A System, says Bucky, is a "conceivable entity" dividing Universe into two parts: the Inside and the Outside of the system. That's it (except, of course, for the part of Universe doing the dividing; he demands precision).
A system is anything that has "insideness and outsideness." Is this notion too simple to deserve our further attention? In fact, as is typical of Fuller's experimental procedure, this is where the fun starts. We begin with a statement almost absurdly general, and ask what must necessarily follow. At this point in Fuller's lectures the mathematics sneaks in, but in his books the subject is apt to make a less subtle entrance! (Half-page sentences sprinkled with polysyllabic words of his own invention have discouraged many a reader.) The math does not have to be intimidating; it's simply a more precise analysis of our definition of system.
So far a system must have an inside and an outside. That sounds easy; he means something we can point to. But is that trivial after all? Let's look at the mathematical words: what are the basic elements necessary for insideness and outsideness, i.e., the minimum requirements for Existence?
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Existence is not the comparison of a form to the absence of a form. It is rather a Comparison between the Outside of an empty container and the Inside. Outside, each and every container represents *not Nothing*. From the outside we associate empty containers with Unity, with that which has no parts. (William Bricken, Iconic Arithmetic Volume I: The Design of Mathematics for Human Understanding (Unary press, 2019), p. 169.)
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Assuming we can imagine an element that doesn't itself have any substance (the Greeks' dimensionless "point"), let's begin with two of them. There now exists a region between the two points – albeit quite an unmanageable region as it lacks any other Boundaries. The same is true for three points, creating a triangular "Betweenness," no matter how the three are arranged (so long as they are not in a straight line). In mathematics, any three noncollinear points define a plane; they also define a unique circle.
Suddenly with the introduction of a fourth point, we have an entirely new situation. We can put that fourth point anywhere we choose, except in the same plane as the first three, and we invariably divide space into two sections: that which is inside the four-point system and that which is outside. Unwittingly, we have created the minimum system. (Similarly, mathematics requires exactly four noncoplanar points to define a Sphere.)
Any material can demonstrate this procedure-small marshmallows and toothpicks will do the trick, or pipecleaner segments inserted into plastic straws. The mathematical statement is unaltered by our choice: a minimum of four corners is required for Existence.
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Fig. 3-1. Six connections between four Events, defining a tetrahedral system.
What else must be true? Let's look at the connections between the four corners. Between two points there is only one link; add a third for a total of three links, inevitably forming a triangle (see if you can make something else!). Now, bring in a fourth point and count the number of interconnections. By joining *a* to *b*, *b* to *c*, *c* to *d*, *d* to *a*, *a* to *c*, and finally *b* to *d* (Fig. 3-1), we exhaust all the possibilities with six connections, or *edges* in geometrical terminology. Edges join *vertices*, and together they generate windows called *faces*.
This minimum system was given the name *Tetrahedron* (four sides) by the Greeks, after the four triangular faces created by the set of four vertices and their six edges (Fig. 3-1). Fuller deplored the Greek nomenclature, which refers exclusively to the number of faces – the very elements that don't exist. ("There are no solids, no continuous surfaces. . . only energy event complexes [and] relationships.") However, he did not fully develop a satisfactory alternative, so we shall have to work with the time-honored convention. What we lose in accurate description of physical reality, we gain in clarity and consistency.
The tetrahedron shows up frequently in this exploration. This and other recurring patterns seem coincidental or magical at first, but soon come to be anticipated – **endless demonstrations of the order inherent in space**.
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We are […] excited about a Schema [→Code] based endless-Form Interaction.
[…] the theory is directed against holographic approaches, as they are offered today in the style of the New Age; somehow a trace of the whole is to be found in everything individual, somehow every element is a part in which the whole is "inscribed". The concept of form is explicitly set against such a holographic New Age mysticism.
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The process is typical of synergetics: we stumble into the Tetrahedron by asking the most elementary question – what is the simplest way to enclose space? – and later, everywhere we look, there it is again, an inescapable consequence of a spectrum of geometric procedures.
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Edmondson, A Fuller Explanation, p. 27
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The tetralemma or catuṣkoṭi wikipedia 
, German sometimes called "Urteils-Vierkant", originates from Indian logic. Here it was first used in jurisdiction. It addresses the question of what positions can be taken with respect to 2 opposing parties.