The last two Posts served as a general introduction to the enneagram, which, when properly understood, can provide powerful insights into all aspects of one's life. This post will explore one of the grand principles involved in the enneagram, known as The Law of Three.
Before diving into the mechanics of how the Law of Three operates in the cosmos, it will be helpful to understand where it comes from and how it appears.
As mentioned in previous lessons, the triangle is the symbol for manifestation, for nothing can appear in the universe unless it carries at least three defining aspects. But from where do these forces come? Moreover, how can they be used with intelligence to achieve a specific outcome?
THE LAW OF TWO
The Law of Three is based on a more fundamental cosmic law known as The Law of Two, which simply states that every force in the universe has a polarity as a fundamental property. That is to say that all forces in the universe demonstrate dualistic properties that appear as opposites.
This observation is quite familiar to us in our daily lives. So much so that it often goes unnoticed. For example, consider our day-to-day familiarity with the following forces in their opposing manifestations:

…and the list could be carried on indefinitely.
Now, one thing that is often missed is that no force can find expression without the presence of some mediating factor. For example, hot and cold cannot exist simultaneously, independent of some medium as a reference point. When referring to the temperature bar, one must first identify the object that is either hot or cold. Next, to draw a contrast, one must recognize the thing's relationship to some other object or state. Hot and cold cannot exist without some mediator and frame of reference. The mediator and frame of reference serve as the neutralizing vector for the opposites in question.
To illustrate the point further, consider that the Law of Three manifests the interaction of two sets of opposing forces. To clarify, consider the following diagram, which shows the intersection of two distinct forces:

It should be noted that although something cannot simultaneously be both hot and cold, nor wet and dry, a thing may be hot and dry or cold and dry or cold and wet or hot and wet. The above diagram perfectly illustrates how one force, subject to the Law of Two, is mediated by another force, also subject to the Law of Two, to create a result. Thus, the intersection of two forces creates a new possibility by virtue of their interaction. In the above example, the effect produced by the meeting of two forces, subject to the Law of Two, is elemental, the manifestation of the four elements: Air, Water, Earth, and Fire.

It should be clear from the above diagram that although abstract forces such as hot/cold or wet/dry do not find direct expression at our level, elements such as fire, earth, water, and air do.
To further illustrate this, consider two more forces that exert a strong influence in all planetary life, including that of the human:

In the above diagram, the horizontal force of life splits the vertical force of gender. This splitting, of one force, by another is the fundamental way the Law of Three operates. This results in four manifestations or four Neutral Vectors:

The above diagram raises some interesting reflections. Consider, for example, that the opposite of a mother is not a father but a son. Similarly, the opposite of a father is a daughter. These observations are based on considerations of polarity alone.
The first thing to note is that there are always two potencies at play on the diagonal vectors. For example, in the "Father" vector, the potencies are +/-. The "Mother" vector contains the -/- forces. The "Daughter" vector corresponds to the -/+ powers, and the "Son" vector adopts the +/+ potencies. In other words, when looking at the results of the Law of Three, all manifestations will either appear as wholly positive, wholly negative, or a combination of the two. Another way to say this is that one vector remains intact (--/++), whereas the other will be split (+-/-+).
We will return to this feature of the Law of Three in upcoming lessons. At this point, let us press forward in our overview of the inner workings of this fundamental law.
THE RELATIONSHIP OF THE LAW OF THREE TO THE OCTAVE.
Let us now return to the above diagram, but this time with the notes of the octave imposed upon the vectors: