The enneagram is a schematic diagram of perpetual motion, that is, of a machine of eternal movement. But of course it is necessary to know how to read the diagram. The understanding of this symbol and the ability to make use of it give man very great power. It is perpetual motion and it also the philosopher's stone of the alchemists.
-Gurdjieff, In Search of the Miraculous
I don't think most people are familiar with why the symbol is designed as it is, so I'd like to cover this topic.
The symbol originated with Gurdjieff, who taught the Fourth Way. He was a spiritual teacher, who saw his work as esoteric Christianity. I believe the reason for this is that he saw very many similarities between what we call Christianity and what Gurdjieff saw while traveling in the Middle East. With this in mind, we can understand why Gurdjieff considered the Enneagram symbol (one of many things he taught) as an illustration of how the "Law of Three" and the "Law of Seven" function in tandem with one another.
The Law of Three is pretty easy to understand. It is reflected in religion as the idea of the Holy Trinity, among other teachings. The Law of Three is simply the idea that every phenomenon is the result of three opposing forces. Gurdjieff called these "Active," "Passive," and "Reconciling." This is similar to an idea in dialectics (thesis, antithesis, and synthesis), as we note that "thesis" and "antithesis" are only labeled as such in relation to one another (and all are "thesis," meaning all forces are "Active"). For Gurdjieff's purposes, this is the law by which learn to see ourselves in terms of three forces.
The Law of Seven is a bit more complex. It is reflected in religion by the Seven Deadly Sins, but similarly the seven heavenly virtues and the seven days of the week. The Law of Seven is basically an explanation of the development of processes. We use the number "7" because it is the law of octaves, which Gurdjieff uses to explain nearly everything. If you're familiar with a musical scale, you'll know that there are seven basic tones (do-re-me-fa-so-la-si), with the fundamental (do) naturally being repeated in the playing of a musical scale. For Gurdjieff, the seven tones represent the Law of Seven, with the fundamental being a repetition of the first note in the another octave. To see how these map on the symbol, here's a figure:
Both of these laws are, at their base, laws of unity. So in sum, we see the symbol as a Triangle (Law of Three), Hexad (Law of Seven), and Circle (both as laws of unity). The circle symbolizes "the eternally returning and uninterruptedly flowing process of the isolated existence of a thing or phenomenon under examination." (source) The triangle is an intuitive representation of the Law of Three, but the Law of Seven needs further explanation.
The hexad is based off a property of the decimal system (which, like everything, is governed by the laws of unity reflected in all phenomena). We use the decimal system because we're adding the Law of Seven to the Law of Three (7+3 = 10). By dividing unity into seven, we get the lines of the hexad repeated infinitely (1/7 = .142857...). In addition, by "theosopical addition" (numerology), all the numbers in the hexad add up to nine, making nine the seventh number (1+4+2+8+5+7 = 27; 2+7 = 9). Additionally, the number 9 as the unifier can be seen in the division of seven into seven (7/7 = .999999...), meaning that the parts of unity cannot be connected without the number nine.
We can also see the Law of Three explained in a similar way (1/3 = .333..., 2/3 = .666..., 3/3 = .999...). And to complete the idea of being based on a decimal system, the symbol as a whole represents the numeral "0."
Ichazo re-explains this symbol through what he calls "Trialectics." If you'd like, I can write a thread discussing that concept as well, but you can get a sense of it by reading here: P2P Foundation
Mapping to the Enneagram of Personality
For brevity, I have left out the fact that Gurdjieff did not link the symbol with 9 personality types. That was Ichazo's doing, five years after Gurdjieff's death. In fact, one of my sources suggests Gurdjieff spoke of 12 personalities linked to the Zodiac (see here), and explicitly not 9 types. However, Ichazo's knowledge is hard to come by, and I've had to go through Naranjo to get a good sense of how these really map to the symbol.
In Naranjo's view of the Enneagram, each ennatype is characterized as being motivated out of a deficiency. In echo of Maslow's self-actualization, Naranjo proposes the dichotomy of abundant love (possible at self-actualization) and deficient desiring (how we are typically motivated). He further divides this deficient desiring into the Buddist three poisons: ignorance, aversion, and craving. Note that these are in contrast to the three poisons given by Ichazo (greed, deceit, and hatred).
Though Naranjo defines each ennea-type by its prevailing fixation and its corresponding passion, Naranjo sets the passion as being the principle component. Put plainly, this means that the 9 types are fundamentally related via the seven deadly sins (plus vanity and fear). It is this formulation that I will use to illuminate the meaning of the connecting lines.
The position of 9 at the top evokes the fact that 9's dominant passion of psychological deadness (acedia/sloth) is the most fundamental, and is the background of all other passions. Along with 3 and 6, it forms the inner triangle, which plots the following psychodynamic cycle: the psychological deadness of acedia (9) deprives the individual of a basis from which to act (loss of inner guidance), leading to fear (6); fear compels us to act from a false sense of self (attachment to the assumed self of personality), causing vanity (3); vanity leads us to repress our true nature (fabricated boundaries of the self), leading us back to the psychological deadness of acedia (9). Additionally, we can see that at each step of the way, there underlies a psychological sloth, with avoidance of fear and craving of vanity ultimately being succumbed to out of sheer laziness.
The hexad points to a similar pattern. Here it is clear that no point is really the first, as Sloth (which underlies all other passions) is not connected by the lines of the hexad. However, I'd like to point out that I did not find the following, complete formulation of these dynamics in Naranjo's work. With that said, let's begin with point 1:
Wrath (1) is a defense against Gluttony (7) through the repression of desire, yet can lead to a hateful Envy (4) of those who freely indulge themselves. Envy (4) is a defense against Wrath (1) through turning anger onto oneself, yet can lead to a romanticization of pain and a corresponding Pride (2) in one's self-destructive suffering. Pride (2) is a defense against Envy (4) through repression of inferiority and lack, yet can transform into an exploitative Lust (8) when the need for others is fully repressed out of consciousness. Lust (8) is a defense against Pride (2) through a denial of the need for their own and others' approval, yet can mutate into a withdrawn, clutching Avarice (5) through the decision to erase others from their life. Avarice (5) is a defense against Lust (8) through a rejection of one's hunger for aliveness, yet can lead to scattered, superficial Gluttony (7) as the need for anything at all is dismissed in favor of a false sense of abundance and love for life. Lastly, Gluttony (7) is a defense against Avarice (5) through a false love of life, yet can lead to an inner deadening, a mounting anger, and finally Wrath (1) as the individual's false love of life shatters, leaving the illusion and denial of destructiveness only on the inside.
There may be a case for the opposite directionality as well, but I think this does not hold up in reality. One may say that those in Wrath are growing by repressing the self-indulgent way of Gluttony, but rarely is it the case that one who is truly wrathful (rather than being compelled by others to exhibit the perfectionistic tendencies at Wrath) can let loose enough to become overly hedonistic. The same may be said of the truly gluttonous becoming avaricious, and so forth.
Additionally, each passion of the Enneagram may be understood as a combination of the two points adjacent to it:
Wrath (1) is the hybrid of drowning out one's perversion of justice (implicit in the moral apathy of acedia, 9) with Pride, insisting that one's actions must be right. Pride (2) is the hybrid of an excessive concern for one's image (vanity, 3) with Wrath (1), insisting that one's image is inherently above others. Vanity (3) is the hybrid of Pride (2) and Envy (4), a union of opposites that neither inflates oneself nor has spite for others, but which has both a love for garnering others approval and and an excessive concern for what one is lacking. Envy (4) is the hybrid of vanity (3) with Avarice (5), with both a concern for one's image and a vindictive need to take what they lack from others. Avarice is a hybrid of fear (6) with Envy (4), leading to fearful grasping for what one deeply lacks. Fear (6) is the hybrid of Avarice (5) and Gluttony (7), again a union of opposites that leads to neither a grasping for what one lacks nor a passion for life, but which has, for that reason, a confusion on a basis for which to act and a compound avoidance of the external and internal worlds (ironically, how these two combine leaves me the most confused). Gluttony (7) is a tempering of fear (6) with Lust (8), leading to a pursuit of life that is superficial and scattered precisely because of the underlying fear it keeps at bay. Lust (8) is the overt expression of the moral apathy seen in acedia (9) being excessively hedonistic, coarse, and careless through its union with Gluttony (7). Acedia (9), at the crown of the Enneagram, is the union of Lust (8) with Wrath (1), forming a character that is neither concerned with being just nor with pursuing life. In fact, this neutralization is the greatest union of opposites, leading to peaceful disposition that denies both destructiveness and the desire for aliveness.
If you'd like to read further, here are some sources I used in writing a more extended version of what I've written here:
Fourth Way - Wikipedia
Unveiling the Enneagram - Introductory Points
Arica School - Wikipedia
Unveiling the Enneagram - Arica Psychology
Claudio Naranjo - Wikipedia
Character and Neurosis
Naranjo's Instinctual Subtypes