Quantum gravity, infinities, black holes, and Leibniz' Harmonic Triangle

[Jess Tauber]

Hi again folks. I was just reading a relatively basic level newsfeed article on black holes and quantum gravity (https://www.yahoo.com/news/holy-grail-theoretical-physics-key-130000300.html), and it occurred to me that the encountered infinities might be handled by Leibniz' Harmonic Triangle, which I've discussed in previous posts some time back.

The structure of the Triangle is similar to that of the Pascal Triangle, whose terms (when doubled) show up in the particle counts of orbital shell components in atomic nuclei (when spherical, and with modifications when possessing ellipsoidal deformations), as I've discussed many times in our group.

All the Harmonic Triangle terms are inverted relative to the Pascal Triangle terms but with a multiplying factor in the denominator identical to the dimensionality that the diagonal of said term resides within. Thus, in the Pascal Triangle, the outer 1's are OD, since they never change value. Then the Natural Numbers are 1D, as they change monotonically (and linearly). The Triangular Numbers are 2D, since they are motivated by close-packing of circles on a plane surface, and the Tetrahedral Numbers depend on close-packing of spheres in a 3D space, so are associated with 3D.

The equivalently positioned Leibniz terms for the Pascal Natural Numbers are simply inverted, so 1/1, 1/2, 1/3, 1/4..... But those for the Leibniz equivalents for the Pascal Triangular numbers (Tri) are 1/2Tri- where the denominator is 2Tri. For the Leibniz equivalent of the Pascal Tetrahedral Numbers (Tet)we get 1/3Tet, and so on into deeper diagonals, each time with the multiplicative factor in the denominator the same as the dimensionality of the diagonal the Pascal term is found within.

And here is where things get interesting- Leibniz himself simply lopped off the diagonal in the Harmonic Triangle equivalent positionally to the outermost 1's in the Pascal system, just because the multiplicative factor for these would be 0, that is terms defined as 1/(0x1). He didn't know how to handle these, deciding they were all undefinable. But you can also simply define them as infinities (as I have). 

In the classical Pascal Triangle, terms start with 1 at the apex and nearest-neighbor terms in rows then sum to give the next lower term between them. But in the Leibniz system, terms are all fractional, and sum UPWARDS to give 1 at the apex. Undefined terms on the outer edge can't do this, but infinities still sum upwards (but obviously don't give 1 at the apex). I had discovered years ago that Leibniz terms relate to the total energy of shell components (as opposed to the Pascal terms, which relate to their particle counts).

Finally, since the singularities at the hearts of black holes are supposed to be points (so 0D), and the Leibniz terms relate to energies (at least in observable quantum systems), and black hole singularities are supposed to be associated with infinity (at least in terms of densities), I propose that in some fashion this relates to the total energies there.

Jess Tauber

tetrahed...@gmail.com