Quadrupole/octupole structure in CML tetrahedron?

[Jess Tauber]

The other day I reminded Eric that I have made no claims about tetrahedral (mathematical -motivational) structure in my electronic and nuclear configurations of shell systems being physically instantiated. Far from it- just being mathematical objects.

But it did get me to thinking. The tetrahedral CML (Continuous Mendeleev's Line) configurations, which have no line breaks in the structures (all EIGHT variants), essentially being a single linear sequence of atomic numbers folded into a tetrahedral shape along symmetrically and antisymmetrically oriented period 'tiles', with each element's atomic number represented by a sphere, close-packed against the others, DOES in fact have 'axes'.

For example, the core and jacket subtetrahedra (inner 20 spheres representing the first four LS periods versus the outer 100 representing the fifth through eighth) meet at so-called 'kissing' spheres (where their surfaces meet directly) at the point where atomic number 20 (the end of te first set of four Janet periods) and 21 (the start of the next set of four) touch. Experiments with physical models I built some years ago showed that if you extended the PT such that the NEXT sets of four periods were similarly 'jacketed' around the tetrahedron, ad infinitum, the set of interconnecting kissing spheres between core and successive jackets was a perfect straight line, something completely unexpected.

But there is other structure as well. For example, when you set the tetrahedron vertically 'edge-on' (as one might a meat cleaver against a cutting board when in use, rather than at rest), then the tetrahedron had an 'equator' dividing it into two symmetrically equal halves. In all eight different configurations, s-block spheres are organized so that either both (in four configurations) or one (in the other four) were on said equator, symmetrically arrayed around the equatorial perimeter (which is laid out as a square).

This is where the idea of quadrupole/octupole structure comes to mind. Many of you will know that the earth's (and the sun's) magnetic poles tend to wander over time. We don't simply get pole-flipping (from North to South or vice-versa). Rather the strong North and South pole local fields divide up into weaker subsidiary poles which then start migrating towards the equator. I imagine they cross the equator pretty much all together from both poles, weakening as they approach it. Minimum field strength occurs when they cross the equator. But then as they approach the opposite pole after they've passed the equator, the subsidiary fields consolidate and strengthen again until finally they become unitary poles once more, but on the opposite side of the planet (or star).

I'm wondering whether or not this can be applied analogically to the configurations of the CML tetrahedra. Is there hidden structure here that might be of some use in chemistry, connections we never could have imagined? I mentioned above how a straight line organized all the positions where the last atomic number of the core four LS periods meets the first of the surrounding jacket (also containing four) etc. etc. for successive jackets. Is there a relationship between the relative positions of the s-block spheres on the tetrahedral equator and this line? Are there other such lines, arranged symmeterically (or antisymmetrically) about the figure- say, organizing where orbitals end or begin, and so on?

Because of the nature of the social world around us, we tend (at least in Western culture) to think in terms of Cartesian coordinates. But Nature may not give a damn about cultural biases.  In my analyses of imitative words in natural languages I've found that angular coordinates are the norm. Perhaps the same sort of thing goes for the abstract representations of the PT as well?

Jess Tauber

tetrahed...@gmail.com