Figurates, Or Something Else? The Family Atomics...

[tetrahedron]

Hi, guys. Still plugging away at my atomic Pascal stuff.

 

Yesterday I noticed a generalization, in terms of numerical differences of suborbital terminals of various types in spherical nuclei, for neutrons.

 

That is, some differences were simple doubled triangular numbers. Some were pairs of every triangular, thus half/double squares. Some were pairs of every other triangular, and so on.

 

So, obviously we are dealing with summed pairs of every Nth doubled triangular, from N=0 on up. Has anyone every addressed this series of sequences in any depth, even if only dealing with single triangulars (the doubling appears to relate to the fact that nucleons pair up immediately with others of the same type (N or P)). I've been looking for online resources but can only find references to the simplexes or squares.

 

And what is the relationship to figurate numbers, if any. The triangulars are found in the (1,1)-sided Pascal Triangle, and the squares in the same diagonal in the (2,1)-sided version.

 

I'm working to see if there is any motivated pattern in assignation of particular N's to particular quantum number differences. Given that a stacked table of generalized Fib sequences generates, directly or indirectly, all the four basic quantum numbers might or might not be related. There are other quantum numbers, so maybe...?

 

In the simple harmonic oscillator model of the atomic nucleus, magic numbers are generated by a function which multiplies the number of double triangular number inputs to the growing string according to the size of the oscillator ratio's polar dimension, while dividing by the equatorial dimension. So multiplication/division, against deformation (usually the x axis in depictions). What I've seen now involves instead the y axis, where energy levels are laid out, and total nucleon counts. This seems instead to yield addition/subtraction as the major motif.

 

So power/root? Metallic Means? More to come...

 

Jess Tauber

[email protected]

[Rade]

Hello Pascal,

 

I have found an interesting relationship between a triangular structure and the first eight alpha and beta stable isotopes of nature.

 

The set of these fundamental stable isotopes is interesting because it is missing the A values 5 and 8, that is, there are no stable isotopes (both alpha and beta stable) with A = 5 or 8 (where A = total of protons P and neutrons N). Numerically this set is {A= 1,2,3,4,6,7,9).

 

These eight stable isotopes are:

 

H1

H2

He3

He4

Li6

Li7

Be9

Be10

 

Of interest is that these eight isotopes can be represented geometrically using a nucleon cluster configuration of P and N for a 10 dot triangular structure composed of four lines of dots:

 

* = line 1

** = line 2

*** = line 3

**** = line 4

 

==

 

In geometric form these four lines yield this structure, and of course all the dots can be connected to represent bonds between P and N nucleons:

 

 

 

==

EDIT CHANGES: To apply P and N nucleons to the dots some selection rules must be applied. Line 1 dot needs to be either a P OR N, and line 3 dots need to be be either [PNP] of [NPN] configuration, depending on the isotope being represented.

 

By connecting dots in the above to form nucleon cluster configurations, we observe that all seven of the fundamental stable isotopes of nature are represented within the geometric structure of 5 Boron 10:

 

line 1 = 1H1 (hydrogen) OR N (neutron)

line 2 = 1H2 (deuterium)

line 3 = 2He3 (Helium 3); selection rule: [PNP] in line 3.

 

combine lines 1,2 = 2He3 (Helium-3); selection rule: Line 1 must be P

 

line 4 = 2He4 (Alpha particle)

 

combine lines 1,2,3 = 3Li6 (Lithium 6); selection rule: Line 1 must be a N and line 3 [PNP]

combine lines 3,4 = 3Li7 (Lithium 7); Line 1 must be P and line 3 [NPN]

 

combine lines 2,3,4 = 4Be9 (Beryllium 9); [NPN] in line 3

 

combine all lines 1,2,3,4 = 5B10 (Boron 10); Line 1 must be a N and line 3 [PNP]

 

==

 

In summary, all of the fundamental stable isotopes of nature are contained within a triangular geometry of Boron 10 when the isotopes are viewed as nucleon cluster configurations of protons P and neutrons N. The required selection rules suggest an important relationship between free N and P and [PNP] and [NPN] clusters required to form stable isotopes. The free N is unstable and decays into a stable P. Free [NPN] is unstable tritium which decays into stable [PNP] helium-3. This representation deviates from the currently accepted view of the atomic nucleus that protons P and neutrons N for the first eight stable isotopes are 'independent entities' existing within energy shells. The above suggests an equally valid representation of fundamental stable isotopes that can exist as nucleon clusters of P and N, nested within a 10 dot triangular geometry of Boron 10.

 

 

==

 

This sequence can be extended:

 

Combining two Boron 10 isotopes yields 10 Ne 20...which is a stable isotope. Thus, a geometric configuration for stable 10 Ne 20 isotope is:

 

 

Combining the two Ne 20 isotopes above yields 20 Ca 40...a stable isotope, the limit of the 5 Boron 10 geometric sequence.

Edited May 10, 2013 by Rade

[tetrahedron]

Well the question then is what is CAUSING all this, assuming there is something like a calculator keeping tabs in fictive or real space, hyper/subspace or whatever.

 

Several people who had done earlier studies on Pascal math in the nucleus created models where the nucleus really WAS double tetrahedral or similar shapes yet we know this isn't true based on properties. Even in my own work it appears there are projections from higher to lower dimensionality going on. Very very strange!

 

Jess Tauber

[tetrahedron]

I've been consulting the OEIS database of sequences. Turns out many of those I've found aren't in it, so I'm going to contribute to the system. Yay. Odd, though, given that they are all ultimately based on doubled triangular numbers. Hmmm...

 

I was right- looking at SINGLE triangular number combinations mostly did the trick. Arrggghh...

 

 

Jess Tauber

Edited May 9, 2013 by pascal

[Rade]

pascal...I made edit changes to my previous post that were required when I applied P and N nucleons to the triangular dots. Of interest is the fact that for the triangular geometry to form stable isotopes two selection rules must be applied (1) dot in line 1 can be either a proton P or neutron N depending on the stable isotope under investigation, and (2) line 3 dots can be either [PNP] or [NPN] clusters. Note the free N=neutron and free [NPN]=tritium are unstable and decay into stable P and [PNP]=helium-3, respectively. So, it appears that using geometry of nucleon clusters to form stable isotopes depends on a very simple selection rule: 3 [NP] = 1 [NPN] + 1 [PNP].

[tetrahedron]

NPN and PNP remind me a lot of the way quarks team up to form baryons: UDU for protons, and DUD for neutrons. Nature repeating the same motif at two different scales.

 

Why might this be important? Because not only is there Pascal-based math in the magic numbers of nuclei for sets of protons and neutrons, there also appears to be linkage between both groups that may be based on more involved, but similar mathematics (for instance that concerning the Metallic Means).

 

With the Golden Mean we have one copy of each generalized Fib number (so Fib, Lucas, or any other like this) adding to one copy of the next number to generate the third in the series. With the Silver mean TWO copies of each generalized Pell number are added to ONE copy of the preceeding. 1.0000... is not usually considered to be a Metallic Mean, but is part of the system if one takes ZERO copies of the second number to add to the first, starting from 1 (though any integer could be used and it will never change), so ultimately we are using the Natural numbers.

 

The N/P ratio in real nuclei appears to largely follow graphically a pattern that incrementally moves from Metallic Mean to Metallic Mean (so 1.0000... for the first elements in stable nuclei, then through Phi, etc. Unstable nuclei seem to jump the gun on the trend, on the one hand, or lag behind, on the other. Whether the Silver Mean will ever be achieved as an N/P ratio is a big question- natural supernova processes don't seem to provide the requisite neutron flux, at least not usually. Perhaps only here would a true 'island of stability' be found (the one they are looking for now just being a southern shoal).

 

Some professional nuclear researchers have found what looks like a link between neutron and proton magics beyond just coincidence of having both- that is if one of the two is magic then it seems to add to the magicity of the other even if the other isn't magic.

 

And, finally, I experimented with counting up and down quarks instead of just neutrons and protons to see if there were any mathematical patterns- there WERE, though I haven't yet explored all this thoroughly enough to make particular claims.

 

It all comes down to layered numerical patterns that somehow dovetail into each other, and all using the same sorts of mathematical motifs (even if not identically hierarchically ranked in terms of the primitives). The whole thing is a coherent system, with subsystems linked to each other even if they have some degree of independence. Brains work the same way, from a systems perspective!

 

Jess Tauber