Johnny is researching the history of Quadrays. As I was saying to fellow faculty in the faculty lounge (virtual):
The way I see it is: Quadrays are a shelf of books in a math library or bookstore, but only a few of them bother with Synergetics. That’s a metaphor as “shelf of books” implies wood pulp, but we’re actually doing all this in the cloud without much paper.
In my version of the narrative, Quadrays are an escape from the linear algebra “hegemon” (spellchecker complains again), but not in such a way as undercut LA in any way. Machine learning is all about LA and I’m one of who teach (some might say preach) ML (my company bosses were near Istanbul — not sure where exactly given I was in a cloud-based company using Google glue).
Some of us (picture a subculture) like exploring in other namespaces with an apparatus that has a lotta family resemblance to the principal Hilbert Space exploration device: the extended Euclidean version of XYZ. Quadrays operate well within the IVM, which ML (NLP) will call a “substrate” for same, per this output from Charlie (Charlie = generic chatbot).
### 3.4 Fuller's isotropic vector matrix (IVM) and the coining of "quadray" **R. Buckminster Fuller (with E. J. Applewhite), *Synergetics: Explorations in the Geometry of Thinking*, Macmillan, Vol. 1, 1975; Vol. 2, 1979.** Full text: [monoskop.org](https://monoskop.org/images/4/46/Fuller_R_Buckminster_Synergetics_1997.pdf); scan at [archive.org](https://archive.org/details/synergeticsexplo0000full_t1i6). Sections 420.01–420.02 describe the isotropic vector matrix (IVM) as "four-dimensional and 60-degree coordinated" — Fuller's own coordinate framework, based on the FCC/CCP (cubic close-packed) lattice, i.e. geometrically the same lattice underlying quadrays and (via the \(A_3^*\) permutohedral identification) GBT in three dimensions. **Fuller's IVM predates the specific "quadray" coordinate formalism by roughly six years** and is the acknowledged geometric substrate on which quadrays were later built, but Fuller's own texts describe vector *directions* and a lattice, not an explicit \((a,b,c,d)\) 4-tuple coordinate algebra with defined arithmetic.
### 3.4 Fuller's isotropic vector matrix (IVM) and the coining of "quadray" **R. Buckminster Fuller (with E. J. Applewhite), *Synergetics: Explorations in the Geometry of Thinking*, Macmillan, Vol. 1, 1975; Vol. 2, 1979.** Full text: [monoskop.org](https://monoskop.org/images/4/46/Fuller_R_Buckminster_Synergetics_1997.pdf); scan at [archive.org](https://archive.org/details/synergeticsexplo0000full_t1i6). Sections 420.01–420.02 describe the isotropic vector matrix (IVM) as "four-dimensional and 60-degree coordinated" — Fuller's own coordinate framework, based on the FCC/CCP (cubic close-packed) lattice, i.e. geometrically the same lattice underlying quadrays and (via the \(A_3^*\) permutohedral identification) GBT in three dimensions. **Fuller's IVM predates the specific "quadray" coordinate formalism by roughly six years** and is the acknowledged geometric substrate on which quadrays were later built, but Fuller's own texts describe vector *directions* and a lattice, not an explicit \((a,b,c,d)\) 4-tuple coordinate algebra with defined arithmetic.
