Jess Tauber: Feb. 2, 2010
After reading more of Bent's book and thinking again about possible relations between unusual kinships, I've got something that might be of use, if not in its current form. In his graphic depiction of highest oxidation state anomalies, Bent tries to demonstrate that He is the (sole? but Be too perhaps) major anomaly for the s block, then the upper/right members of the p block, similar types of anomalies in d and f. The numbers of actual elements involved grow horizontally with increasing l.
But there seems also to be the beginnings of a pattern when it comes to diagonals (only involving s and p transitions), diagonals (only involving d and p transitions), etc.
There is no mismatch when only considering H, He, Li, Be.
So is the pattern somewhat dyadic in nature? 0 steps to the right around periods 1 and 2 in s, 1 step to the right around periods 3, 4 in s (diagonals), then shifting 2 steps to the right around periods 5 and 6 in s (knight's moves, though only easily seen in d and p blocks), and hypothetically and ideally (numerologically) 3 moves to the right around periods 7 and 8 in s.
There might be interferences for the f block that prevent the 3 moves to the right from showing up fully for period 7.
As for multidimensionality, I've been thinking in terms of all-spatial dimensions, but this may be a mistake. The real world is 3 spatial and 1 temporal (plus other hidden dimensions that may add up to 16 in total, with 4 timelike and 12 spacelike, of which normal spacetime might only be a quarter of the total).
If a fourth dimension for the periodic system is timelike, then perhaps it is also quantized. Could this explain numbers of steps to the right with increasing dyadic number? And as relativistic effects accumulate on electrons for higher elements, relativity being more of a continuous thing and somewhat at cross-purposes to quantization, could this be a possible source of interference preventing a 3-step from fully showing up?
Could these effects (including aufbau anomalies) all be linked through distortions of TIME?
Now, if there is some sort of tension between quantization on the one hand, and continuous variation on the other, might we see it in the increasing gentleness of changes of properties with increasing n, l, etc.? In my angled ring system, the s block is a mostly vertical axis running 4 rings deep through the tetrahdron from the center of one edge through to its perpendicular mate on the other side, but as one goes out to the f-ring numbers of rings in the stack decrease from 4 in s, to 3 in p, to 2 in d, and just 1 for f. Does this number change iconically describe smoothness of property changes as numbers of m sub l states increases?
How do these facts pattern in TIME is considered? Does it take time for electrons to exchange (since they aren't fixed into particular slots)? Remember that if energy and mass can interchange according to Einstein, the space and time must be able to as well (time-dilation, foreshortening, but also some sort of weird rotation I recently read about).
Would the addition of a time dimension also smooth out some of the angularity of the tetrahedral system (at least mine anyway)? Things to think about.
Roy Alexander: Feb. 2, 2010
If the possibility exists, as Jess suggests, that "the periodic system is capturable in its entirety only in multiple dimensions", the problem may not be that a 3DPT will be able to "get everything", but whether the professionals will be able to conclude just what constitutes "everything".
You all don't seem to be getting any closer to agreement.
Jess Tauber: Feb.2, 2010
Well, IF it turns out that the periodic system is capturable in its entirety only in multiple dimensions, then there aren't any 2 or 3D representations that will get everything- one will have to choose what goes in, and what stays out. However, one might be able to create an algorithm that takes one on a well-motivated journey through the various lower dimensional projections.
Jess Tauber: Jan. 31, 2010
In Henry Bent's New Ideas
Fresh Energy, on p.xv in the Short Abstract Amplified section, he defines n, the Orbital's Principle Quantum Number, as n=r+l, where r=orbital's radial quantum numbers. He also defines n+l to be the Madelung parameter r+2l.
If n=r+l, then n-l= r.
So my (n-l) has physical meaning, and isn't just numerology? In the angled ring system, the two upper tetrahedral faces map (n+l) where upper is defined by where H is, and the lower two faces map (n-l), near element 120. Each face pair is symmetrical around one edge, and these two edges are oriented perpendicularly to each other. Does this have physical meaning, relating (n+l) at right angles to (n-l)? Magnetic fields are perpendicular to electrical ones.
Roy Alexander: Jan. 31, 2010
The idea of a "block [being] a parallelogram instead of a rectangle" is evident in de Chancourtois' and the Alexander Arrangement. It surprised me at the time that this feature (it needs only the p-block slanted to align periods and hold to an unbroken Mendeleev's Line throughout the whole table) was available for patent in the early `70s but no longer appears a wonder when one sees how doggedly the professionals cling to flat tables. It is clear that it takes an "amateur" as Philip Stewart refers to Charles Janet, who "derived his table from a helix arrangement ", to provide forward motion, and provide a fresh view of element relationships.
As to the "knight's moves" to which Melinda Green and Jess Tauber refer, the knight's agility is multiplied when one jumps around a corner to another plane completing a tertiary kinship (http://allperiodictables.com/kin)
impossible in the flat table. As an amateur myself, I cannot be aware of all the "knight's moves" (or new bishop's moves) which, as Melinda says "could lead to ideas for discovering some hidden features of atoms and their interactions".
Jess Tauber: Jan. 31, 2010
Valery- in the angled ring system secondary periodicity can be read directly as vertical apposition- the odd period orbitals are atop the other odd ones in sequence, and the even ones other evens.
The rings register with each other laterally- so that (going outward from the central axis) the first members of s, p, d, and f connect in lines of spheres, on one side of an edge, and the last ones on the other side.
Jess Tauber: Jan. 31, 2010
OOPS! I meant to write that knight's moves go from upper left to lower right. Been a long week
Jess Tauber: Jan. 31, 2010
I don't remember whether I've written this already, but if one takes the LSTP, as a system of linked rectangular blocks, and shifts the upper edges to the right so that what were vertical relations now engage the element below and next right (so each block is now a parallelogram instead of a rectangle), then one at least geometrically motivates some of the nontraditional kinships.
In this slanted format periods are unaltered, while normal groups now go from lower left to upper right, diagonals are now vertical, and knight's moves go from upper right to lower left.
Has anyone thought of this already?
Jess Tauber: Jan. 29, 2010
Well, I noted here as well as elsewhere that in an angled ring system (at least- what about ADOMAH?) (n-1) is just as valid as (n+1). Both terms give ordered, coherent patterns, patterns that are perfectly complementary.
Could other interactions between quantum numbers motivate some of the unusual kinships? Or aufbau anomalies? Has anyone ever looked? We already see in secondary periodicity some sort of preference for even-even or odd-odd period connection. Is there any similar kind of thing going on for groups?
I've started considering whether the packed-sphere tetrahedron's own internal structure acts as a kind of `computer', of neural network type. Each sphere would count as a `node', with a minimum of 3 connections, for vertices, then 6 for edge members, 9 for faces, and maximally 12 for sphere in the central subtetrahedron. Could property skews from expectations pattern based on some give and take between the spheres both locally and globally?
Jess Tauber: Jan. 28, 2010
All the interesting secondary, tertiary, and other unusual linkages, kinships, etc. in the periodic relation show there is context involved, since these links are NOT across the board, but deal only with particular single interactions or sets of interactions.
Valery Tsimmerman: Jan. 28, 2010
I believe that your belief that tetrahedral arrangement of the periodic table is not objective is certainly subjective. It has been established long ago that physical world follows certain rules, that are called physical laws, irregardless of our perception. Why would people like you exclude the Periodic System from this is beyond my understanding.
Perhaps you could come up with alternative explanation why atomic numbers of Be, Ca, Ba correspond to every other tetrahedral number and Mg, Sr and Ra correspond to arithmetic means of those tetrahedral numbers? Also, why n+l diagram extended into 3rd dimension to show quantum number "ml" becomes tetrahedron?
ADOMAH PT goes beyond putting elements in boxes and drawing pretty pictures. It provides simplified algorithm for deriving electron configurations, that makes it naturally symmetric by the way. Can you become objective for a moment and admit this simple fact?
Melinda Green: Jan. 28, 2010
@ Valery: I believe that a tetrahedral layout is just as subjective as any other, including the standard PT. None of these inventions are god-given. Some are simply more or less useful for various needs of ours. I wouldn't be surprised if intelligent beings on other planets come up with some of the same diagrams that we do, but that wouldn't indicate anything special about the universe, but rather something common about the goals of beings similar to ourselves. But rather than say that they're all subjective, I'd prefer to say that none of them are objective.
@ Jess: Regarding the 4D equivalent of the tetrahedron, it has five 3D hyper-faces, not projections. It's best not to call this a 4D tetrahedron because there's nothing "tetra" about it. You could call it a pentachoron but better is probably just "4-simplex". (The tetrahedron is a 3-simplex, and the triangle is a 2-simplex). My main point is that there's nothing particularly dimensional about elemental relationships. The relationship graphs that I talked about simply describe the connections between pairs of elements, and elements are related to each other in a lot of ways, not just two. Steps along a row or column of the standard PT describe one kind of relationship but there are more. Knight's moves describe more graph connections, etc.
The game that I think we're playing on this blog involves two steps: First, choose a graph of elemental relations that we think are the most fundamentally important to our needs, and second, to find the most symmetric presentation of that graph. Arguments about how He connects to other elements involve the first step whereas arguments about the best diagrams involve the second step.
The cubic and other lattices in various spatial dimensions that you talk about, really just describe some particularly symmetric graph layouts. I won't say that it's folly to try to map a popular elemental graph onto one of these regular graphs. If you do find a good mapping, then it could lead to ideas for discovering some hidden features of atoms and their interactions. The tetrahedral sphere packing model that you and others advocate is intriguing because of it's high symmetry but chemistry is not my field so I just have no idea whether it means anything. It sure is a fun game though!
Jess Tauber: Jan. 27, 2010
Response to Melinda
From what I've read online, a 4D hypertetrahedron has 5 TETRAHEDRAL 3D projections. I'm no mathematician, just plodding along. But since one tends to lose information from higher dimensions when projecting to lower ones, I've thought that one might utilize this possiblity for the unusual linkages in the PT, such as knight's moves.
In a regular triangular slice made up of close-packed spheres (using Pascal diagonals for count) there are maximally three different rows of spheres (maximally because for ex. the vertices only have access to two). In the traditional rectangular box depiction you only have two linear sequences- rows and columns. But because the blocks are of different heights and widths connectivities are a bit freaky- there are a good number of ways to stack or align/justify which account for part of the unusual connectivities PT workers are familiar with.
In a tetrahedron of close-packed spheres there are, for body-internal spheres, a maximum of 12 linear sphere sequences passing through the sphere- the minimum, again for vertices, is 3.
Unless I'm wrong, a 4D tetrahedron (or other solid) will have even more connectivity. Vertices will have at least 4 (2 for triangle slice, 3 for tetrahedron so 4 for hypertetrahedron?). Internally things are even more interesting: 3 for triangular slice, 12 for tetrahedron- anybody know what the number will be for the hypertetrahedron?
My tetrahedral model already connects, in linear fashion, many of the unusual PT memberships that don't fit this way in the traditional 2D tables, but not all. With an extra dimension maybe these others fit too.
Melinda Green: Jan. 26, 2010
Jess Tauber writes:
> In the angled ring tetrahedron both (n+1) and (n-1) appear to be meaningful, and pattern in complementary fashion on the rings and subtetrahedral faces. This makes me think that perhaps ml and ms might have a similar relationship, though ms, as ±1/2, seems hard to integrate. If ms was an integer (or if the other numbers were made into halves) things might fall into place better.
> As for continuity and dimensionality, everyone seems to avoid 4 dimensions, which have enough linkage capacity to cover all known relations. Although a static, single representation in 4D is hard to imagine, I've seen dynamic hypertetrahedral depictions, but not using spheres (or hyperspheres?), nor linkages between these. A hypertetrahedron has 5 conventional 3D projections. My guess (which I'm finding hard to visualize) is that such a set (along with the inter-projection transformations) would carry all the things we argue about, and perhaps more nobody has noticed yet.
> Jess Tauber
I didn't follow your first paragraph, but regarding the second, I don't see what 4 dimensions has to do with any of this unless you're talking about some 4 special dimensions of atomic qualities such as atomic number, atomic weight, etc. That's because graphs don't have any particular dimension. You can always flatten any graph onto a 2D plane, or into any other dimension for that matter. I.E. all dimensions have the same linkage capacity.
We are looking for good or even ideal visualizations of the elemental relations graph, so in that regard you might find a highly symmetric embedding of that graph into 4 dimensions. There are lots of ways to visualize a 4-space, so you shouldn't have to worry about that. It's enough to just find a good embedding.
Also I have no idea what you mean when you say that "A hypertetrahedron has 5 conventional 3D projections." There are any number of ways to project any 4D object into 3-space.
Jess Tauber: Jan. 22, 2010
In the angled ring tetrahedron both (n+1) and (n-1) appear to be meaningful, and pattern in complementary fashion on the rings and subtetrahedral faces. This makes me think that perhaps ml and ms might have a similar relationship, though ms, as +/-1/2, seems hard to integrate. If ms was an integer (or if the other numbers were made into halves) things might fall into place better.
As for continuity and dimensionality, everyone seems to avoid 4 dimensions, which have enough linkage capacity to cover all known relations. Although a static, single representation in 4D is hard to imagine, I've seen dynamic hypertetrahedral depictions, but not using spheres (or hyperspheres?), nor linkages between these. A hypertetrahedron has 5 conventional 3D projections. My guess (which I'm finding hard to visualize) is that such a set (along with the inter-projection transformations) would carry all the things we argue about, and perhaps more nobody has noticed yet.
Jess Tauber: Jan. 21, 2010
Well, then a tetrahedron is ideal- it has symmetry, hierarchical structure, a texture, heft (you can hold it in your hand, and throw it at the teacher!). You can take it apart and put it back together. One can fashion it into jewelry, or candy with something else in the center. Makes a great paperweight. Lots of possibilities.
Jess Tauber: Jan. 20, 2010
January 20, 2010 at 7:04 pm
The every-other-tetrahedral-number fact, when related to a tetrahedral modeling of the system, seems to relate to which particular ways the model can be cut up into periods, etc.
In a face-based tetrahedron of close-packed spheres, each layer's sphere count conforms exactly to the Pascal triangular numbers, the sums of which give the Pascal tetrahedral numbers.
Relevant triangular numbers are: (0),1, 3, 6, 10, 15, 21,28, 36. Note that intervals between each triangular numbers are the integers, which are the next earlier Pascal diagonals. The triangular numbers pair in odd-odd (1,3), (15,21) alternating with even-even (6,10),(28,36).
IF one wanted to cut up the tetrahedron to be a model of periodic relations, one could simply use the known face-based triangular-number layering. But this would force each period to be different in sum from every other- no repetitions as found in the `real' periodic relation. So period 1 would be H only (1), and then period 2 would contain He,Li,Be (2,3,4) to sum at that point to 4=sq2. Period 3 would contain B,C,N,O,F,Ne, and miss the 3s set, while period 4 would contain both 3s and 4s. And so on.
The above is the periodic system one would get if it were forced to use every tetrahedral number.
However, there ARE other ways to cut the tetrahedron. The nuclear folks appear in many cases to be just fine with the double tetrahedral notion, where two virtual close-packed sphere tetrahedra actually intersect, which means that the total sum is NOT the sum of two independent tetrahedra, thus taking into account the weird fact that the number of neutrons doesn't start to outstrip the number of protons until after Ca.
In the angled ring model I'm still developing, the rings sum to period doublets, like taco shells- that is, triangular faces that share an edge, so their sum is that of two triangular faces minus an edge. Very similar in spirit to the double tetrahedron idea for the nucleus, but pushed down to lower dimensionality.
Another way of looking at the `taco shell' double triangular face is that it is mathematically identical to adding sequential triangular layers from the face-based tetrahedron together along one edge. But now the two triangles are identical and symmetrically oriented along the shared edge. Robbing Peter to pay Paul. Two equal triangular faces together in this fashion means that the odd tetrahedral numbers are `out', and forces us to use two equal intervals between the even tetrahedral numbers.
Valery Tsimmerman: Jan. 20, 2010
One more thing. Irregardless of how periodic table is presented, the n+l rule remains very important part of the Periodic Law. I do not think that Dr. Scerri in 2nd edition of his book will go as far as to deny its importance. Therefore, my argument that Alkaline group of elements is special among all other groups because n+l value changes only between the alkaline earths and the elements of the first groups of spdf blocks remain valid and irrefutable.
Also, the fact that atomic number of every other alkaline earth element corresponds to every second tetrahedral number and the atomic numbers of intermediate alkaline earths are arithmetic means of those tetrahedral numbers is very interesting and is telling us that Periodic Law somehow is related to tetrahedral stack of spheres. I think that significance of this fact needs to be discussed and looked into, instead of being flatly rejected.