Essay:Number theory or numerology in the atomic realm?

I sometimes wonder how close a line divides number theory from plain old crank numerology. I see friends doing the same sorts of work as I do traversing this line all the time, usually not TOO far over, but enough that I worry.

Many years ago, as a young undergraduate chemistry student, I was toying with a depiction of the periodic table. I noticed suddenly that if one cuts out the blocks of the figure out and lines them up vertically you get a rough, distorted tetrahedron. The system could be improved by several small changes: 1) move He over the alkaline earths, which share its s2 electronic configuration (despite He's nobility), 2) move the s1 and s2 elements bodily to the right of the table, despite metallic behavior of most of these elements which chemists use to motivate its placement on the left, 3) move the f-block elements, usually shown in the West as a 'footnote' to aid printing, and 4) fill the system out to atomic number 120, which is in the alkaline earth group. If one does all this then all the blocks are rectangular, and follow an interesting rule: the sum of column height and row half-length always equals 9. That is, for the s-block 8+1, p-block 6+3, d-block 4+5, and f-block 2+7. I used this to create my first tetrahedral periodic table, which of course nobody was interested. But it was the late '70s, and folks were either too stoned or busy getting dates to care. I moved on to other interests, but often used what I had found in lectures to show that Nature still had mathematical surprises for us, even in areas considered 'done'.

Now we time-travel to just a few years ago. I'd given up generally looking for new postings on the internet about tetrahedral periodic tables several years before, since the only things I ever found were my own. But just after reading about yet another new high-mass element being synthesized, on a whim I took a deep breath and looked once more. This time, however, I discovered a blog posting by someone who claimed to have discovered the tetrahedral relation. I looked him up and eventually found his blog, and saw my own discovery. Worried that I'd been robbed, I contacted him and we discussed the issue. I showed him old posts from years before his work, and he showed me how he developed his model. Realizing that we had each independently discovered this through different routes, we started working together. The give and take has been quite valuable for both of us idea-wise. You can visit his pages at: www.perfectperiodictable.com

I had never been particularly happy with my own tetrahedral model: halving the horizontal rows seemed an arbitrary choice, but it did give a rational distribution of blocks vertically in the tetrahedral space that matched the way orbitals were laid out. The problem is that this is not the way they fill with electrons. Plus any cell in the system has to contain 2 elements. My new research partner had improved over what I had done by replacing square cells by spheres. These are laid out in rectangular arrays in layers through the tetrahedron, despite the fact that from other perspectives the spheres are close-packed. I started looking for ways to re-rationalize the system so that it used all the spheres, one per element. Eventually I discovered many new structures that filled the tetrahedral space this way, some much more aesthetically pleasing and symmetrical than others.

At this time I started my own blog, at tech.groups.yahoo.com/group/tetrahedronT3, which acted as an interactive blackboard for the development of the concepts related to all this, though it hasn't been much used of late. While looking up the properties of tetrahedra on Wikipedia I read about the relation of tetrahedra to the Pascal Triangle. It suddenly hit me while looking at the Triangle's diagonals that every other tetrahedral number, 4,20,56,120 was identical to every other alkaline earth atomic number, and remember that these are rightmost in my reworked periodic system from the late '70's. I learned that in fact the reworked table had first been published back in 1929 or so by an elderly French polymath named Janet, who called it the 'Left-Step Table'. It has since been independently rediscovered by many investigators, yet never accepted by chemists, who seem to prefer surface chemical facts over deeper quantum design. The alkaline earths end period analogues in the Janet table.

The link between the tetrahedral numbers and the periodic relation has to do with period lengths. In the Janet table, ALL periods are organized so they form duals, that is, two sequential periods of the same length. In the traditional periodic table the first period fails to be so doubled, so people don't get the larger pattern. Each period length is a half- or double-square: 2,8,18,32..., with duals containing numbers of elements which are squares of even integers. In the Pascal Triangle's tetrahedral diagonal each number is a running sum of squares, either of odds or evens. Since the periodic systems' duals are based on squares of evens, only every other Pascal tetrahedral number matches the alkaline earth atomic numbers.

As time marched on, however, even further Pascal relations popped up. It was found that, counting leftwards FROM the alkaline earth that using just Pascal triangular numbers (0),1,3,6,10,15,21,28... within any horizontal Janet period ALWAYS landed on a position where the quantum number ml is equal to zero, that is, the two centers of orbital half-rows within any block. And these positions were all vertices in the tetrahedral model I developed using all spheres, which was organized around skew rhombi stacked to create ever larger tetrahedra. Rhombi made of close packed spheres contain the same number of spheres as square arrays, thus two same-length periods worth. People had previously worked out square pyramidal periodic table models- the new tetrahedral model is the most compact possible, and the most symmetrical. In both the s-block element form a tall central column, surrounded by a shorter jacket of p-block elements, then d- and finaly f-. It is possible in the tetrahedral space to create several mappings where Mendeleev's Line, the unbroken string of atomic numbers, folds neatly and symmetrically.

And yet things get even weirder. The earlier Pascal Triangle's normal diagonals have their roles in motivating the periodic relation: the natural numbers for Mendeleev's Line, the triangular numbers defining ml=0 positions in the half-orbital row centers (which form vertices in the tetahedral models), the tetrahedral numbers which mark vertical positions in both the Janet table and the tetrahedral mapping. I have not yet managed to find any relevant mapping for the pentatope and higher diagonals. Maybe I'm just not thinking far enough outside the box. In any case there are also, in the Pascal Triangle, the so-called 'shallow' diagonals, which cut across the others. Samplings from the normal diagonals sum to create the FIBONACCI numbers. Interestingly, each normal diagonal is associated with a dimensionality (that is, the 1's are 0D, naturals 1D, triangulars 2D, tetrahedrals 3D and so on)- and in creating the Fibonacci numbers it is observed that the shallow diagonals either sample from all-even or all-odd dimensioned normal diagonals. There is also structure in the Fib sequence itself, with two odd Fib numbers for every even Fib (that is the smallest nonzero number of evens of odds (2xodd)), versus the smallest nonzero number of odds of evens (1xeven).

Well, just out of curiosity I decided to take Fib numbers AS atomic numbers, to see what they did. Amazingly, EVERY SINGLE FIB NUMBER UP TO 89, WITHOUT EXCEPTION mapped to LEFTMOST positions in atomic orbital half-rows. Further the odd Fibs mapped to the first (left) half-row, where we have the first singlet electron, and the even Fibs mapped to the second (right) half-row, where the first doublet valence electron appears (the orbital lobes have to fill with singlets before doublets are allowed).

Then I found that the related Lucas numbers (2,1,3,4,7,11,18,29,47,76...) mapped to RIGHTMOST positions in orbital half-rows, with last singlets (half-filled) or last doublets (completely filled), though less perfectly than the Fib trend. The same odd/even bias is observed. Fib gets off track at 144, but this is much further out than the known elements (perhaps further than we CAN get). The Lucas mapping veers off at 29 and stays off. Both 29 (copper) and 47 (silver) are interestingly in the SAME column/group, and have 'anomalous' ground state electronic configurations (so do about two dozen other elements)- but by taking a filled s2 orbital and donating one electron to the nearly filled d9 (both 29 and 47 are one move left from where the Lucas mapping *should* put them), both elements end up with s1,d10 configurations, both of which fulfill the Lucas trend configurationally, if not positionally in the table. Osmium, element 76, gets around its worse misplacement (d6) by behaving like a noble gas (p6). The Lucas numbers can be generated in the sister of the Pascal Triangle where one side replaces 1's with 2's. This same triangle gives the Fibonacci sequence on the other side, but moved up one position, so that 1 appears once, not twice. It may be that this sister is the more basic motivating object for the periodic system, but then one still has to account for the triangular and natural numbers behaviors.

The Fib and Luc numbers of course are part of an entire system of related series, all of which see the ratio of successive pairs of numbers converge on the Golden Ratio, which is itself part of a still larger system of Metal Means, which take different numbers of successive numbers in sequences to give the next result. It also is found that the Metal Means relate to the sister Pascal Triangle, in that the equations which define the powers of the Metal Means have power terms with numerical coefficients, and when you line all these equations up, the coefficients turn out to be the numbers from the sister Pascal Triangle diagonals (whether someone noted this before me is unknown, and attempts to find out have come up empty).

The nuclear realm is no slouch in terms of the Pascal relations, though in the nucleus we find DOUBLED triangular and tetrahedral numbers of nucleons in shell filling biases, organized and motivated differently from the electronic system. There are distortions, just as in the electronic side. But most of the mappings are quite simple, no fancy equations, no weird hedges. And interestingly, as atomic number increases we see that the ratio of N/P converges on the Golden Ratio at the limit of normal nuclear stability (that is U or Pu, the highest elements found in natural conditions for any significant time). Other related numbers appear to be approximately 100x Golden Ratio related numbers (which can be formed from sums or differences, etc. of Fib and Luc numbers). So for example ...38,62,100,162,262.. paralleling ..0.381..,0.618..,1.000,1.618,2.618... Nickel 62 (with a Fib number, 34, of neutrons (and 28 or 4x7, both Luc), has the greatest binding energy of any nucleus in the entire periodic system.

Such things show up all over the place, far more often than expected. So two questions arise. First, how real are these mathematical phenomena in Nature, above and beyond the electronic and nuclear realms. All seem to be related to packing and growth issues. There are biases towards Fib and Luc numbers in the relative proportions of elements in natural compounds of certain types, and in alloys of metals. We see these also in quasicrystals, which have 5-symmetry and were thought to be nothing but a mathematicians fancy until real materials were discovered with these behaviors, and recently quantum behaviors were discovered having a Golden Ratio bias as well.

The second question is how far can one be allowed to explore with eyes attuned to perceived mathematical regularities before one starts to verge on crank status, or worse? Some of the folks I exchange mails with about the above topics get into what I see as real Numerology, looking for patterns in the Bible, Torah, Koran and so on. Personally I can't go that far, though I do like to look for new patterns in natural systems that have been less examined. I have been attacked for even attempting the latter, by real scientists who are happy with the hodgepodge equations their fields have come up with, even though by separating out the different threads the same equations can be broken up into very simple parts. An example would be the interaction of quantum mechanics with relativity, which increasingly affects the structures and properties of elements with increasing atomic number. Many of these properties appear to be the main reason chemists won't abandon the traditional periodic system, even though by abstracting out the purely motivational quantum mechanical you arrive at a tetrahedral system. Sure the surface system of observables is much messier, because the simple underlying motivations interact. Is that any reason to throw out the underlying system(s)? Linguistics had this argument almost a century ago, when they invented the notion of an underlying phonemic level as opposed to the more surface and interactive phonetic level. But even today this dichotomy gets debated, so I suppose I'll have to give chemists some leeway here.

In any case new research keeps finding Golden Ratio based behaviors in natural systems. It was recently discovered that the overall balance of nucleotide bases in DNA has two attractors that are defined by the Phi-related value of .381..., but divided by two, as against .5 (or 1.000 divided by two). Remember the series ..0.381..,0.618..,1.000.., etc. Other work suggests that the distribution of brainwave frequencies is determined by mixtures of Fib and Luc numbers, and transformations of these (since there are many ways to write out any one of them) that allow the waves to pass each other noninterferingly. And a new doctoral thesis in linguistics explores putative Golden Ratio behavior in syntax. How many of these will be found to be flights of fancy remains to be seen. I suspect that the more strained the argument, the more complex the dance one has to do to justify the mappings, the more likely the researchers have crossed the line into numerology. And on the other side we have a large percentage of folks who refuse to even look for such patterns, believing any such effort to be unprofessional, and unscientific. Where does one strike a balance, and how does one manage to stay there?

Gonefission (talk) 14:04, 15 March 2012 (UTC)