One Alternate View of the Periodic Table

[Raphie]

aka "How to Rearrange the Elements Without Engaging in Alchemy"

Below is one view on what the Periodic Table might look like if it were "callibrated" in geometric fashion to the Atomic Numbers of the Earth Alkaline Metals. (red cells)

4, 20, 56 and 120 are all 4 times the sum of squares (1 + 4 + 9 + 16 + 25...). Multiply by 7/4 ...and you get 7, 35, 98 and 210, integers which are row sums of a Rydberg-Ritz array associated with the Hydrogen Emission Spectrum.

Subtract 2*2^2 from 20 to get 12. 2*3^2 from 56 to get 38, and 2*4^2 from 120 to get 88.

Note: The Color Coding for elements 113 - 120 are all tentative predictions. These elements have not yet been discovered. Ditto for 170 & 220.

RELEVANT ESSENTIAL INFORMATION REGARDING ALKALINE EARTH METAL ELEMENTS & HELIUM

ELEMENT---- --> SHELL CONFIGURATION --> ATOMIC NUMBER
HELIUM----- --> 02 -------------------------------> 002
BERYLLLIUM --> 02,02 ------------------------------> 04
MAGNESIUM --> 02,08,02 ----------------------------> 12
CALCIUM---- --> 02,08 08,02 -------------------------> 20
STRONTIUM --> 02,08,18,08,02 ---------------------> 38
BARIUM----- --> 02,08,18,18,08,02 ----------------> 56
RADIUM----- --> 02,08,18,32,18,08,02------------> 88
UNBINILIUM --> 02,08,18,32,32,18,08,02------> 120 (one hypothesized "Island of Stability")
Note the symmetrical structure...

What I find particularly interesting about this is that it incorporates within it's structure the rare Earth elements.

Here is a link to the standard periodic table from Chemistry.com with atomic numbers included:
http://www.chemicalelements.com/show/atomicnumber.html"

And here is a link to a page on Wikipedia that describes other alternative versions of the periodic table: http://en.wikipedia.org/wiki/Alternative_periodic_tables"

 

[Dadface]

I think the standard periodic table has evolved into a design classic and can stand beside other icons such as the map of the London underground.The alternatives are nowhere near as pretty(to my mind)but are they better in other ways?

Oh,and welcome raphie

 

[Raphie]

I think the standard periodic table has evolved into a design classic and can stand beside other icons such as the map of the London underground.The alternatives are nowhere near as pretty(to my mind)but are they better in other ways?

Oh,and welcome raphie

First, many thanks for the welcome, Dadface.

Second, I don't believe it's a question of better or worse, but simply different. Consequence of the manner in which this table is constructed, namely geometrically, I see potential for linkage to many areas of higher mathematics.

e.g. That last value, for instance, 117... Multiply by 70 (the last "z" coordinate of the Leech Lattice) and then by dimension (24) and, voila, you get 196560, the number of vertices of the Leech Lattice.

Furthermore, if there is anything to this formulation, which I am not saying there is, but see as a possibility, it is also predictive.

e.g. Although I would not suggest myself to be well-versed in chemistry (this was a somewhat accidental formulation in relation to other investigations), it is plain to see that Rutherfordium (Atomic Number 104) and Hafnium (Atomic Number 72) should share properties with the Rare Earth Metals. And if they have shared properties, both of those elements, in some sense, could be considered "bridging elements"

Additionally, the table suggests the families that elements 113 --> 120 should belong to, although, quite honestly, I would not expect it to be 100% accurate. No more so, say, than Mersenne's guesses as to Mersenne Prime exponents.

And one other point... if you look at the way the numbers are "slotted," the entire "kit and caboodle" can wrap back around on itself, so, essentially, it could be constructed as one continuous curve.

2 --> 3, 12 --> 13, 20 --> 21, etc.

Just a few examples amongst many I could offer in relation to how this table, in current or modified form, could be potentially "useful," not as a replacement, but as a supplement to both the Standard and Janet Periodic tables.


Raphie

 

[Raphie]

In terms of this framing of the "alternative view" periodic table, which I will simply refer to as the "Alkaline Earth" Periodic Table, here is the relevant progression from Sloane's On-Line Encyclopedia of Integer Sequences:

========================================================================
Sloane's A169603: http://www.research.att.com/~njas/sequences/A169603
First p terms of array from (odd rows) numerators of Rydberg-Ritz spectra of hydrogen atom.
0, 0, 7, 0, 11, 24, 0, 15, 32, 51, 0, 19, 40, 63, 88, 0, 23, 48, 75, 104, 135, 0, 27, 56, 87, 120, 155, 192, 0, 31, 64, 99, 136, 175, 216, 259, 0, 35, 72, 111, 152, 195, 240, 287, 336

COMMENT
Rows from Lyman A005563, Paschen 061039, Pfund A061043, Hansen-Strong A061047, .. : 0; 0,7; 0,11,24; 0,15,32,51; 0,19,40,63,88; 0,23,48,75,104,135; last two (and after in a(n)) from Richard Mathar,April 29. Sum of each row: 0,7,35,98,210,=7*(A000330 square pyramidal numbers which appear in Janet periodic table via A002492(n+1)=4*A000330= 4(Be),20(Ca),56(Ba),120;see (*),page 39).
========================================================================

Below is what the array looks like in triangular form, partitioned into segments...

NULL
--------
Periodic Table
--------------------------------
"Extended Periodic Table"
--------
??

-----------------------------------------------
ROWSUM = 7* Sum of Squares
-----------------------------------------------

Bold indicates value divisible by 7

0 = 0
-------
0 + 7 = 7
0 + 11 + 24 = 35
0 + 15 + 32 + 051 = 98
0 + 19 + 40 + 063 + 88 = 210
--------------------------------------
0 + 23 + 48 + 075 + 104 + 135 = 385
0 + 27 + 56 + 087 + 120 + 155 + 192 = 637
0 + 31 + 64 + 099 + 136 + 175 + 216 + 259 = 980
0 + 35 + 72 + 111 + 152 + 195 + 240 + 287 + 336 = 1428
-----------------------------------------------------------------------------
0 + 39 + 80 + 123 + 168 + 215 + 264 + 315 + 368 + 423 = 1995And here is a table that gives least common multiples for the Atomic Numbers of (Beryllium, Calcium, Barium and Unbinilium) and the rows sums of the above...

-----------------------------------------------
ROWSUM = 28 * Sum of Squares
-----------------------------------------------

0 = 0
-------
0 + 28 = 28
0 + 44 + 96 = 140
0 + 60 + 128 + 204 = 392
0 + 76 + 160 + 252 + 352 = 840
--------------------------------------
0 + 92 + 192 + 300 + 416 + 540 = 1540
0 + 108 + 224 + 348 + 480 + 620 + 768 = 2548
0 + 124 + 256 + 396 + 544 + 700 + 864 + 1036 = 3920
0 + 140 + 288 + 444 + 608 + 780 + 960 + 1148 + 1344 = 5712
------------------------------------------------------------------------------------------------
0 + 156 + 320 + 492 + 672 + 860 + 1056 + 1260 + 1472 + 1692 = 7980Values in each row can be partitioned into "blocks" divisible by 28...

-----------------------------------------------
ROWSUM = 28 * Sum of Squares
-----------------------------------------------
Partitioned into blocks divisible by 28
and sorted in ascending order...

0 = 0
-------
0 + 028 = 28
0 + 140 = 140
0 + 392 = 392
0 + 252 + 0588 = 840
-------------------------------
0 + 392 + 1148 = 1540
0 + 952 + 1596 = 2548
0 + 700 + 0924 + 1036 + 1260 = 3920
0 + 140 + 1148 + 1344 + 1512 + 1568 = 5712
--------------------------------------------------------------
0 + 672 + 1260 + 1792 + 2072 + 2184 = 7980Which can then be "reduced"...

-----------------------------------------------
ROWSUM = Sum of Squares
-----------------------------------------------
Previous divided by 28
and sorted in ascending order...

0 = 000
----------------
0 + 001 = 01
0 + 005 = 05
0 + 014 = 14
0 + 009 + 021 = 030
---------------------------
0 + 014 + 041 = 055
0 + 034 + 057 = 091
0 + 025 + 033 + 037 + 045 = 140
0 + 005 + 041 + 048 + 054 + 056 = 204
-----------------------------------------------------
0 + 024 + 045 + 064 + 074 + 078 = 285

... to partition the sums of squares in a manner I can't say I've ever seen before.

Unique Values: 0, 1, 5, 9, 14, 21 || 25, 33, 34, 37, 41, 45, 48, 54, 56, 57, 64, 74, 78
Repeat Values: 5,14,41 || 45
Prime Numbers in this array: 5, 37 & 41RaphieNOTE:
Further review would seem to indicate that the "Extended Periodic Table" predicted by scientists "slots in" much better to a fifth block that is 6 x 6, not 5 x 5. Thus, the "block hierarchy" would go 4*(1^2, 2^2,3^2,4^2,6^2), a schema that dovetails nicely with the thinking behind the initial formulation of this table.

e.g.
Let... n be an element of {N} | d(p_n -1)*k = n for the special case of k = 1.

for d(x) the divisor function, and p_n = nth prime number.

Solutions: 2, 3, 5, 7, 13 = p_1, p_2, p_3, p_4, p_6.
I believe this to be a finite set...

2,3,5,7, & 13 are the set of Mersenne Prime exponents highlighted by Frampton and Kephart in their 1999 paper:
Mersenne Primes, Polygonal Anomalies and String Theory Classification
http://arxiv.org/abs/hep-th/9904212

 

[LudusRex]

Thank you for sharing your alternate view of the periodic table. It is always fascinating to see different perspectives and approaches to organizing the elements. Your geometric calibration to the atomic numbers of the Earth alkaline metals is certainly an interesting concept. However, it is important to note that the current periodic table is organized based on the elements' electronic configurations and properties, not just their atomic numbers. Additionally, there is a lot of experimental evidence and scientific research that supports the current layout of the periodic table.

While it is possible to rearrange the elements in different ways, it is not necessary to engage in alchemy to do so. The periodic table is a representation of the naturally occurring elements and their properties, and it is constantly evolving as new elements are discovered and our understanding of them deepens. As scientists, it is important to base our theories and hypotheses on empirical evidence rather than purely mathematical or geometric patterns.

That being said, exploring different ways to organize and visualize the elements can lead to new insights and perspectives. Thank you for sharing your unique viewpoint on the periodic table.