In my view, a lot of readers of Synergetics get off on the wrong foot in thinking Fuller’s “4D” is carbon copied from say either Einstein or Coxeter.
But from which one, Einstein or Coxeter? Does it matter? Didn’t they mean the same thing? No.
Einstein with his “time is the fourth dimension” and Coxeter with his n-dimensional polytopes (e.g. “tesseract”), are not speaking the same language, as Coxeter himself makes clear on page 119 of Regular Polytopes (Dover edition):
Little, if anything, is gained by representing the fourth Euclidean dimension as time. In fact, this idea, so attractively developed by H.G. Wells in The Time Machine, has led such authors as J. W. Dunne (An Experiment with Time) into a serious misconception of the theory of Relativity. Minkowski’s geometry of space-time is not Euclidean, and consequently has no connection with the present investigation.
H.S.M. Coxeter. Regular Polytopes. Dover Publications, 1973. pg. 119
I’d say what Fuller learned from observing his intellectual milieu is that the word “dimension” is all over the place and is not especially nailed down forever, anywhere, even to this day.
Mathematics is allowed to, and does, produce new usage patterns for old words and this is what Fuller engages in a lot: crafting his own meanings, inventing his own language games.
[ Coxeter studied under Wittgenstein a bit, the “language game” guy, but then opted out of philosophy while still allowing the meetup to use his living quarters for their intimate meetings. Source: Siobhan Roberts, The King of Infinite Space ]
Fuller’s “4D” aside from serving as a kind of logo, a brand, refers to the irreducible relationship between our concept of “containment” (container) and the tetrahedron or simplex, the simplest container, the first most minimal “box” with an inside and outside.
We learn as kids in elementary geometry: it takes four points to establish volume whereas three points only establish a plane, a triangle. Observing that triangle (imagining it from the “outside”) is already adding a fourth point (the vantage point). Four points make the four planes of a tetrahedron, each with its own perpendicular. Here is where Fuller’s “4D” anchors…