A fundamental discovery

[Rene]

I seem to have uncovered a structural duality in the periodic system that appears to be mathematically intrinsic, not just a quirk of how we arrange the elements.

So, writing the positive integers in a spiral arrangement on a square lattice yields:

65-64-63-62-61-60-59-58-57

 |                       |          

66 37-36-35-34-33-32-31 56

 |  |                 |  |

67 38 17-16-15-14-13 30 55

 |  |  |           |  |  |

68 39 18  5--4--3 12 29 54

 |  |  |  |     |  |  |  |

69 40 19  6  1--2 11 28 53

 |  |  |  |        |  |  |

70 41 20  7–-8–-9-10 27 52 etc 

 |  |  |              |  |  |

71 42 21-22-23-24-25-26 51 84

 |  |                    |  |

72 43-44-45-46-47-48-49-50 83

 |                          |

73-74-75-76-77-78-79-80-81-82

Note that the lengths of the double periods in the LST (4-16-36-64) run in a NW direction, and that the lengths of the double periods in the conventional PT (CPT) (10-26-50-82) run in a SE direction.

The LST and CPT are often viewed as competing forms, but the spiral arrangement suggests that they may actually be two halves of a single whole:

• the NW (LST) direction follows quantum principles first—grounding periodicity in the fundamental (n + l) rule;
• the SE (CPT) direction follows chemical periodicity, prioritising group similarities and valence behavior;
• these two competing principles are actually opposing-yet-complementary.

I had previously drawn such a table but had never until now realised there was a mathematical basis to it:

https://www.meta-synthesis.com/webbook/35_pt/pt_database.php?PT_id=1252

This duality reminds me of:

• the Taoist yin-yang symbol: two forces in constant motion, balancing each other; and
• wave-particle duality: the electron’s nature shifts between orbital structure (LST) and empirical behavior (CPT).

René

[Jess Tauber]

This reminds me a little of my 'shell stack' arrangement of spin-orbit split orbital partials from the atomic nucleus. That is, if you arrange orbital partials HORIZONTALLY for SPHERICAL shells (for example 1s1/2 over 1p3/2, 1p1/2 over 1d5/2, 1d3/2, 2s1/2, over 1f7/2, 1f/52, 2p3/2, 1p1/2 etc., always keeping same-spin values aligned vertically in columns, then simply by taking straight-line cuts through this matrix gives you constituents of NON-spherical shells. Organized as above, positive angles of cuts to the horizontal give the constituencies of prolate nuclear shells, and negative angles to the horizontal give the constituencies of oblate nuclear shells. The only caveat has to be that  the horizontal spacing of orbital partials in the horizontal rows has to be varied in order to align constituents between shells, but this adjustment is entirely regular and rule-based mathematically/graphically.

Jess Tauber

--
You received this message because you are subscribed to the Google Groups "Periodic table mailing list" group.
To unsubscribe from this group and stop receiving emails from it, send an email to PT-L+uns...@googlegroups.com.
To view this discussion visit https://groups.google.com/d/msgid/PT-L/9D72C896-80B9-4ED7-A03D-330E77F3E86C%40iinet.net.au.

[Rene]

Dear Jess, John, Julio, and colleagues

As a follow-up, below are some additional observations on the mathematical basis for both the left-step periodic table and the conventional periodic table.

I hope these observations clarify my reasoning and provide a fresh perspective on periodicity. I look forward to your thoughts.

René

The square spiral has form and structure

In 1963, Stanisław Ulam used the square spiral to visualise the distribution of prime numbers. The result was a striking pattern of prominent diagonal, horizontal, and vertical lines, each containing a high density of primes. The appearance of such lines is not unexpected, as lines in the spiral correspond to quadratic polynomials. Certain polynomials, such as Euler's prime-generating polynomial x^2−x+41, are known to produce a high concentration of prime numbers. See https://en.wikipedia.org/wiki/Ulam_spiral for more details.

My point is that the square spiral is not arbitrary; it has a strong mathematical foundation.

Refining the continuum of periodic tables

Do you recall Eric Scerri’s continuum of periodic tables? At the Platonic end of his continuum was the left-step (Janet) periodic table (32-columns wide), which I would call the "physics end" of the continuum. At the other end was Rayner-Canham’s "unruly" Inorganic Chemist’s Periodic Table, which I would refer to as the "chemistry end". The conventional periodic table occupied the middle of the continuum.

A reinterpretation based on the square spiral

I instead suggest the following:

• The Platonic/physics end of Eric’s continuum is essentially correct.
• The other (chemistry) end of the continuum should be the conventional periodic table rather than Rayner-Canham’s table.

This dichotomy aligns with the NW and SE sequences of the square spiral, forming a yin-yang-like physics-chemistry opposition or complementarity.

At the physics end, the left-step periodic table is "fixed", dictated purely by quantum mechanics. However, at the chemistry end, there is no single conventional periodic table—rather, there exists a family of 18-column tables, all sharing the same double period length structure (10, 26, 50, 82) but differing in details such as the placement of H, He, and the location and composition of Group 3. This variability is consistent with the more fuzzy nature of chemistry.

Rayner-Canham’s Inorganic Chemist’s Periodic Table then becomes a variant of the conventional periodic table. Other such variations include:

• An 18-column table with H over F.
• The IUPAC table with its 15-wide "f-block".

An Sc-Y-Lu-Lr table would move one step towards the physics end of the continuum, as would an 18-column table with He over Be.

Meanwhile, Julio, your binodic periodic table becomes a variant of the left step table.

Positioned somewhere near the middle of the continuum is the step pyramid periodic table of Jensen (1986), here https://www.meta-synthesis.com/webbook/35_pt/pt_database.php?PT_id=1039. It attempts to integrate the electronic structure focus of the left-step table with more of the chemical periodicity emphasised in conventional 18-column tables. Unlike conventional tables, which footnote the f-block, it retains the full 32-column width, ensuring that all electron subshells are represented in their natural sequence. At the same time, it preserves the double-period structure (10, 26, 50, 82), maintaining continuity with chemical periodicity trends.

One of its insightful features is its handling of hydrogen and helium, which are both shown as s-block elements. Hydrogen has a primary relationship to lithium while also preserving a secondary relationship to fluorine—an elegant resolution to the long-standing debate over its placement. In a similar fashion, helium has a primary relationship to the noble gases, while also exhibiting an electronic relationship to beryllium, reflecting its filled 1s² configuration. This hybrid approach demonstrates how the step pyramid periodic table seeks to blend theoretical electronic structure with observed chemical periodicity.

Conclusion

These observations suggest that the structure of the periodic system may be more deeply rooted in mathematics than previously thought. The square spiral provides a natural basis for understanding the relationship between different table forms, uniting physics and chemistry in a way that has not been previously recognised.

Jensen, W. B. (1986). Classification, symmetry and the periodic table. Computers & Mathematics with Applications, 12(1-2), 487–510. doi:10.1016/0898-1221(86)90167-7

[johnmarks9]

Dear René et al.,

I like your physics-chemistry spectrum, from Janet to Rayner-Canham. Perhaps PTs should be accorded a Janet-Canham index on this dimension?

Where would you place Mendeleyev revisited (Leach no. 1279)? Having just 14 groups, I presume it would lie between IUPAC´s conventional PT and Rayner-Canham´s.

My problem with 18-group PTs is their elevation of the A subgroups (the transition metals) to the same level as main groups while the B subgroups are relegated to a footnote. I think this is largely for historical reasons and their electric and magnetic properties, along with their rapidly increasing use, warrant their treatment more equally, at least with the A subgroups. MR (1279) also illustrates Brauner´s internal periodicity rather well.

Regards,

John

Ref: 

Brauner B “Über die Stellung der Elemente der seltenen Erden im periodischen System” Z. Elektrochem. 1908, 14: 525-527

[Julio gutierrez samanez]

Rene.

Great your discovery of the square spiral and your conclusion that there are two proposals in opposition and complementarity, a dialectical opposition, between LSPT and the standard table. In the Andean world there is something similar that we call “Yanantin” it is like the mythical couple of man and woman. They are not equal, they are opposites and they complement each other to form a dual unit and generate children. As in the even periods, one is smaller than the other, despite having the same size or number of elements. The second period covers the other.

In my 3D spiral, this exact mathematics of nature is clearly seen. A simple law 2x^2, gives the pattern of the radii or divisions of each pair of periods or (binode, dyad). That is, it gives the “structure” or background (Background): 2, 8, 18, 32…. (spaces or radii) for each pair of periods, one smaller or, so to speak, “female” and another larger “male” (in Andean terms). The binode spaces are double circular crowns, each divided by 2, 8, 18, 32 radii. On this background divided by increasingly numerous radii, the spiral passes, with two volutes per binode, placing the elements according to the Binodic law: 4x^2, that is: 4 in the first pair of periods, 16 in the second, 36 in the third and 64 in the fourth, etc.

Since the spirals add up, in the same way as your square spirals, then the sequence is 4(sum x^2) = 0, 4, 20, 56, 120, 220… , incredibly that function had been Z the series of atomic numbers, divided proportionally, by binodes, or pairs of periods, fundamental and primary reason for the “periodicity”.

I am pleased to know that we have arrived at the same goal, by different paths. That is the mathematics that has been sought since Mendeleev, Janet, Baca Mendoza, etc. and, furthermore, in that function or equation we can substitute x by n (principal quantum number) it can also be substituted by B, (number of the binode or pair of periods) without the function varying.

It is very pleasant to contribute a grain of sand to this formidable collective construction of science. I believe that the puzzle will be completed with the fifth quantum number, which will identify each of the complementary even periods, since these pairs of periods have the same four quantum numbers.

Greetings to all

Julio

Enviado con Gmail Mobile

--
You received this message because you are subscribed to the Google Groups "Periodic table mailing list" group.
To unsubscribe from this group and stop receiving emails from it, send an email to PT-L+uns...@googlegroups.com.

To view this discussion visit https://groups.google.com/d/msgid/PT-L/e4161066-53d9-42d2-8ee9-62baaa2be67dn%40googlegroups.com.

[Rene]

Thank you Julio

It was good to be able to accomodate the binodic table since it was your focus on period doubling that prompted me to look closer at the lengths of the double periods.

I do not see the need for a fifth quantum number, since the number of a binode is given by (n(lowest) + n(highest))/3.

Thus, in the first binode, the lowest n = 1 and the highest = 2 = 3/3 = 1.

In the second binode, lowest n = 2 and highest n = 4 = 2 + 4 = 6/3 = 2, and so on.

I had not known anything about Andean Yanantin but I see it is often referred to as a "dualism of complementary terms" or, simply, a “complementary dualism", and that it is much like Chinese Taoism.

Very appropriate.

René

[Rene]

I see now that the formula for the lengths of the double periods in the LSPT can be given as 4x^2, while for the CPT, it can be expressed as 4x^2 + 4x + 2. The two formulae feature three sets of 4 and 2 between them—a structural resonance that seems remarkable.

If the finding has legs—as it seems to—it would ostensibly represent a new perspective on the periodic system that ties mathematics, physics, and chemistry more closely together.

René

PS The 4, 2, 4, 2, 4, 2 pattern was a surprise to me when I stumbled upon it. I suspect Douglas Adams, who wrote The Hitchhiker's Guide to the Galaxy, in which 42 was the "Answer to the Ultimate Question of Life, the Universe, and Everything" would be amused.

[Rene]

Thanks John.

Bear in mind that Eric Scerri was the originator of the periodic table continuum, from Janet (left) to Rayner-Canham (right), with the CPT somewhere near the middle. The notion of an index for PT’s in the context of the continuum sounds interesting but how to develop such a thing is not something that immediately comes to my mind.

In considering Leach #1279, I note that the continuum Mk 2 has the following navigation points:

Janet table..?..Jensen Pyramid table..?..CPT

32-col          2 to 32-col              18-col

The question marks mean that Jensen’s table is somewhere near the middle. Looking again at his table I see that its double period lengths are the same as those of the CPT.

Now #1279 reduces the columns to 14. The double period lengths are 16-18-18-22-18 i.e. there does not seem to be a pattern to them.

It might best be thought of as an offshoot rather than a direct participant in the continuum?

I suggest the treatment of the B subgroups occurs for pragmatic (space-saving) reasons.

René

--
You received this message because you are subscribed to the Google Groups "Periodic table mailing list" group.
To unsubscribe from this group and stop receiving emails from it, send an email to PT-L+uns...@googlegroups.com.

To view this discussion visit https://groups.google.com/d/msgid/PT-L/e4161066-53d9-42d2-8ee9-62baaa2be67dn%40googlegroups.com.

[Julio gutierrez samanez]

Well René:
Of course, there is a “close relationship between mathematics, physics and chemistry.” And there is “viability and a new perspective.”
Regarding the explanation of “lowest n” and “highest n” and the division by 3, really, it seems like numerology to me, there is nothing rational about it, and I don’t think anyone would “digest well” this knowledge that does not come from Quantum Mechanics. I think it is, rather, a numerical juggling act. Also, it is not understood why it should be divided by three.
And, where does the “pattern” 4, 2, 4, 2, 4, 2… come from? In the binodes the “pattern” changes according to 2x^2. = 2, 8, 18, 32…, which is doubled to make the pair or binode and define the number of elements: 2, 2; 8, 8; 18. 18; 32, 32… and this duplication is 4, 16, 36, 64… and its sum is: Z = 4, 20, 56, 120… where: Z and n are related to the binodes and not to simple periods. Through a second degree parabolic curve.
And this relationship Z= f(n), was not known before, it does not exist in the literature on the Periodic Table, nor in the texts. There is no “authority” to rely on. But, it provides “viability and a new perspective”.
Greetings.
Julio

--

You received this message because you are subscribed to the Google Groups "Periodic table mailing list" group.
To unsubscribe from this group and stop receiving emails from it, send an email to PT-L+uns...@googlegroups.com.

To view this discussion visit https://groups.google.com/d/msgid/PT-L/97582ED0-EF0E-4400-9BAD-C27F7364CC7F%40iinet.net.au.

[Rene]

Thanks Julio. Your points are very interesting.

1. Regarding the "close relationship between mathematics, physics and chemistry" there has never been (until now), as far as I know, such a relationship between the left step periodic table (or binodic table), and the conventional periodic table. This relationship certainly is a new perspective.

2. Regarding the explanation of "lowest n" and "highest n" and the division by three, there is no numerology, only some maths that establishes a relationship between two values of n. There is no need for a 5th quantum number when an existing quantum number already works to establish the row number. Alternatively, the number of each double period is given by the value of n at the start of the double period:

1122

2222223333333344

333333333344444455444444444455555566

4444444444444455555555556666667755555555555555666666666677777788

As another alternative, the number of each double period is given by the number of block types it traverses.

3. The pattern "4, 2, 4, 2, 4, 2" comes from the quadratic formulae for the lengths of the double periods in the LST or binodic table (4x^2) and the CPT (4x^2 + 4x + 2). The 4’s and 2’s are an outcome of the way physical laws happen to work in our universe, in the same way that a 2 occurs in E = mc^2. These numbers are not metaphysically inevitable—they simply describe the reality we find ourselves in. In an alternate universe with different fundamental principles, the corresponding equations could look entirely different.

4a. Yes, in the left step table or binodic table the lengths of each pair of periods or binode is 4, 16, 36, 64 = 120, given by the function 4x^2. The sums involved are:

4 = 4 = Be

4+16 = 20 = Ca

4+16+36 = 56 = Ba

4+16+36+64 = 120 = Ubn

These elements are the alkaline earths occurring at the end of each double period or binode. The Z's are generated by the formula (4/3)d^3 + 2d^2 + (2/3)d, where d is the double period or binode number.

4b. In the conventional periodic table the lengths of each pair of periods is 10, 26, 50, 82 given by the formula 4x^2 + 4x + 2. The sums involved are:

10 = 10 = Ne

10+26 = 36 = Kr

10+26+50 = 86 = Rn

10+26+50+82 = 168 = Uho

These are the group 18 elements occurring at the end of each double period. The Z's are generated by the formula (4/3)d^3 + 4d^2 + (14/3)d, where d is the double period number.

5. The relationship Z = f(n) was indirectly alluded to by Bent (2006, p. 5) in his discussion of how to generate the LSPT from the Z’s of the alkaline earths:

"Write down the integers 1 2 3 4, each one twice; (II) square them; (III) double the squares; and add each double to the previous sum, [underline added] starting with the first 2.

(I)     1   1   2  2   3   3   4   4

(II)    1   1   4  4   9   9  16  16

(III)   2   2   8  8  18  18  32  32

(IV)  0   2   4  12 20  38  56  88  120

Bent H 2006, New Ideas in Chemisty from Fresh Energy for the Periodic Law, AuthorHouse, Bloomington, IN

René

[Jess Tauber]

As a reminder of the link between the electronic and nuclear shells, just note that in the latter system we have not pairs of two periods of the same length, but instead each shell has its own unique length. BUT, in common with the electronic LST nuclear shells come in pairs sharing the SAME NUMBER OF ORBITALS (1,1), (2,2), (3,3), (4,4). And while the electronic system revolves around tetrahedral numbers (every other atomic number of s2 LST period terminals), the nuclear system shells (in the simple harmonic oscillator model which does not take spin-orbit coupling into account) has shell lengths all DOUBLED triangular numbers 2, 6, 12, 20, 30, 42, 56, 72.... so that the harmonic oscillator magic numbers are all doubled TETRAHEDRAL numbers, 2, 8, 20, 40, 70, 112, etc.

Any suggestions as to how we can express all this mathematically?

Jess Tauber,

tetrahed...@earthlink.net

--
You received this message because you are subscribed to the Google Groups "Periodic table mailing list" group.
To unsubscribe from this group and stop receiving emails from it, send an email to PT-L+uns...@googlegroups.com.

To view this discussion visit https://groups.google.com/d/msgid/PT-L/E7DA8F48-DFB4-4D8A-B454-4DBCE959AB4A%40iinet.net.au.

[Julio gutierrez samanez]

Rene.

What a great thing Bent did, a visionary. All that was missing was for him to add the even numbers in (III)

To obtain the squares of the even integers.

(III) 2 2. 8. 8 18. 18. 32. 32. 50. 50

(Ivb). 4. 16. 36. 64. 100

And if he then added the terms successively, he would have:

(V). 0. 4. 20. 56. 120. 220…. Z= 4 [Sum n^2]

Where, Z is the last element of each binode. But in the graph, all Z is defined in terms of n, (or x or B).

I think this is my main contribution. Because it mathematically defines or shows the geometry of periodicity: it mathematically defines the Periodic Law.

 

This is complementary to your proposal for the standard table. For the Orientals it would be the yin yang and for the Andeans the yanantin or the mythical couple. (Male and female, complementary opposites). To define the inner difference of these dualities it would be the fifth quantum number. Remember that Pauli got out of the dilemma by “duplicating” with the “spin” the three previous quantum numbers and achieved “the closure of the periods”: 2n^2= 2, 8, 18, 32, 50…. From there (1924) there was not a step forward. A new “doubling” with “even periods” produces 4n^2= 4, 16, 36, 64, 100…

And their sum will be: 4, 20, 56, 120, 220…

That fifth quantum number is needed, just as Pauli needed to invent, create or devise the “spin” (angular momentum) which was nothing other than doubling what was already there until that moment (1924).

If the periods are even, it is because they have the four equal quantum numbers, only a fifth quantum would save the exclusion principle.

This is clear from the standard table itself:

The third period should have 2xs, 10xd, 6xp = 2 + 10 + 6= 18 elements. But it only has 8 elements, like the second. What happened there?

 

The fourth level should have, according to quantum mechanics, 32 elements, but it has 18.

The fifth level should have 50 elements, but it has 18, like the previous one.

The sixth level should have 72 elements, but it has 32

The seventh level should have 98 elements and it has 32.

 

For this reason, quantum mechanics cannot explain the periodic table, it could not.

But, the periodic table is showing the solution. The fruit of experimental or empirical activity, “shows” the “theory” how it should be interpreted, above conventional nomenclatures.

Is there someone, like Bent, who could have anticipated and observed this phenomenon? Maybe Rydberg, or Janet, Mazurs…

Julio

 

Rene.

Qué bueno lo de Bent, todo un visionario. Sólo faltó que sume los pares en (III)

Para obtener los cuadrados de los números enteros pares.

(III)  2  2. 8. 8  18.  18.  32. 32. 50.  50

(Ivb).  4.     16.     36.       64.       100

Y si, luego,  sumaba los términos , sucesivamente, tendría:

(V). 0.   4.   20.      56.       120.       220….Z= 4 [Sum n^2]

Donde, Z es el ültimo elemento de cada bínodo.Pero en el gráfico se define todo Z en función de n, ( o x o B).

Creo que este es mi aporte principal. Porque define matemáticamente o muestra la geometría de la periodicidad: define matemáticamente la Ley periódica.  

 

Esto se complementa con tu propuesta para la tabla estándar. Para los orientales sería el  yin yang y para los andinos el yanantin o la pareja mítica. (Macho y hembra, opuestos complementarios). Para definir la diferencia interior de estas dualidades sería el quinto número cuántico. Acuérdate que Pauli, salió del dilema “duplicando” con el “spin” los tres números cuánticos anteriores y consiguió “el cierre de los periodos”: 2n^2= 2, 8, 18, 32, 50…. De allí (1924) no se dio un paso adelante.

Una nueva “duplicación” con “periodos pares” produce 4n^2= 4, 16, 36, 64,  100…

Y su suma será:  4, 20, 56, 120, 220…

Hace falta ese quinto número cuàntico., como hacía falta que Pauli invente, cree o idee el “spin” (momento angular) que no era otra cosa que duplicar lo que ya se tenía hasta ese momento (1924).

 Si los periodos son pares, es que tienen los cuatro números cuánticos iguales, sólo un quinto cuántico salvaría  del principio de exclusión.

Esto se desprende de la propia tabla estándar:

 El tercer periodo debía tener  2xs, 10xd, 6xp = 2 + 10 + 6= 18 elementos. Pero sólo tiene 8 elementos , como el segundo. ¿Qué pasó allí?

 

El cuarto nivel debía tener, según la Mecánica cuántica, 32 elementos, pero tiene 18.

El quinto nivel debía tener 50 elementos, pero tiene 18, como el anterior.

El sexto nivel debía tener 72 elementos, pero tiene 32

El séptimo nivel, debía tener  98  elementos y tiene 32.

 

Por esta razón la mecánica cuántica no puede explicar la tabla periódica, no lo podría.

Pero, la tabla periódica está mostrando la solución. El fruto de la actividad experimental o empírica, le “muestra” a la “teoría” cómo debe interpretarse, por encima de las nomenclaturas convencionales.

¿Habrá alguien, como Bent, que pudo anticiparse y observar este fenómeno? Quizá Rydberg, o Janet, Mazurs…

Julio

 

Enviado con Gmail Mobile

[Jess Tauber]

In Pascal Triangle math, terms sum pairwise horizontally to produce the next term midway below each nearest-neighbor pair, without exception. Each edge-parallel diagonal is dimensionally one unit higher than the previous one closer to the edge. Thus the outer edge terms are all 1- they never change, and can thus be considered 0D. The natural numbers one diagonal further in are 1D, since they increase monotonically/linearly by 1 with each lower term. The triangular numbers are 2D, relating to the close-packing of circles in a plane. The tetrahedral numbers are 3D, as they relate to the close packing of spheres in a space/volume, and so forth. In Pascal math, the sums of natural numbers in sequence give triangular numbers- 1, 1+2=3, 3+3=6, 6+4=10, 10+5=15, 15+6=21, 21+7=28, 38+8=36, 36+9=45, 45+10=55, and so forth. Then the running sum of sequential odd numbers gives the square numbers- 1, 1+3=4, 4+5=9, 9+7=16, 16+9=25, 25+11=36, 36+13=49, and so on. And the number of spin-opposed electron PAIRS in orbitals is of course equal to s=1, p=3, d=5, f=7, etc., that is, odd numbers. Similar math in the simple harmonic oscillator model of atomic nuclear, when such nuclei are spherical and thus maximally symmetrical, except in the case of nuclei shells all have to be of the same parity,, so every OTHER orbital (thus HALF the total available). And yet they sum to DOUBLED tetrahedral numbers in sequence- 2, 8, 20, 40, 72, 112... I've feld for a while now that this doubling of half the possibilities is some sort of compensatory effect- but what could possibly motivate it? We also have the fact that while in the nuclear system nucleons pair up (in opposite spin) immediately as they are added to orbitals (not lobes as in the electronic system, but is there some sort of geometrical layout one could refer to??), in the ELECTRONIC system all electrons being added to orbital lobes singly must have the same spin orientation, and only pair up with opposite spin partners after all lobes have been filled singly. Lastly we have the behavior of nuclear 'intruder' levels, the highest spin spin-split orbital partials of the next-higher shell which energetically drop down into that of the previous shell, leading to the spin-orbit magic numbers.  In the electronic system, by contrast, the lowest spin component, s-electrons, add themselves to the next higher LST period to produce the traditional periods. It is known that spin-orbit coupling works in opposite directions between electronic and nuclear systems. What I'm hoping to see is a simple mathematical description that can deal with both systems simultaneously, as well as some hypothesis as to why this opposition exists (and is perhaps necessary for the maintenance of material reality as we know it).

Jess Tauber

--
You received this message because you are subscribed to the Google Groups "Periodic table mailing list" group.
To unsubscribe from this group and stop receiving emails from it, send an email to PT-L+uns...@googlegroups.com.

To view this discussion visit https://groups.google.com/d/msgid/PT-L/CAB_%2BVbCBDLHH645OXcbkjnWj%3DTASiL4hiQDgMsb_Qr7WuLZOTA%40mail.gmail.com.

[Rene]

Thanks Julio

Where you wrote...

"I think this is my main contribution. Because it mathematically defines or shows the geometry of periodicity: it mathematically defines the Periodic Law."

…please bear in mind that there are different ways to mathematically express the periodic law.

Hsueh & Chiang (1940) did so in their article "Periodic properties of elements",  published in the Journal of the Chinese Chemical Society, vol. 5, no. 5, pp. 253–257. They referred to the eight groups divided into A and B series.

Tomkeieff in 1951 published a formula for determining the length of each period in his article, "Length of the period of the periodic system", Nature, vol. 167, p. 854, https://www.nature.com/articles/167954b0. He was referring to a conventional 18-column table.

In the case of the left step periodic table (a) or the binodic form (b), the series is 4, 16, 36, 64 as you have noted for (b).

Fifth quantum number?

In quantum mechanics, each of the four quantum numbers corresponds to a measurable property of an electron:

1st. Principal quantum number (n). Defines the energy level (or shell) of the electron and its average distance from the nucleus. Energy levels can be observed in spectral lines through absorption and emission spectra, making n a physically measurable quantity.

2nd. Azimuthal (or angular momentum) quantum number (l). Determines the shape of the orbital and contributes to the electron's total angular momentum. Orbital shapes can be inferred from electron density distributions (e.g., through scanning tunneling microscopy in some cases), and angular momentum itself is a well-defined physical quantity.

3rd. Magnetic quantum number (ml​). Specifies the orientation of the orbital in space relative to an external magnetic field. The Zeeman effect (splitting of spectral lines in a magnetic field) directly reveals the influence of ml​, making it experimentally observable.

4th. Spin quantum number (ms​). Represents the intrinsic spin of the electron, which can be either +1/2​ or −1.2​. Spin is measured using techniques such as the Stern-Gerlach experiment or through spin-resonance methods e.g., electron spin resonance, nuclear magnetic resonance.

The 5th quantum number you have proposed does not correspond to a measurable property of an electron.

That does not matter since a 5th quantum number is not required.

So, I suggest that rather than saying..

"We describe the periodic law as an increasing function of the principal quantum number (n)" as you did in https://doi.org/10.1007/s10698-020-09359-3

...it would be entirely accurate to say:

"We describe the periodic law, as manifested in the Left-Step Table or binodic form, as an increasing function of the principal quantum number (n), where n increases at the start of each double period or binode."

This pattern can be seen in the values of the principal quantum numbers for the 120 elements in the binodic form, followed by the numbers of elements in each binode:

1122 [4]

2222223333333344 [16]

333333333344444455444444444455555566 [32]

4444444444444455555555556666667755555555555555666666666677777788 [64]

René

<IMG_4420.jpeg>

Enviado con Gmail Mobile

-- 
You received this message because you are subscribed to the Google Groups "Periodic table mailing list" group.
To unsubscribe from this group and stop receiving emails from it, send an email to PT-L+uns...@googlegroups.com.

To view this discussion visit https://groups.google.com/d/msgid/PT-L/CAB_%2BVbCBDLHH645OXcbkjnWj%3DTASiL4hiQDgMsb_Qr7WuLZOTA%40mail.gmail.com.

[Rene]

Dear Jess

There’s a lot to unpack here, but to be honest, the dense formatting makes it hard to follow. With respect, and in my humble opinion, you’re doing yourself no favours by posting a 454-word block of text without paragraph breaks—it would be much easier to engage with if it were structured into key points or sections.

I’m not guaranteeing any responses, but as it stands, your post appears (to my eye) simply too hard to process.

sincerely, René

[Julio gutierrez samanez]

I think that's fine, René.

In fact, it is a preliminary form to the "Periodic Law" to have found that Z, the Atomic Number, is a function of the principal quantum number n, and in passing to discover that this number n, works for two even periods, that is, it is also defined as the number of the binode.

The periodic law would be derived from these proportionally defined lengths as: 4, 16, 36, 64, 100... and then added together: 4, 20, 56, 120, 220... It is notable that, as you have shown in the diagram, the binode changes when another azimuthal appears that is added to the copies of the azimuthals of the previous binode, and then it is doubled to make the even period.

I believe that it is experimentally proven that these even periods exist, with all the methods used until today. If the table shows them, it is because they are found in reality in the atoms themselves.

If we superimpose two periods of the same number of components, how do they manage to not mix and stay separated, but together? They must have different polarity or movement. If they are layers with the same components (different periods, although the same size) there is no doubt that something electromagnetic separates, differentiates or particularizes them, and, surely, those potentials have already been measured; both, if it is about protons, as well as electrons. I can only speculate. I am not a laboratory scientist. The fact is that, as I showed you in my previous email, it is not possible to apply Quantum Mechanics to the periodic table, if it is not accepted that it must be in even periods, as shown by the Periodic Table (standard, LSPT or Binodic, which is not the same as LSPT). I repeat, the nomenclature can change.

The Periodic Law itself is achieved by making the chemical properties (colored by Bent, which you yourself shared with us) increasing functions of Z, (atomic number). which are spirals with increasing whorls in one of my graphs.

 

Well, anyway, I agree with your conclusion:

"it would be completely accurate to say:

 

"We describe the periodic law, as it is manifested in the Table of Left Steps or binode form, as an increasing function of the principal quantum number (n), where n increases at the beginning of each double period or binode."

 

This pattern can be observed in the values of the principal quantum numbers for the 120 elements in the binode form, followed by the numbers of elements in each binode."

 

I would only add that the "background or division pattern" is 2n² and the elements are distributed under the 4n² norm.

Regards

Julio

Me parece bien. René.

En realidad, es una forma preliminar a la "Ley Periódica" el haber encontrado que Z, el Número Atómico, es función del número cuántico principal n, y de paso descubrir que este número n, funciona para dos periodos pares, es decir se define, también, como el número del bínodo. 

La ley periódica derivaría de esas longitudes definidas proporcionalmente como: 4, 16, 36, 64, 100... y luego sumadas: 4, 20, 56, 120, 220,,,,Es notorio que, como lo has mostrado en el esquema, el bínodo cambia cuando aparece otro azimutal que se suma a las copias de los azimutales del bínodo anterior, y luego se duplica para hacer el periodo par. 

Creo que experimentalmente está probado que existen esos periodos pares, con todos los métodos utilizados hasta hoy. Si la tabla los muestra, es que así se encuentran en la realidad en los átomos mismos. 

Si sobreponemos dos periodos del mismo número de componentes. ¿Cómo hacen para no mezclarse y mantenerse separados, pero juntos? Deben tener polaridad o movimiento distinto. Si son capas con los mismos componentes. (Periodos distintos, aunque del mismo tamaño) no hay duda que algo electromagnético los separa, diferencia o particulariza, y, seguro que ya se tienen medidos esos potenciales; tanto, si se trata de protones, como de electrones. Yo solo puedo especular. No soy un científico de laboratorio. El hecho es que, como te lo mostré en mi correo anterior, no es posible aplicar la Mecánica cuántica, a la tabla periódica, sino se acepta que debe ser por periodos pares, tal y como nos muestra la Tabla periódica (estándar, LSPT o Binódica, que no es lo mismo que LSPT). Repito, la nomenclatura puede cambiar.

La Ley periódica propiamente dicha se consigue haciendo que las propiedades químicas (coloreadas por Bent, que tú mismo nos compartiste) sean funciones crecientes de Z, (número atómico). que son espirales con volutas crecientes en uno de mis gráficos.

Bueno, como sea, estoy de acuerdo con tu conclusión:

"sería completamente exacto decir:

 

"Describimos la ley periódica, tal como se manifiesta en la Tabla de Pasos a la Izquierda o forma binódica, como una función creciente del número cuántico principal (n), donde n aumenta al comienzo de cada período doble o binodo".

 

Este patrón se puede observar en los valores de los números cuánticos principales para los 120 elementos en la forma binódica, seguidos por los números de elementos en cada binodo".

Solo acotaría que el "patrón" de fondo o de división es 2 n² y los elementos se distribuyen bajo la norma 4n².

Saludos

Julio

[Rene]

Question: Does anyone disagree with the presumption that nature has selected the most elegant and efficient combination of laws, leading to a world that is both simplest in hypotheses and richest in phenomena?

The context for my question is that Weiss (2013) provides the number of elements in each row (n) of the conventional periodic table (CPT) as:

((−1)^n(2n + 3) + 2n^2 + 6n + 5)/4 = 2, 8, 8, 18, 18, 32, 32

He also gives the atomic numbers at the end of each row (i.e. the noble gases) as:

Z = ((−1)^n(3n + 6) + 2n^3 + 12n^2 + 25n − 6)/12 = 2, 10, 18, 36, 54, 86, 118

(See John D. Cook's blog: https://www.johndcook.com/blog/2022/06/08/infinite-periodic-table/)

If nature truly adheres to the principle of maximum impact with minimal sufficient complexity, then these two formulae should "cut the mustard" for CPT mathematics:

• Number of elements in each double period:
• 4n^2 + 4n + 2 = 10, 26, 50, 82 

• Atomic numbers at the end of each double period:
• Z = [Sum n^2] = 10, 36, 86, 168

I’d be interested in hearing any thoughts or counterarguments.

René

[Jess Tauber]

For me, since I have both the electronic and nuclear systems to compare, it seems (again, to me) that both the final systems are somewhat derived. If you start with the simple harmonic oscillator layout of the spherical atomic nucleus, you get something very similar in spirit to the LST in the electronic system. That is, each shell in the nucleus here i 'Left-Step, with higher-spin orbitals to the left of the lower spin ones. This then gets more complex when you add in spin-orbit effects, which insert the highest spin orbital partials into the (at least ENERGETIC) structure of the previous shell. I don't believe these 'intruder levels are actually a part of the previous shat all (despite their apparent effects jacking up magic numbers terminating these shells relative to the shell terminals of the harmonic oscillator sphere). This is because all intruders have opposite PARITY versus their new 'host' shells. Sorta like inviting a guy over to an all-female pajama party.

In the electronic system, of course, we face the opposite effect- s-electrons from the PREVIOUS shell being promoted to become part of the next higher period, thus giving the traditional PT structure. I asked in my last post whether there were some way to relate these opposite effects mathematically into a single rule (say with + versue - terms, for clarity). I would very much like to know how the different hybrid orbitals reconcile same-versus-opposite parity issues, since we see both. And does something similar ever happen in the nucleus, but we've missed it?

Jess Tauber

--
You received this message because you are subscribed to the Google Groups "Periodic table mailing list" group.
To unsubscribe from this group and stop receiving emails from it, send an email to PT-L+uns...@googlegroups.com.

To view this discussion visit https://groups.google.com/d/msgid/PT-L/2A672C56-962F-4995-A20D-AA517BDC55FD%40iinet.net.au.

[Julio Gutiérrez Samanez]

Hello René and colleagues: I am sending you this communication in PDF format, as it contains tables and graphs.
Regards
Julio

--
You received this message because you are subscribed to the Google Groups "Periodic table mailing list" group.
To unsubscribe from this group and stop receiving emails from it, send an email to PT-L+uns...@googlegroups.com.

To view this discussion visit https://groups.google.com/d/msgid/PT-L/2A672C56-962F-4995-A20D-AA517BDC55FD%40iinet.net.au.

[Julio Gutiérrez Samanez]

Sorry, Rene and colleagues, I sent an earlier PDF file. This is the last one I worked on to share with you.

Julio

[Rene]

Thank you Julio. The new pdf is better.

Where you wrote:

"However, the series: 10, 26, 50, 82, 122 … Does not describe the size of the double periods in the standard table. It should be: 2, 8, 8, 18, 18, 32, 32, 50, 50."

I am not sure what you are saying here. The double periods in the standard table have lengths of:

Period 1 + period 2 = 2 + 8 = 10

Period 3 + period 4 = 8 + 18 = 26
Period 5 + period 6 = 18 + 32 = 50

Period 7 + period 8 = 32 + 50 = 82

You’re right when you wrote, "I think you made an unintentional mistake Rene, in the second function, it should be like this…". Thanks for pointing out that error.

As you say, it is Z = {Sum (4n^2 +4n +2)} = 10, 36, 86, 168 

What you wrote here is not right:

"However, these values are not from the last elements of the double periods, but the last elements “of the first periods of the double periods”, that is, the intermediate ones. The last elements of the double periods have to be: 2He (which is unitary), 18Ar, 54Xe, 118 Og, 218… 

For example, the 2nd double period in the standard table runs from Na to Kr i.e. periods 3 and 4. Thus, Kr (36) is the last element of the double period. And the last element in the 1st double period (H to Ne) is Ne.

The lengths of individual periods is not the primary issue. Instead, the primary issue is the approximate periodicity of physicochemical properties occurring among the elements and how this can be optimally displayed in a convenient form. Both the LST and binodic form, and the standard table have mathematical underpinnings. I suggest the standard periodic table has the advantage in that it achieves this at the same time as showing a clear left to right trend in metallic to nonmetallic character.

René

To view this discussion visit https://groups.google.com/d/msgid/PT-L/CAA-xvT6ynAD6CNxr31p7E7oKJG1B2RaqTQHhZ5jSNjPQGo%3DJqg%40mail.gmail.com.
<Hi René and colleagues all.pdf>

[Jess Tauber]

Interestingly, 50 and 82 are nuclear magic numbers for spherical nuclei (the most symmetrical and so most stable) under the model which includes spin-orbit coupling, for proton number in tin and lead respectively. These elements have more stable isotopes than others, iirc. And 26 of course is iron, the heaviest element capable of being produced exothermically by normal stellar nucleosynthesis.  Once iron accumulates in the stellar core, the star only has moments to live, becoming a supernova that tears it to pieces, with the core collapsing either to a neutron star or black hole. Curiously, the iron isn't formed directly, but first becomes nickel, with a magic number of protons, which decays to iron.

To view this discussion visit https://groups.google.com/d/msgid/PT-L/0BF0493B-E3BA-429D-9598-DA98840B27AA%40iinet.net.au.

[Rene]

Dear Julio

It is worth recalling, in my view, that the PT is a classification and that the particulars of a classification depend on the criteria used to arrange or organise it. Scerri (2010, p. 70) refers to the PT as, “the supreme example of a scientific system of classification.” And later: “Recall that the periodic table…is not a theory, at least for the vast majority of authors.” (Scerri 2012, p. 283).

Each classification (e.g. LSPT or binodic form, or standard table) highlights different aspects of periodicity, showing that the PT is not a single fixed entity but a flexible representation shaped by the organising principles we apply.

The LSPT and binodic form are both regular in appearance, as informed by quantum mechanics. As you reported, the binodic form (which is equivalent to the LSPT) has an underlying mathematical basis in terms of the lengths of its double periods.

The standard form with its two turrets; the failure of its first period length to recur; and its gaps between the s-block and the p-block, has a less regular structure as informed by chemico-physical considerations. It nevertheless has an underlying mathematical basis in terms of the lengths of its double periods.

In turn it is rather surprising that the underlying mathematical bases of the two classifications are related via their complementary positions on the square spiral, i.e. the LSPT/binodic form to the northwest and the standard form to the southeast. Thus, a perfectly straight NW-to-SE line can be drawn through the lengths of the respective double periods: 64-36-16-4 and 10-26-50-82, as seen in the attached square spiral image.

The number 2 bridges the two sequences. It represents duality, balance, and opposition—like yin-yang in Taoism. In the periodic table, 2 represents helium (He), the first noble gas. Helium is the only element that fits two classifications:

• in the LST, He is placed over Be (following quantum mechanics);

• in the standard table, He is placed over Ne (following chemical periodicity).

In both the LST and standard table, the first period is always two elements long.

The 2 in the center of the sequence is thus significant mathematically, chemically, structurally, and conceptually (it reinforces the idea that the periodic system is built on an underlying balance of competing principles).

If I have not already said it I credit you with prompting me to consider the lengths of the double periods in the standard table.

René

Scerri E 2010, Explaining the periodic table, and the role of chemical triads, Found Chem. 12, 69–83

—— 2012, A critique of Weisberg’s view on the periodic table and some speculations on the nature of classifications, Found Chem, 14, 3, 275–284.

[Julio Gutiérrez Samanez]

Thanks René.

We have a slight difference regarding the concept of "double period". Baca Mendoza considered "double periods or binodes" those that had the same number of elements. In his system he excepted H, and considered the first period from 2 He, to 9 F, (8 elements). The second period, doubled from the previous one, was from 10 Ne to 17 Cl. Also 8 periods) making a first binode of 8, 8 elements). The second binode from 18 Ar to 35 Br (18 elements), and from 36 Kr to 53 I. (18 elements). The third binode of 32, 32 elements, etc.

The standard table has a unitary period, without a pair, of only two elements. Then come the even periods with equal numbers of elements: 3 Li to 10 Ne (8 elements) and 11 Na to 18 Ar (8 elements); 19 K to 36 Kr (18 elements) and 37 Rb to 54 Xe (also 18 elements); 55 Cs to 84 Rn and 87 Fr to 118 Og. (32, 32 elements).

Janet's table has complete pairs of periods (2, 2 elements); (8, 8 elements); (18, 18 elements) and (32, 32 elements), etc.

The binodic system has many double periods of: (4, 16, 36, 64, 100 elements) and these values mark the sizes of the binodes, exactly as in your square spiral (2n^2).

If we add the terms one by one successively, we will have the entire series of Atomic Numbers Z = (4, 20, 56, 120, 220 ...), segmented into proportional parts that justify the "periodicity" by binodes "based on n". (Principal Quantum Number). (4 Be, 20 Ca, 56 Ba, 120, 220 ...). Something important that emerges from this is that: every two periods a new azimuthal quantum number is generated, which forms a new binode, in increasing progression. This is what is confusing in the standard table; because, "so that the metallic elements are on the left and the non-metallic ones on the right" the alkaline metals and the alkaline earth metals are kept in two columns separate from the rest of the table. What is defended by chemists cannot be explained by Quantum Mechanics, because it is known that the sequence of appearance of the sub orbitals or azimuthals is not the anomalous form: s...f, d, p, but … f, d, p, s.

In the series:

2 + 8 = 10

8 + 18 = 26

18 + 32 =50

32 + 50 = 82.

There are no even or double periods but one small and one large (2, 8); (8, 18); (18, 32); (32, 50)… Apparently they are the “magic” numbers, they must have another important connotation that must be investigated.

I think that the “double periods” have to be, in addition, symmetrical or with the same number of elements, (2, 2), (8, 8), (18, 18), (32, 32)… Or as in the standard table: (2), (8, 8), (18, 18), (32, 32)…

There we can find, as a new alternative, that simple and beautiful mathematics that is being sought, without the need to discard the standard table so popular and so dear to chemist friends.

That is my opinion and I think that many of us understand it that way.

Greetings

Julio

[Rene]

Thank you Julio.

I agree we have a slight difference regarding the concept of what is a double period.

You think that each period in a double period should have the same number of elements.

In fact there is no fundamental requirement for this to be so.

In the standard table, the single period lengths follow the sequence:

(2, 8), (8, 18), (18, 32), (32, 50), etc.

…where each pair of single periods is grouped into a double period.

This pattern reflects how each new period (corresponding to a new value of n) is followed by a period that is either longer than the previous one or of the same length, following a regularly alternating sequence.

Your comment that "That is my opinion and I think that many of us understand it that way" seems to reflect cognitive dissonance. The idea that double periods must be of equal length is not a strict requirement of quantum mechanics or periodicity—it is a conceptual choice. The standard table follows an alternating pattern in period lengths (2, 8), (8, 18), (18, 32), etc., which does not strictly fit a "double period" framework in the way you describe. So if many people "understand it that way," I wonder if that is because they are holding onto a preference for symmetry rather than an underlying necessity. What do you think?"

René

[Julio Gutiérrez Samanez]

Dear Rene and colleagues.

Thank you for attributing me some merit in these conversations, coming from an enlightened and scholarly person that you, I receive which a medal.

The periodic table, you say: "It is not a unique fixed entity but a flexible representation." And it is a great truth, because it is something in constant construction, as is science. It may not be a theory, but several theories. At least, there are two opposite, but complementary theories.

Regarding symmetrical periods, it is true that "there is no fundamental requirement for that to be," it simply is how they appear; product of empirical practice or research: in both constructions there are double symmetrical periods, with the same azimutal and with the same number of elements.

And there are always two: in the binodic form, or, with a unit period and all the following pairs of the same number, in the standard form. And this even way is due to the appearance of new azimutals that determine the growth of double and symmetrical periods. Now I see that, too, double periods can be made, two by two, with the periods of the standard table. Good, because it also has its logic.

Yes, I am also surprised that his "square spiral" shows us, as taken by the hand, the two classification models, to all those who, like me, "cling to a preference for symmetry" and those who prefer to cling to "an underlying need."

At this point in our dialogue, it seems that nature itself paves the way to suggest that we are facing a dialectical duality of "complementary opposites", just like two faces of a coin. In one there is symmetry and in another no; And, for that difference, they complement each other in the provisions or locations that take the numbers in the square spiral.

As I wrote in a previous communication, there is no need to rule out the standard form. And, now, complete the idea saying that we could not rule out the symmetrical and binodic form. That we both have to understand how: the East Yin Yang, the Andean Yanatin or as the materialistic dialectic of the unity and opposition of contradictory principles. This is of course "reflects a cognitive dissonance", without forgetting that it is only ways to approach reality and represent it, knowing that reality will always be more complex and multiform.

It is true that science, the more mathematics it contains, it will be more rigorous and accurate. But, until now, there is no mathematics that describes and explains everything. What we have are barely any quantitative growth parcels, towards an asymptotic truth, in a certain way, unattainable.

For our friend Jess I send this numerical speculation:

… 100, 64, 36, 16, 4, 2, 10, 26, 50, 82, 122, 170 …

     … 36   28  20 12  2   8  16   24  32  40    48…

           8    8      8     8    6   8   8    8    8      8

            0      0      0    2     2   0    0    0    0

Subtracting 100 - 64 = 36 to 4 - 2 = 2

Then 10 - 2 = 8 to 170 - 122 = 48

36 - 28 = 8 up to 12 - 2 = 8

Then 8 - 2 = 6, 16 - 8 = 8, up to 48 - 40 = 8           etc.

Julio Gutierrez 

[Rene]

Julio wrote:

"This is what is confusing [bold added] in the standard table; because, "so that the metallic elements are on the left and the non-metallic ones on the right" the alkaline metals and the alkaline earth metals are kept in two columns separate from the rest of the table. What is defended by chemists cannot be explained by Quantum Mechanics, because it is known that the sequence of appearance of the sub orbitals or azimuthals is not the anomalous form: s...f, d, p, but … f, d, p, s."

Comments

The two sequences s…f, d, p and … f, d, p, s are the same. The only difference is where the "cuts" are made.

Since both versions contain the same sequence of orbital filling, the choice of where to make the cuts is arbitrary, depending on whether one prioritises chemical trends (standard table) or quantum mechanics (left-step table).

I can remember being introduced to the periodic table in high school, and no one found the gap between the alkali metals and alkaline earth metals (in periods 2 and 3) and the rest of the table to be confusing.

By contrast, Gary Katz, an advocate of the left step table, noted: "Chemical educators are known to complain that the left-step table is too confusing for students".

Consistent with these observations, Scerri (2021, p. 132) observed that the level at which a science operates is a question for its practitioners and the deepest most fundamental bases are not necessarily the best for all purposes.

René

* Katz G 2019, Left-step periodic table, in "Reactions", Chemical & Engineering News, vol. 97, no. 9, https://cen.acs.org/physical-chemistry/periodic-table/Reactions/97/i9

* Scerri ER 2021, The Periodic Table: Its Story and Its Significance, 2d ed., Oxford University Press, New York

Postscript

In the attached 1993 article, the authors say there are four kinds of table in which Z increases when moving from L to R.

They say there is no "most appropriate" table, and recommend the following:

Physicists---an fdps Madelung style table.

Chemists---an sfdp familiar style table.

Crystallographers---dpfs or psfd.

They conclude by saying that the four tables complement each other, and that all of them can be applied equally well to describe and explain the nature of the chemical elements.

[Julio Gutiérrez Samanez]

THIS VERSION IS CORRECTED

… 100, 64, 36, 16, 4, (2),  10,  26,  50, 82, 122, 170 …

     … 36  28  20   12   |    8    16   24   32   40    48…

            8    8      8       |         8    8    8     8      8

              0      0            |             0   0    0    0

Subtracting 100 - 64 = 36 to 16 - 4 = 8

Then 10 - 2 = 8 to 170 - 122 = 48

36 - 28 = 8 up to 20 - 12 = 8

Then 16 - 8 = 8, to 48 - 40 = 8

8 – 8 = 0

[Rene]

Julio

In this sequence...

… 100, 64, 36, 16, 4, 2, 10, 26, 50, 82, 122, 170 …

     … 36   28  20 12  2   8  16   24  32  40    48…

           8    8      8     8    6   8   8    8    8      8

            0      0      0    2     2   0    0    0    0

…I believe that line 3 should read 8  8  8 10  6  8  8  8  8  8

René

[Rene]

Thank you Julio

I believe the correct version, with eight periods on each side, is:

64  36  16  4 (2) 10  26  50  82

  28  20  12  ...   16  24  32

    08  08    ...     08  08

      00                00       

 

Quite pretty! I hope I have this right now!

René

[Larry T.]

I think it wouldn't be too confusing to explain to students that there are four distinct types of metals:

alkalis and alkaline metals, p-block metals, transition metals and lanthanides/actinides.

 It is just too simplistic to lump them all together and that is what traditional PT does.

--
You received this message because you are subscribed to the Google Groups "Periodic table mailing list" group.
To unsubscribe from this group and stop receiving emails from it, send an email to PT-L+uns...@googlegroups.com.

To view this discussion visit https://groups.google.com/d/msgid/PT-L/E7B6D2BC-6107-41FA-8AD6-9E2D180DC4F7%40iinet.net.au.

--
You received this message because you are subscribed to the Google Groups "Periodic table mailing list" group.
To unsubscribe from this group and stop receiving emails from it, send an email to PT-L+uns...@googlegroups.com.

To view this discussion visit https://groups.google.com/d/msgid/PT-L/E7B6D2BC-6107-41FA-8AD6-9E2D180DC4F7%40iinet.net.au.

[Rene]

On 10 Mar 2025, at 00:15, Larry T. <ora...@gmail.com> wrote:

I think it wouldn't be too confusing to explain to students that there are four distinct types of metals:

alkalis and alkaline metals, p-block metals, transition metals and lanthanides/actinides.

 It is just too simplistic to lump them all together and that is what traditional PT does.

Hi Larry

I partly agree with you about four types of metals, and disagree with you that the traditional table "lumps" them all together.

Four distinct types of metals?

I tend to agree with you about explaining four distinct types of metals.

Traditional teaching practice is to contrast the alkalis (and sometimes the alkaline metals too) with the halogens. What then happens is that the contrast between types of metals and nonmetals can’t be continued since there don’t seem to be nonmetal analogues for the Ln/An, transition metals, and p-block metals. Add in the notion of metalloids as an "in-between" type and confusion reigns.

Confusion arises due to placing too much emphasis on the electronic basis of the metals, at the expense of their chemistry and reactivity. Focusing on the latter properties enables everything to come together:

Reactivity  Metal types      Nonmetal types        Reactivity

    ^     sf-metals*      Halogen nonmetals         ^

    |     d-metals (most)  H, C, N, O, P, S, Se      |

    |     p-metals      metalloid nonmetals       |

    |     noble metals     noble gases               |

   * Please see my note about lumping s- and f-metals together, after my sign off

Of course, the electronic basis of the PT is important for understanding its underlying structure but it does not do so well for the purposes of grasping the 4 + 4 counterpart symmetry among the metals and nonmetals.

Lumping together of metal types in the traditional PT?

Where is the lumping together?

Here is such a table showing, from L to R, the s-block metals, the f-block metals, the d-block metals, and the p-block metals. A picture of clarity (with the exception of the usual needless confusion surrounding the metalloids),

The traditional table also preserves the widely taught and well-known horizontal, vertical and diagonal trends.

René

* sf-metals?

While bringing the s- and f-block metals under one umbrella may seem questionable the similarities between the two sets are well documented in the literature. Thus, the Ln are described as behaving largely as if they were trivalent versions of the +2 cations Ca2+, Mg2+ of group 2 (Jones 2017, p.1). More generally, King[i] associates the chemistry of the Ln in the +3 state with the chemistry of the alkali and alkaline metals. The actinides are similarly reactive[ii] noting the early members show a wider range of oxidation states. Further:

“Many significant trends are apparent in the structures of the halides and their physical and chemical properties…The majority of pre-transition metals (Groups 1, 2) together with Group 3, the Ln and the An in the +2 and +3 oxidation states form halides that are predominately ionic in character, whereas the non-metals and metals in high oxidation states (≥ +3) tend to form covalent molecular halides.”[iii]

         [i] King RB 1995, Inorganic Chemistry of Main Group Elements, VCH, New York, p. 289

         [ii] Parish RV 1977, The Metallic Elements, Longman, London, p. 163

         [iii] Greenwood NN & Earnshaw A 2002, Chemistry of the Elements, 2nd ed., Butterworth-Heinemann, Oxford, p. 823

[Jess Tauber]

Where the cuts are made conforms to my claim of a kind of "frame shift" in the linear sequence (z), and that this shift relates to the inverse effects of spin orbit coupling in the nuclear system. What I want to know is whether there is some known order of preference in electronic hybrid orbitals and how easily they participate in different types of bond. Sp bonds seem to be mostly involved in covalent bonding, while bonds involving d (f?) orbitals are far weaker.

--
You received this message because you are subscribed to the Google Groups "Periodic table mailing list" group.
To unsubscribe from this group and stop receiving emails from it, send an email to PT-L+uns...@googlegroups.com.

To view this discussion visit https://groups.google.com/d/msgid/PT-L/E7B6D2BC-6107-41FA-8AD6-9E2D180DC4F7%40iinet.net.au.

--
You received this message because you are subscribed to the Google Groups "Periodic table mailing list" group.
To unsubscribe from this group and stop receiving emails from it, send an email to PT-L+uns...@googlegroups.com.

To view this discussion visit https://groups.google.com/d/msgid/PT-L/E7B6D2BC-6107-41FA-8AD6-9E2D180DC4F7%40iinet.net.au.

[Larry T.]

How about making the cuts where the maximum n+l number changes its value? 

Actually n+l signifies none other than the Heisenberg uncertainty principle. Electrons with high "l" value are more localized but have greater velocities, while electrons with high "n" value are smeared more, but slower. So, n+l expression basically tells us that you can not measure an electron's position and velocity with the same precision. If it's more localized, it moves much faster. If it moves slower, its position is less definite. 

 Just a thought.

Best Regards,

V. "Larry" Tsimmerman

To view this discussion visit https://groups.google.com/d/msgid/PT-L/CAO6d3hK9tw-yUDzbqGp92SLY7A7UghkeGMBCTQuG9skGeCwtLw%40mail.gmail.com.

[Rene]

On 12 Mar 2025, at 03:09, Larry T. <ora...@gmail.com> wrote:

How about making the cuts where the maximum n+l number changes its value? 

Actually n+l signifies none other than the Heisenberg uncertainty principle. Electrons with high "l" value are more localized but have greater velocities, while electrons with high "n" value are smeared more, but slower. So, n+l expression basically tells us that you can not measure an electron's position and velocity with the same precision. If it's more localized, it moves much faster. If it moves slower, its position is less definite. 

 Just a thought.

Best Regards,

V. "Larry" Tsimmerman

Thanks Larry. That’s a fascinating observation you made about the Heisenberg uncertainty principle (HUP).

If the HUP is being referred to then presumably the n+l values would refer to the electron in the (idealised) highest occupied orbital rather than to the orbital occupied by the (idealised) differentiating electron?

The reason for this is that, as I understand it, the Heisenberg uncertainty principle is most relevant to the electrons that dominate the physical and chemical properties of an atom, which are typically found in the highest occupied orbital. Thus, the spread (delocalization) or localization of an electron wave function depends more on the highest occupied orbital because this orbital governs the atom’s size, ionization energy, and chemical reactivity. The n + l value of the highest occupied orbital then better reflects how an atom balances energy minimisation and spatial distribution for its electrons.

If the cuts were made where the maximum value of the n+l values for the highest occupied orbital change value, they would occur after the following elements

He

Be

Mg

Zn

Cd

Hg

Cn

I’ve attached the resulting psfd table, which includes the relevant n+l values.

The 1983 article I attached earlier in the "A fundamental discovery" thread recommends the psfd table for crystallographers.

Summarising:

• cuts made when n increases = conventional table = good for chemistry;
• cuts made when n+l for differentiating electron increases = left-step or ADOMAH table = good for more fundamental quantum structure; and
• cuts made when n+l for highest occupied orbital increases = crystallographers' table.

Who would have thought that the above psfd table has a quantum mechanical basis?

René

[Larry T.]

Since all electrons are identical and indistinguishable (per Gibbs) and can be found anywhere at any time, instead of looking at a particular orbital, or an electron, how about looking at all filled, or partially filled orbitals of an atom and identifying one with the highest n+l value and assigning that atom to a period of the same order? 

  Just a thought.

VT

[Rene]

Larry

I believe that if looking at all filled, or partially filled, orbitals of an atom and identifying the one with the highest n+l value and assigning that atom to a period of the same order, the result is a left step table.

I have not previously seen such a formulation for generating the LSTP.

Congratulations for your apparent discovery!

It would be good if someone else could confirm your finding.

René

On 13 Mar 2025, at 11:50, Larry T. <ora...@gmail.com> wrote:

Since all electrons are identical and indistinguishable (per Gibbs) and can be found anywhere at any time, instead of looking at a particular orbital, or an electron, how about looking at all filled, or partially filled orbitals of an atom and identifying one with the highest n+l value and assigning that atom to a period of the same order? 

  Just a thought.

VT

On Wed, Mar 12, 2025 at 8:17 PM Rene <re...@iinet.net.au> wrote:

On 12 Mar 2025, at 03:09, Larry T. <ora...@gmail.com> wrote:

How about making the cuts where the maximum n+l number changes its value? 

Actually n+l signifies none other than the Heisenberg uncertainty principle. Electrons with high "l" value are more localized but have greater velocities, while electrons with high "n" value are smeared more, but slower. So, n+l expression basically tells us that you can not measure an electron's position and velocity with the same precision. If it's more localized, it moves much faster. If it moves slower, its position is less definite. 

 Just a thought.

Best Regards,

V. "Larry" Tsimmerman

Thanks Larry. That’s a fascinating observation you made about the Heisenberg uncertainty principle (HUP).

If the HUP is being referred to then presumably the n+l values would refer to the electron in the (idealised) highest occupied orbital rather than to the orbital occupied by the (idealised) differentiating electron?

The reason for this is that, as I understand it, the Heisenberg uncertainty principle is most relevant to the electrons that dominate the physical and chemical properties of an atom, which are typically found in the highest occupied orbital. Thus, the spread (delocalization) or localization of an electron wave function depends more on the highest occupied orbital because this orbital governs the atom’s size, ionization energy, and chemical reactivity. The n + l value of the highest occupied orbital then better reflects how an atom balances energy minimisation and spatial distribution for its electrons.

If the cuts were made where the maximum value of the n+l values for the highest occupied orbital change value, they would occur after the following elements

He

Be

Mg

Zn

Cd

Hg

Cn

I’ve attached the resulting psfd table, which includes the relevant n+l values.

<psfd table.jpg>

[Julio Gutiérrez Samanez]

Rene and colleagues.
Perhaps this is a fifth regular form of the periodic table or system; the divisions correspond to a mathematical expression Z as a function of the principal quantum number n. Uniform growth is observed in pairs, with new suborbitals increasing with the change in n. Geometrically, they form a quadratic parabola. I suppose this is a good mathematical basis for chemical periodicity and all its complex variety, which may not be as random as it seems.
Regards
Julio

Rene y colegas.
Quizá esta sea una quinta forma regular de tabla o sistema periódico, las  divisiones corresponden a una expresión matemática Z en función del número cuántico principal n. Se observa un crecimiento uniforme por pares en los que van aumentando nuevos sub orbitales con el cambio de n. Geométricamente conforman una parábola cuadrática. Supongo que esta es una buena base matemática para la periodicidad química y toda su compleja variedad que quizá no es tan aleatoria como parece.
Saludos
Julio

--
You received this message because you are subscribed to the Google Groups "Periodic table mailing list" group.
To unsubscribe from this group and stop receiving emails from it, send an email to PT-L+uns...@googlegroups.com.

To view this discussion visit https://groups.google.com/d/msgid/PT-L/9618862E-4B8D-421B-B126-4DB160ABD9B7%40iinet.net.au.

[Jess Tauber]

WIth regard to the quadratic parabola mentioned in Julio's last posting, note that the terms of the quadratic formula are reproduced in the rows of the Pascal Triangle, something exploited for many decades now in the analysis of NMR spectra (I used to do it myself as part of my work in laboratory chemical characterization).

Jess Tauber

[Julio Gutiérrez Samanez]

Qué bien René, es como llegar un puerto seguro después de navegar en Mar tormentoso. Me gustaría que escribamos juntos un artículo que describa cómo inquietudes opuestas llegan a ser matemáticamente complementarias en tu espiral cuadrada. Creo que es conocimiento nuevo que debemos compartirlo con la comunidad científica. Yo puedo hacer mi parte y compartirte.
Julio

[Rene]

Here is a translation of Julio’s post:

“Great, René. It's like reaching a safe harbor after sailing through stormy seas. I'd like us to write an article together describing how opposing concerns become mathematically complementary in your square spiral. I think this is new knowledge that we should share with the scientific community. I can do my part and share it with you.”

On 16 Mar 2025, at 12:49 AM, Julio Gutiérrez Samanez <kut...@gmail.com> wrote:



[Larry T.]

Thank you, Rene.

I ran it by Eric Scerri a few times calling it (n+l)max rule, but he ignored it because he was concentrating on the classical Madelung rule based on differentiating electrons. Perhaps he misunderstood it.  

It's been on the home page of Perfectperiodictable.com for many years.

Best Regards,

Valery

[Rene]

On 16 Mar 2025, at 14:51, Larry T. <ora...@gmail.com> wrote:

Thank you, Rene.

I ran it by Eric Scerri a few times calling it (n+l)max rule, but he ignored it because he was concentrating on the classical Madelung rule based on differentiating electrons. Perhaps he misunderstood it.  

It's been on the home page of Perfectperiodictable.com for many years.

Best Regards,

Valery

Larry

You’re welcome.

Eric’s reaction is interesting.

Looking at Pb, for example, it's configuration is [Xe]4f^14  5d^10  6s^2  6p^2 in which case the n+l values are 7, 7, 6, 7, so (n+l)max = 7. The differentiating electron is 6p.

Looking at the rest of the elements, it seems like the differentiating electron always has a value of n+l that is equal to the maximum even though other electrons may have as high a value, as in the Pb example.

It then seems easier to refer to the classical Madelung rule differentiating electron, since this also gives the block type of the element involved. So Pb, with its 6p differentiating electron is a p-block element. Thus, the differentiating electron:

• encodes the highest-energy state of the atom

• directly determines the element’s notional block type

• automatically follows the Madelung filling sequence

That said, it remains curious that two different ways of determining an element’s notional placement in the periodic table yield the same result.

René

-- 
You received this message because you are subscribed to the Google Groups "Periodic table mailing list" group.
To unsubscribe from this group and stop receiving emails from it, send an email to PT-L+uns...@googlegroups.com.

To view this discussion visit https://groups.google.com/d/msgid/PT-L/CAM1noHV3U6CqqWZGFfbsAQcGBK_S_MDZ-r%3DU8vdyzHj66HhVmQ%40mail.gmail.com.

[Larry T.]

The difference between two methods is in the way you look at an atom:

1) As an assembly of nucleus and individual electrons. This view, where each electron is residing in its orbital and is acting independently of others, might not be in line with the well known QM paradoxes.

2) As a whole, where any electron can occupy any orbital, which is more in line with the idea that electrons are indistinguishable, non local and entangled.

VT

[Rene]

Thank you Julio. I have not forgotten this post and intend to respond to it when I am able to do so.

René

To view this discussion visit https://groups.google.com/d/msgid/PT-L/CAA-xvT4Yz4w%2BNkQKpkH0F7ixH1an3N9RMaU%2Bg7DWRh5d4eKTVA%40mail.gmail.com.
<1742017723486.jpeg>

[Rene]

Hi Julio

What did you mean by the "fifth" regular form of the PT?

I know there is:

• the left step table (1928)
• ADOMAH (2006)
• Scerri (2006) https://www.meta-synthesis.com/webbook/35_pt/pt_database.php?PT_id=20; and
• Beylkin (2018) https://www.meta-synthesis.com/webbook/35_pt/pt_database.php?PT_id=1202

Bear in mind that none of these tables are good for showing the left to right trend in metallic to nonmetallic character.

On the other hand, the conventional periodic table (CPT) does show this trend. 

And it is perhaps appropriate that the CPT — shaped by chemical priorities and practical classification — does not exhibit complete regularity. Chemistry, after all, thrives in the realm of functional definitions and fuzzy boundaries. Yet when its periods are paired into binodes, even this more intuitive, human-oriented table discloses a hidden mathematical structure: the sequence 10, 26, 50, 82, described by the quadratic formula 4x^2 + 4x + 2.

[Julio Gutiérrez Samanez]

Understood, Rene.

I don't mean to contradict what you say. The PCT is useful and necessary. It also works for the needs of chemistry teachers and in the laboratory. It shows "the left-to-right tendency in metallic to non-metallic character." Thus, it remains "sanctified" and eternal. That's not my problem.

What I want to make clear is that there is "also" another mathematical and geometrical way (if you will, physical, not chemical), of approaching the Periodic Table (the System, rather than the "Table"), and I wrote "fifth form", because the article "Four Regular Forms for the Periodic Table of Chemical Elements" (Russia, 1992) by G. G. Filipov and A. I. Gorbonov, which was published and commented on our website, shows "four forms."

Likewise, N. S. Imyanitov. In his article "The Periodic Law. Formulations, Equations, Graphic Representations" (Russia, 2011), he repeated those same "four forms," which I even colored to differentiate the quantum azimuthals (s, p, d, f, etc.).

My proposal, unlike the other four, which I attach as attached figures, demonstrates or makes visible a functional mathematical relationship between the principal quantum number n (made up of binodes or pairs of periods of "equal size") and the Z series, which is the atomic number of the chemical elements. This work is as valid and useful as your proposal of pairs of periods of different sizes, and other proposals by other authors.

Similarly, I have colored the proposals by Janet (1929), La ADOMAH, Scerri (2006), and Beylkin (2018), to make the sequences: s, p, d, f, evident.

Julio

[Rene]

On 18 Mar 2025, at 00:08, Larry T. <ora...@gmail.com> wrote:

The difference between two methods is in the way you look at an atom:
1) As an assembly of nucleus and individual electrons. This view, where each electron is residing in its orbital and is acting independently of others, might not be in line with the well known QM paradoxes.
2) As a whole, where any electron can occupy any orbital, which is more in line with the idea that electrons are indistinguishable, non local and entangled.
VT

Thanks for that perspective Larry. You gave me a lot to think about.

I’d gently push back on the idea that using your method 2 i.e. max(n+l) is more consistent with QM principles like indistinguishability or entanglement.

While it’s true that electrons are indistinguishable and the total wave function is entangled, n+l values refer to orbitals, not to electrons themselves. For example, in lead (Pb), the 4f, 5d, and 6p orbitals all have n+l = 7, but they are distinct in terms of energy, radial extent, and chemical role. The fact that they share an n+l value reflects the Madelung filling order, not quantum entanglement between those orbitals.

Entanglement ensures that we can’t assign specific electrons to particular orbitals in a classical sense, but orbital occupancy remains meaningful. This is supported by measurable consequences in spectroscopy, ionization energies, chemical reactivity—and even by direct visualization of orbital shapes, as seen in recent AFM studies:

https://tacc.utexas.edu/news/latest-news/2023/05/15/seeing-electron-orbital-signatures/

That’s why I think the differentiating electron offers a more structured and informative way to describe atomic organization. It tells us:

• the maximum value of n+l for the atom,
• the atom’s block (s, p, d, or f),
• and its position within that block.

In contrast, scanning for the maximum n+l across all occupied orbitals gives only a partial picture—and relies on the same orbital ordering assumptions to begin with. Method 1 (based on the differentiating electron) is fully consistent with quantum mechanical principles and significantly more informative in terms of both the structure of the periodic table and chemical behavior.

Upon refection, I feel this is likely to explain why Eric ignored your references to max(n+l).

René

[Larry T.]

Rene,

This is what I read: "...n+l values refer to orbitals, not to electrons themselves."

Then you switch to: "Madelung filling order" which everybody agrees is not how orbitals get filled and Eric himself was a big critic of Madelung rule.

Then you slip back to "differentiating electron offers a more structured and informative way to describe atomic organization" 

To that I say that the way you describe it informatively doesn't necessarily mean describing it correctly.

  Yes, you can see new orbitals added to some (not all) atoms, but you have no way to say how many and which electrons occupy that orbital and where they are in those orbitals. Using your lead example all elements from La to Pb and to Ra have the same maximum value of n+l that corresponds to the periods in the spectroscopic chart that is also called the Left Step Periodic Table. Coincidence? I don't think so.

VT

[Julio Gutiérrez Samanez]

Hello everyone, back here again. I fell ill and had appendicitis surgery, which was successful, and I'm still recovering. Best regards.
Julio

--
You received this message because you are subscribed to the Google Groups "Periodic table mailing list" group.
To unsubscribe from this group and stop receiving emails from it, send an email to PT-L+uns...@googlegroups.com.

To view this discussion visit https://groups.google.com/d/msgid/PT-L/CAM1noHXA3pLYZZ%3DaVODd8SaiTh3sroOsgwxc%2BBUcxZ%2B7BZiMNA%40mail.gmail.com.

[Larry T.]

We wish you speedy recovery.

Best Regards,

V. "Larry" Tsimmerman

[Jess Tauber]

Ditto, Julio.  

Anyone here willing to venture a guess as to the nature of the relationship between shell structures (at least with regards to LST and nuclear equivalent, minus spin-orbit coupling)? I've outlined the differences numerous times here on our list. For example, in the electronic system orbitals alternate parity WITHIN shells, whereas in the nuclear system parity alternates BETWEEN shells (thus a 'horizontal-, row-bound trend versus a 'vertical', column-bound one). Then there  is the repetition of LST period lengths, versus the non-repetition in the nuclear equivalent. Yet because of the parity issue, and the way that Pascal Triangle math works in the two systems, magic numbers in the nuclear LST are TWICE as large as those in every other electronic LST period. And so on.. Finally we have mirror-image additions to LST period/shell lengths- from lowest-spin UNSPLIT s-block electrons that rise in energy to join in the electronic structure of following LST periods versus the highest-spin component of SPLIT nuclear orbitals falling in energy enough to join the shell structures in the nucleus.

I'm also wondering whether there is a bit of a 'spinor' structure to the LST layout of the electronic system- in that we get pairs of same length periods here- just as you have to rotate the system TWICE with a spinor to get to where you used to be in the system. In the nuclear system under the LST harmonic oscillator model, only number of component orbitals within a shell doubles, so 1, 1; 2, 2; 3, 3; 4; 4,

Jess Tauber

[Rene]

Good to hear from you Julio. Here’s to a speedy recovery!

cheers, René

PS I owe you a response to your post of Mar 31, 2025, 8:54:28 AM

PS Larry: I owe you a response to your post of Apr 2, 2025, 3:25:13 AM

[Julio Gutiérrez Samanez]

Interesting, Jess.

What do you think of these sequences?

1,1,  2, 2,  3,  3,  4, 4…

1²,    2²,      3²,     4²,…….    =   1 n²= 1, 4, 9, 16…

2(1²,    2²,      3²,     4²,…….)= 2 n² = 2, 8, 18, 32…

4(1²,    2²,      3²,     4²,…….)= 4 n² = 4, 16, 36, 64…

 

All three apply to the atomic configuration (LST) and are a family of nested quadratic parabolas. They are very simple mathematical functions. They reduce physics and chemistry to mathematics. In fact, it is related to Pascal's triangle in the plane and to the tetrahedron in space.

If we add the terms one by one, we have:

 

1, 5, 14, 30…

2, 10, 28, 69…

4, 20, 56, 120… =Z

Julio

 

[Jess Tauber]

I posted earlier<Then there  is the repetition of LST period lengths, versus the non-repetition in the nuclear equivalent. Yet because of the parity issue, and the way that Pascal Triangle math works in the two systems, magic numbers in the nuclear LST are TWICE as large as those in every other electronic LST period.>     It occurs to me that the doubling of same length periods in the LST is a VERTICAL TREND, while the doubling of the magic numbers (cumulative for nuclear LST-equivalent shells) is a HORIZONTAL one. So, perhaps the way shell/orbital parity works in columnar versus row relations between the electronic and nuclear systems are in some way in COMPLEMENTARY DISTRIBUTION- one makes up for any issues with the other?

Jess

[Rene]

On 10 Mar 2025, at 11:35, Rene <re...@iinet.net.au> wrote:

On 10 Mar 2025, at 00:15, Larry T. <ora...@gmail.com> wrote:

I think it wouldn't be too confusing to explain to students that there are four distinct types of metals:

alkalis and alkaline metals, p-block metals, transition metals and lanthanides/actinides.

 It is just too simplistic to lump them all together and that is what traditional PT does.

Thank you Larry.

In Nature (2019), a machine-based analysis looked at the proximity of element names across 3.3 million abstracts published between 1922 and 2018 in more than 1,000 journals.

It showed six natural clusters of metals, illustrated in the attached figure. For now, if you focus just on the shapes and colours in the legend (ignoring the dashed borders and circled numbers), you can see:

• alkali metals
• alkaline earth metals

• transition metals

• post-transition metals (p-block metals)

• lanthanides

• actinides

So yes, the alkalis and alkaline earths are close enough to be treated as one type. That gives five types: alkalis + alkaline earths; transition metals; p-block metals; lanthanides; and actinides—very close to your suggestion.

If we then bring the dashed borders and circled numbers into play, the classification collapses to four types of metal:

• s/f-block metals: s-metals (alkalis + alkaline earths) and f-metals (lanthanides + actinides)
• d-block metals (non-noble)

• p-block metals

• noble metals (a subset of the d-block)

The beauty of this scheme is that it translates symmetrically to the nonmetals:

5. halogens

6. macrogens (C, N, O, P, S, Se, H)

7. metalloids

8. noble gases

So instead of the current asymmetrical tenfold taxonomy, we get a 4+4 balance. My presumption is that this symmetry offers a cleaner and more intuitive framework. As you’re an aficionado of symmetry, I imagine you’d agree?

I further presume that for students, the initial focus on the s/f block metals would be on the alkalis and alkaline earths, with the f-metals being deferred for more advanced studies.

René

[Jess Tauber]

8 basic groupings reminds me of the vertices of a cube, in which can be inscribed two co-centered tetrahedral each facing in opposite directions.  This cube can then be inscribed within higher Platonic solids such as the dodecahedron or icosahedron, which are duals of each other, just as the cube is the dual of the octahedron. The tetrahedron is its own dual, and the opposing ones inscribed in the cube can partially represent this duality, ignoring size and orientation. More numerology...

--

You received this message because you are subscribed to the Google Groups "Periodic table mailing list" group.
To unsubscribe from this group and stop receiving emails from it, send an email to PT-L+uns...@googlegroups.com.

To view this discussion visit https://groups.google.com/d/msgid/PT-L/45F30BB4-CA47-441F-9E33-04DE82F55BDF%40iinet.net.au.

[ERIC SCERRI]

Hi Jess

Thanks for the further numerology, although you are in good company!

Kepler already anticipated you with a version of the inscribed Platonic solids and GN Lewis, not unreasonably connected the 8 of the periodic table with his pre quantum electronic configurations of up to 8 electrons on the 8 corners of a cube. 

Eric Scerri

On Aug 30, 2025, at 5:31 AM, Jess Tauber <tetrahed...@gmail.com> wrote:



<PastedGraphic-14.png>

--
You received this message because you are subscribed to the Google Groups "Periodic table mailing list" group.
To unsubscribe from this group and stop receiving emails from it, send an email to PT-L+uns...@googlegroups.com.
To view this discussion visit https://groups.google.com/d/msgid/PT-L/45F30BB4-CA47-441F-9E33-04DE82F55BDF%40iinet.net.au.

--
You received this message because you are subscribed to the Google Groups "Periodic table mailing list" group.
To unsubscribe from this group and stop receiving emails from it, send an email to PT-L+uns...@googlegroups.com.

To view this discussion visit https://groups.google.com/d/msgid/PT-L/CAO6d3hLYCq82WcqMxszbS8LGstXE6NyozJiqYxpjov_qrdYgvA%40mail.gmail.com.

<PastedGraphic-14.png>

[Jess Tauber]

But wait, there's MORE! There are eight LS period taking us to Z=120, and in the simple harmonic oscillator nucleus, there are similarly eight LS-style shells, taking into account quantum numbers 0-7, just as the 120 element LS system gives us 0-3, but with repeated lengths of all periods ending in s2.