Four Triangles Out of Two
Two triangles can and frequently do associate with one another, and in so doing they afford us with a synergetic demonstration of two prime events cooperating in Universe. Triangles cannot be structured in planes. They are always positive or negative helixes. You may say that we had no right to break the triangles open in order to add them together, but the triangles were in fact never closed because no line can ever come completely back into itself. Experiment shows that two lines cannot be constructed through the same point at the same time. One line will be superimposed on the other. Therefore, the triangle is a spiral – a very flat spiral, but open at the recycling point.
By conventional arithmetic, one triangle plus one triangle equals two triangles. But in association as left helix and right helix, they form a six-edged tetrahedron of four triangular faces. This illustrates an interference of two events impinging at both ends of their actions to give us something very fundamental: a tetrahedron, a system, a division of Universe into inside and outside. We get the two other triangles from the rest of Universe because we are not out of this world. This is the complementation of Universe that shows up time and again in the way structures are made and in the way crystals grow. As separate actions, the two actions and resultants were very unstable, but when associated as positive and negative helixes, they complement one another as a stable structure.
Our two triangles now add up as one plus one equals four. The two events make the tetrahedron the four-triangular-sided polyhedron. This is not a trick; this is the way atoms themselves behave. This is a demonstration of synergy. Just as the chemists found when they separated atoms out, or molecules out, of compounds, that the separate parts never explained the associated behaviors; there seemed to be “lost” energies. The lost energies were the lost synergetic interstabilizations.
Excerpt from: Synergetics Explorations in the Geometry of Thinking, by R. Buckminster Fuller in collaboration with E.J. Applewhite, Volume I pgs. 4-5 © 1975 R. Buckminster Fuller