Knowing this thread speaks to mathematics and physics (which I don’t profess to know anything – but interested) and having read this thread a few times over, following some of the links posted too, including the esoteric properties you discuss Ark, this send me off looking up a few things, and as happened, went to Fulcanelli.
In your picture of the Groupoid, could not help thinking about interlocking gears with the central circle representing the influence over the others as it transforms their triad aspects. In reading out of Fulcanelli’s ‘The Dwellings of the Philosophers’, firstly on Algebra, this came up related to X (unknown quantity):
Furthermore, almost all meanings revealed by the sign X have a transcendent or mysterious value, and this fact is singular. In algebra, X is the unknown quantity; it is also the problem to be solved, the solution to be discovered; it is the Pythagorean sign of multiplication, and the element to cast out the nines in arithmetic; it is the popular symbol of mathematics in what concerns higher or abstract development. It characterizes that which, generally is excellent, useful, remarkable.
Secondly, in thinking about the Groupoid depiction you posted, chemistry came to mind, and in particular Fulcanneli’s words below. It is noted that in this case my thinking is that the central circle of the Groupoid, as it relates to the chemical description, is like X (what is the nature of the influence, how does it transform the others and is there something in the others that either does nothing or energetically changes its nature?).
Light --- rarified and spiritualized fire --- possesses the same chemical virtues and power as elementary crude fire. [whatever light is, in this case assigned it to the central circle 1] An experiment, with the object of synthetically creating hydrochloric acid (HCl) from its components, amply demonstrates it. If we put equal volumes of chlorine and hydrogen gas in a glass flask [circle 0,0 & circle 1,0], the two gases will keep their own individuality as long as the flask that contains them is kept in darkness. With some diffused light [circle 1], they progressively combine. But if we expose the vessel to direct solar rays, it explodes and shatters violently.
“exchange transformation” is denoted by 1.
Probably barking up the wrong tree here in thinking about chemical reactions and Groupoids, however, Fulcanelli discusses some unusual principles of triad interactions in many instances.
Hope to learn more.