Terahedron as its own polyhedral dual

[Jess Tauber]

As many of you likely know, classical Platonic solids, the first historically well-explored polyhedra, come in pairs, called 'duals', simply by exchanging the faces and vertices. The cube is the dual of the octahedron, and the dodecahedron is the dual of the icosahedron.

But the dual of the tetrahedron, the simplest polyhedron, is also a tetrahedron. 

And this leads me back to the tetrahedral models of the periodic system. In these models, the individual elements are represented by close-packed spheres. And pairs of LST periods of same length sum to give square numbers- 2+2=4; 8+8=16; 18+18=36; and 32+32=64, all squares of even integers. And pairs of THESE then sum to tetrahedral numbers (out of the Pascal Triangle): 4+16=20; 36+64=100. 

Now, if the initial PT model is based on close packing of spheres, what would this imply in the dual tetrahedron? Spheres have their own equations, but if we're somehow exchanging the coordinates of the dual (faces for vertices), then what would the elements be represented by in the dual here? More spheres, or perhaps hyerboloids? And how would these be packed into the structure? In the tetrahedral models I and others have proposed, spheres are in nearest-neighbor contact (i.e. 'kissing'). But there is also rhombic structure present, and where one axis of the rhombus presents the kissing spheres, its perpendicular partner axis has the spheres separated by a gap traversing the center of the figure.

Would packing of hyperboloids focus on close contact as with kissing spheres, or more on longer-distance connectivity. I use the analogy here of the way the human cerebral cortex is structured and processes information- the typical left hemisphere deals more with nearest-neighbor connectivity (and in language more with grammatical issues), while the right deals more with long-distance connectivity and associations. This is pertinent to discussions on the way iconic vocabulary (the kind I study) is processed.

Anyway, I'm just beginning to explore these issues- any first impressions or suggestions would be welcome. With regard to the PT, would the dual tetrahedral representation of the LST structure show linkage between more far-flung memberships than the simple period and group ones that flat representations are able to, such as, for example, knight's move, etc.?

Jess Tauber

tetrah...@gmail.com

[ERIC SCERRI]

Dear Jess

Your talk of tetrahedra comes at an interesting time for me. Just yesterday at the international society for the philosophy of chemistry here in Antwerp, Jozsef Garai gave an interesting talk on the PT and tetrahedra in the nucleus. 

My problem is that I find it hard to accept this something as small as a p or n should even have solidity that would feature in close packing.  

What is the response to this?

He also published an article in a volume I edited with G. Restrepo. From Mendeleev to Oganesson, OUP. 

Regards 

Eric Scerri

www.ericscerri.com

On Jul 12, 2025, at 2:13 PM, Jess Tauber <tetrahed...@gmail.com> wrote:



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[Jess Tauber]

Welcome back. I have never made any claims that the Pascal Triangle relations have any sort of MATERIAL realization- just as I've never made such claims with regard to energy relations and the Leibniz Harmonic Triangle. These are mathematical observations that come from the experimental observations of other workers. I don't think Albert Einstein claimed that e=mc^2 was written in the fabric of spacetime, either. Whatever is giving our universe its marching orders (be it Divine Intervention, or just the combinatorics of material interactions), the computations don't seem to be observable to us mortals. They just ARE, whether we have an issue with them or not- reality doesn't really give a **** what we think about it.  I think we can all agree that there is no PT with all the elements sitting in nice, neat columns and rows for us to discover. It too is an abstraction. A model.

Our models of mathematical motivation, on the other hand, DO have relations to close-packing. Triangular numbers (which relate (when doubled) to the sizes of subshells in atomic nuclei) can be visualized as close-packing of circles in a plane. And tetrahedral numbers (again doubled) relate to the close packing of spheres in a volume. And so on, up through as many higher dimensions as you care to imagine.

Interestingly, spatiotemporal  'packing'(distributional) issues come up also in my research on imitative words in natural languages. In Japanese, for example, there are tens of thousands of such words in the lexicon. The longer forms are split into two parts- one of which details the mass-energy relationships of the phenomena/object being described by the form. The other part details its spatiotemporal distribution. Short forms can go either way. The human brain itself has tracts specialized for both these functions.

Jess Tauber

tetrahed...@gmail.com