Disambiguation: this story is about the American Transcendentalist Synergetics (two volumes, Macmillan, 1970s), by Bucky Fuller, not about Haken’s 1982 Springer-Verlag Synergetik.
I wrestle with whether to classify Fuller as “strictly Euclidean” bouncing off Karl Menger (dimension theorist) writing in another context (not about Bucky) that another way to deviate from the Euclidean program (and hence to not be a Euclidean) is to disavow axioms separating points, lines, planes, solids on the basis of their being “differently dimensioned” somehow.
Topologically, we can distinguish a ball (big or small) from a rod (line) from a board (plane or slab)… they’re all considered “solids” in the sense of “space-occupying” (even purely conceptual space, or call it Kantian space or as Bucky called it: “pre-frequency” space).
Menger called this “everything takes up space” geometry a “geometry of lumps”.
Menger goes on in the same essay to say denying the infinitude of any of these objects (no “infinitely thin, infinitely extensive planes”) is likewise a deviation from Euclidean thought patterns.
I quote Menger on his lumpy geometry in my slide deck on Synergetics (slide 47), the deck I encourage teachers to just go out there and use with my permission (then I give examples of me using it, on Youtube). The D’Arcy Thompson quote, from a 1918 letter to Whitehead, is also gold (slide 46).
I argue this “geometry of lumps” paradigm is precisely where Synergetics begins, with the tetrahedron our paradigm “lump” (you can shape it into any other shape so far mentioned, poke holes, whatever).
The suggestion is: there are other ways to be non-Euclidean than by fiddling with the 5th postulate.
On the contrary, those doing anything multi-dimensional on the metaphor of mutual orthogonals ala Coxeter are practicing “extended Euclideanism”.
