In geometry, an enneagram is a ninepointed 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: 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.
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:
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 beingPartkdolg 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 NirioonossianWorldSound" (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 soundproducing 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: 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.)  
Here is the Enhanced Enneagram with all possible lines

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 wellthought 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. 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 twostep process (Piandjoehary first moves and then is converted into Exioehary by Harnelmiatznel) occurs at exactly that phase of the sixphase 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) 
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 (18561943), 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.”

Vortex Math also works for the Fibonacci Series as shown by Maff Math:

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.

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 (adapted from Keith William Horwood's program that plays standard notes.) 
derived from the table above (showing exact digits) 
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: 