/2018 by Dylan Stephens
This is discussed in Lecture 2 on You Tube :

"The Secret Enneagram of Gurdjieff and its relationship to Universal Vortex Math"

Dylan Stephens © 2018

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 P. D. Ouspensky's "In Search of the Miraculous", Chapter 14 where Gurdjieff gave a lecture showing this symbol. This was before the founding of the Institute for the Harmonious Development of Man in Fontainebleau and the writing of Beelzebub's Tales. 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. It is the Ouspensky and John G. Bennett followers who use this term, but it should be correctly called the Gurdjieff Enneagram. The so-called 'scientific' studies of the "Fourth Way Enneagram" that try to predict events by using the six vectors of the repeating decimal seem to be no more useful than games of chance.

Astrology already has twelve types, but they are of the essence, whereas these nine types in the Enneagram_of_Personality 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.

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 ... 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. "

When I read this excerpt, I said to myself, this must be a mistake and it should be "Sevenfoldness" as in the "Law of Heptaparaparshinokh", especially in the context of the seven notes in the octave. If it was meant to be the one and only reference to the nine points of the Enneagram, it is strange, since the unique figure really consists of seven lines that directly relate to the repeating decimal being the division by 7. The other two points are from the inner triangle which is often left out.

It is more likely that the mentioned 'Law of Ninefoldness' refers to something more complex than the Enneagram and was not known by Gurdjieff at the time of his writing. This "Law of Ninefoldness" most likely refers to the recently discovered (or shall I say remembered) Vortex Math that will be demonstrated in this lecture. In Vortex Math the number nine has a unique relationship, 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
  • 12,345 (1+2+3+4+5=15=6==12,345+9=12,354 which is 1+2+3+5+4=15=6
  • ..

It is important to note that the process of "reduction", as used throughout this presentation, is to add all digits greater than nine together to demonstrate their essence as 1-9. This is a common practice in Numerology.

This 'Law of Ninefoldness' has many other amazing features such as perpetual motion as I will show. Disappointingly, "Beezebub's Tales to His Grandson" Chapter 6 Perpetual Motion does not reveal this secret, except to describe the spaceship that they are traveling in as an analogy of being-Partkdolg duty. In the Chapter 41, the excerpt of Gurdjieff as Beelzebub mentioning the 'Law of Ninefoldness' is, unfortunately lost in a tangential story on the effect of sound vibrations on the physical body that is clearly about his failure to prevent his wife's death from cancer.

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 by Vortex Math of "The Nirioonossian-World-Sound" as described in The Law of Heptaparaparshinokh, Ch 40: "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 Gurdjieff Enneagram, as shown to Ouspensky, is formed by this mathematically elegant process:

  • Mark nine points on a circle numbered 1,2,3,4,5,6,7,8, with 9 at the top
  • Divide the numbers 1 through 6 by 7 for the 'Law of Sevenfoldness' yielding
    1/7, 2/7, 3/7, 4/7, 5/7, 6/7 giving: .142857142857142857..., .285714285714..., .428571..., .571428..., .714285..., .857142 ...
  • Note that these results contain the same repeating decimal
  • Draw a line connecting these numbers on the circle (Note that the repeating decimal numbers when added together equal 9 (1+4+2+8+5+7=27=9)
  • The missing numbers 3, 6, and 9 can then be connected to form an inner triangle with it apex at the top. ****Thus the famous symbol appears***
    (We will see that Vortex Math has a similar triangle where 3 and 6 continually swap places.)
  • More importantly, is the assignment of the 7 positions of the octave as follows: 're' to position 1, 'mi' to position 2, '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'. Thus the Gurdjieff Enneagram incorporates the law of Seven (Heptaparaparshinokh), the law of Three (Triamazikamno), the law of Ninefoldness, the sacred law of Six (the inner lines), and the law of Twelve (the intersection of these lines with the inner triangle).

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

In my first lecture: "The Key to Soul Is Within The Missing Stopinder in the Food Octave", I ignore the inner vector lines and use only the notes and the triangle of the Enneagram to 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 'la' was intentionally left out in order to demonstrate the action of the vectors in the Enneagram. He supports this by showing that Beezebub's six "Voyages to the Earth" and other processes in that book follow the Enneagram order, even though he agrees that Enneagram was not explicitly mentioned. His interpretation of the Enneagram process is of interest even though I believe that my solution of the missing 'la' is simpler.

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. 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 are swapping natures alternately. The devil and the angel pull each other around the circle. In the dance I show the angel walking from 'la' to 'ti' to allow for the problem of the missing position 'la' in the Food Octave. Other than that the diagrams match the author's.

six sets of "par de deux"

Left section (8-5-7-1) of eneagram controls

Right side of triangle controls

Right section (1-4) of eneagram controls

Bottom side of triangle controls

Right section (4-2-8) of eneagram controls

Left side of triangle controls

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.

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

  • Starting with 1 doubled = 2 shows the line is drawn from 1 to 2
  • 2 doubled = 4 shows the line is drawn from 2 to 4
  • 4 doubled = 8 shows the line is drawn from 4 to 8
  • 8 doubled =16 which is then reduced to 7, thus shows the line is drawn from 8 to 7
  • 7(16) doubled = 32 which reduces it to 5, thus shows the line drawn from 7 to 5
  • 5(32) doubled is 64 which reduces to 10 and then reduced to 1, thus shows a line drawn from 5 to 1
  • The triangle points have a unique quality that, when added or 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
  • Imagining a line drawn down from 9 at the top vertex or apex to the middle of the bottom line demonstrates
    the horizontal symmetry which divides the circle into mirrored halves. Looking across then 8+1=9; 7+2=9; 6+3=9; and 5+4=9
The Gurdjieff Enneagram can be derived from the Vortex figure

1-2-4-8-7-5-1                                     not 1-4-2-8-5-7-1
This derivation shows that the two systems are related, except for having opposite circular motions.

Vortex and Enneagram overladed

The Marko Rodin Vortex also works by halving. In this case the circular direction is reversed which clears up the conflicting lines 4-2 and 5-7.


Clearly the Vortex and the Enneagram are related. As we will delve deeper into Vortex Math,
we will find that it is is a gold mine of mathematical derivations.

The are certainly critics of the Vortex Math presented by Marko Rodin. They claim that it is the obvious result of the base-Ten mathematical system and point out that it will not work for other base number systems. This is, however an absurd complaint 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.”

Tesla's Map to Multiplication - Composites and Primes as a Self Organized System

Tesla observes that:

  • "2 and 10 act as doublers alternating between the doubling of prime positions next to them and across
  • 3 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, like the Gurdjieff Enneagram, is 9-3-6.

Pythagoras grouped numbers as to patterns (for instance, the triangle is made up of triangular numbers):

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:

In the next table of Vortex numbers, all numbers are multiplied and then reduced (RED), showing each number has its reversed twin. (See the linked statements in the top row.)


Thus the Vortex diagram can be divided into right and left, positive is on the right and the negative on the left.

NegativePositive


Vortex and Enneagram overladed

Since the Enneagram has similar mirror image, we can surmise that it also has a positive and negative side. The action of the Enneagram can be considered to be of the spiritual realm and the Vortex to be of the material realm because the crossing point of the Enneagram is closer to God ('do') at the top than the Vortex which serves the physical body and is lower down. Also, the Vortex follows the side tracks and the Enneagram takes shortcuts and the returns back like the peaks and troughs of a wave which allows it to cross at a higher point.

The influence of the triangle points at 3 and 6 are similar in both systems. In the Vortex Math, 6 and 3 swap places continuously. In the Enneagram, the triangle influence is important to help bridge the two discontinuities of 'mi' and 'fa' and 'sol' and 'la' as both system lines of each are in opposition. (Incidentally, this supports the symmetrical action at 'sol' and 'la' instead of the expected discontinuity between 'ti' and 'do' - further discussed in the in the Food Octave in Talk 1.)

The next figure shows the Enneagram's mirrored nature. (What is amazing is the affinity of mi' and 'la' to the triangle at 6 and 3.)

NoteActivePassiveNeutralAffinitySignificance
re1427 (1+4+2) Actual line reversed
fa4285 (4+2+8) Lower arc of circle clockwise
mi2856 (2+8+5) Triangle connection left side
ti8572 (8+5+7) Actual line reversed
sol5714 (5+7+1) Lower arc of circle counterclockwise
la7143 (5+7+1) Triangle connection right side

Next, let us study the relationships of the notes using Vortex Math.
This table below derives the hertz values of the Circle of Fifths
using the basis numbers that created the vortex diagram above.

The first column of the table below:
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) ...
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) ...
the column for C is the unreduced number and 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 universal derived note vectors with the Gurdjieff Enneagram overladed:

Here we can see some notes that are opposite to the direction of the Enneagram and some places they enforce the direction of the Enneagram. And the two places actually where they are opposed and supported will be the laurels of the devil and the angel in our final diagram.

The line E ('mi') to B('ti') from seven natural notes of the Twelve Note diagram (above) reinforces one line of the Enneagram while line D ('re') to A ('la') is being opposed by the Enneagram line. These are the two laurels of the angel and the devil. Comparing the functions to the corrected Food Octave in lecture 1:

In Food octave 're' was Digestive System (stomach), 'mi' was Circulatory System (liver), 'fa' was Nervous System (cerebellum), 'sol' was Thinking Center (cerebrum) (These were swapped from what Gurdjieff has shown.), and 'la' was the Endocrine System (hormones, "heart") (that was the vibration that was left out of the Food Octave by Gurdjieff), and then the final one was the Reproductive System (the sex organs).

Here is the triumph of ourstidy

  • The devil is on the right using the negative influence of matter in opposition to the angel on the left using the positive influence of spirit.
  • The unopposed path 'mi' to 'ti' is the devil taking the energy from food in the stomach powered by nerve system (cerebellum) and sending it out from the liver and bile and blood to the sexual center, thus fulfilling the purpose of life to procreate.
  • The opposed path from 're' to 'la' is overcome by the musical path of the angel that draws energy from the food in the stomach to join the sexual energy that is powered by the cerebrum and combining with the endocrine system of the heart's higher blood to form the Kesdjanian Body and Soul.
  • These paths cross at a place lower than the higher 'do' (God) which is the human form of God as indicated by their crossed laurels. It is the place of the Holy Planet Purgatory.
  • (The two unrelated musical vectors: A ('la') to E ('mi') and G ('sol') to D ('re') show that one should guard against hormonal energy being dissipated in the liver and spleen and the cerebrum being too attached to material food instead of the spiritual.)
  • (Note that the cerebellum and cerebrum are related, but opposite with the cerebrum supporting the Kesdjanian Body and Soul.)



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