Amy C. Edmondson A Fuller Explanation

Chapter 7, Vector Equilibrium pages 85 through 89

Spatial Considerations
A symmetrical distribution of "energy events" involves a large number of equivalent separation distances. For a more tangible image than provided by gas molecules, picture a large room full of people asked to spread themselves out for stretching exercises. If the room is sufficiently crowded, a more or less triangular pattern in the distribution of people can be observed, as a result of individuals' trying to maximize the area of their territory. Each person feels he or she has more space when the distances between people are maximized, which is the case when all distances are as close to equivalent as possible. If it's hard to see why that equivalence implies a triangular pattern, read on.
Think about baking cookies on a tray. Intuition rather than geometrical training tells you that the cookies in successive rows should be offset to maximize the number which can fit on each tray without spreading into each other. Observe in Figure 73 that a square distribution with the same minimum separation between cookies wastes considerable tray area, resulting in fewer cookies than a triangular pattern.

Fig. 73 Triangular pattern enables one more row of cookies than square pattern with the same minimum separation distance between cookies. Click on thumbnail for larger image.

In the same way, people in a room naturally (and quite unconscious of the advantages of triangular distribution) milling around until each carves out a desirable comfort zone can end up by increasing the overall symmetry. This organization does not require a director at the head of the room. Nature behaves in the same manner, seeking the most comfortable resting position. Forces continue to push or pull until counterbalanced, and in the absence of other influences, symmetrical considerations dominate. (There is, in Fuller's words, an "a priori absolute mystery" of why nature behaves this way, which is beyond explanation. The question is thus how Universe operates.)
The advantageous balances suggested by the term equilibrium can be expressed in terms of symmetry. The properties of space are necessarily behind all events and reactions in nature; hence Fuller's assertion that forces in both "macrocosmic and microcosmic structures interact in the same way." Space is the same on every scale, embracing and molding the "most economic equilibrious packings."
With the conceptual foundation in place, we can now describe the model proposed by Fuller to represent equilibrium. In his words, we seek the simplest "omniaccommodative system" able to model the behavior of complex systems. Basically, we want to draw, or better yet, build that muchdiscussed balance of forces. To accomplish the desired result, a model must incorporate two aspects of Fuller's geometry: first it must consist of vectors, and secondly it must cover all directions. In short, we want to illustrate an equilibrium of vectors in space.
Planar Equilibrium
Spatial configurations tend to be difficult to visualize, whereas fiat patterns are not, so we start with the page. Draw a vector of some arbitrary lengthwhich we designate "unit length"—in any direction. To counteract that force, we position a second vector directly opposite the first, head to head (Fig. 74a). They have the same magnitude and opposite direction and are therefore balanced.

Fig. 74. In search of equivalent radial and circumferential vectors. Click on thumbnail for larger image.

It is an unstable balance however, easily knocked out of equilibrium by a force from any other direction. Suppose that the original two forces push on a body with equal strength from east and west. A force from the north or south, even if smaller than the eastwest pair, can easily destroy that unstable equilibrium. So how can we most efficiently insure the stability of the body in question?
Suppose instead that the forces are directed outwardly from the body (Fuller's "explosive potential," as quoted in the first paragraph of this chapter). To begin with, imagine four equal vectors, heading north, south, east and westthat is, in the positive and negative directions of the X and Y axes. To counteract the four explosive forces, we need equivalent restraining forces ("implosive" or "embracing"). Fuller's "embracing" vectors are not technically part of the conventional language of vectors. Having neither head nor tail, their effect is simply restraining, like a net, and their magnitude is still assumed to be represented by their length. As strength is graphically depicted by vectorlength, we soon find that there is no easy way to draw four embracing vectors of unit length. In Figure 74b, the ends of our four explosive vectors are interconnected, but these new lines are approximately 1.414 times as long as the outward forces. The longer vector lines represent more powerful forces and thus overpower the explosive potential, meaning that the whole display must collapse inwardly.
We might then choose to add more outward forces, maintaining symmetry by an additional unit vector in between each of the original four (Fig. 74c). Now we have eight equal forces emanating from one point, and the resulting eight embracing lines are only 0.765 times as long as the unit lengthtoo small to restrain the explosive forces. This imbalance leads to outward dispersal of the hypothetical system.
The soughtafter balance will be achieved by "omnisymmetry", that is, maximum symmetry. The desired array must consist exclusively of unitlength vectorsboth explosive and embracing.
One way to solve the problem is to picture a square grid of "energy events," interconnected by vectors. Squares provide an easy starting point because they make up the basic framework of current mathematics: the XY coordinate system. As before, the length of vectors which connect "events" represents the strength of their interattraction. Because the distance between diagonal corners is 1.414 times the distance between adjacent loci, the attractive forces represented are that much greater. Imbalance (or lack of equilibrium) in a diagram of forces represents motion. As a result of the attractive forces between neighboring energy events, they push and pull on each other until all the disparate separation distances became equal. Again, like the cookie tray, a pattern of equilateral triangles is established. Every single energy event is a uniform distance apart from each neighbor, and there are 60degree angles between all vector connections. The forces are finally balanced, and the resulting array informs us about the fundamental symmetry inherent in a flat surface.
Going back to the radial vector diagram, we now arrange six vectors emanating from a point. The embracing vectors will be the same length as the "exploding" group. An inescapable consequence of this fact (obvious in retrospect) is that angles between all vectors are also identicalnot just those between radial vectors, but also those between circumferential and radial vectors (Fig. 74d). No other arrangement has this property, because 60degree angles are integral to equilateral triangles.
A diagram of radial vectors can be thought of as an apple pie divided into some number of pieces. These pieces are always triangular, with two radial edges and one circumferential edge. Because of the fact that the angular sum in every fiat triangle is 180 degrees, the only way for a triangle to have all angles the same is with three sixtydegree angles. Therefore, the only angular measurement that will allow us to divide the vector pie "omnisymmetrically" is 60 degrees, requiring that unity (360 degrees) be divided six ways. The procedure is straightforward so far. In the plane, equilibrium is demonstrated by a hexagon (Fig. 74d).
Now we make the leap into space, with its accompanying leap in complexity. It may be difficult to visualize a spatial array, especially noncubical configurations, but taking it step by step, we shall be able to develop and understand Fuller's model.
In terms of vectorial dynamics, the outward radial thrust of the vector equilibrium is exactly balanced by the circumferentially restraining chordal forces: hence the figure is an equilibrium of vectors. All the edges of the figure are of equal length, and this length is always the same as the distance of any of its vertexes from the center of the figure. (430.03)
A "geometry of vectors," Fuller reasoned, must be "omnidirectionally operative" —hence, radially oriented and omnisymmetrical. Following the planar example, we want some number of vectors emanating from an origin, situated so that the distances between vector end points (vertices) are not only all equal to each other, but also exactly equal to the length of the radial vectors.
