As a young undergraduate chemistry student 30 years ago, I had discovered that if one cut out the s, p, d, and f blocks from the traditional 2-D periodic table, and assumed symmetry so that the s block was 8 elements deep, the p 6, d 4, and f 2 (8s2 makes for a symmetrical system of 120 elements of periods 2, 8, 18, 32, 32, 18, 8, and 2, which you won't see again for another 100 elements, which physical facts can't support), then the stack would be a tetrahedron, though distorted.

I found that if one halved the width of each block, though, then the blocks would stack as a equilateral tetrahedron, though there would be intervening blank spaces. And the sum of this half width plus the depth of each block was always 9 (1+8 for s,3+6 for p,5+4 for d, and 7+2 for f). Note that the depth goes up as odds, and the half-width goes down as evens.

The chemistry community at that time seemed utterly disinterested, and I moved on to other interests, and a different career path.

A few months ago I learned that another fellow, Valery Tsimmerman (see www.perfectperiodictable.com) had a few years ago independently rediscovered this relation, and improved upon the model by showing that one could display three of the four atomic quantum numbers as divisions along edges of the tetrahedron. In addition he redesigned the tetrahedron with 120 close-packed spheres.

I had never been happy with having two elements per cell (here spheres) while leaving an equal number as spacers or filler. Bad use of real estate. So I started making new models to see if I could improve on this- several more obvious solutions were not very elegant, since they either were asymmetrical or disconnected with regard to block membership. But a few weeks ago I hit on a radically different solution that mixed quantum numbers, and was able to create a completely symmetrical system.

The new periodic tetrahedron is different from anything previously published. All the periods are there, all the orbitals, all the known groups. But in addition to what traditional depictions show, it is able to capture all four quantum numbers, as well as less well-known aspects such as secondary periodicity, diagonals, and knight's mover connectivity, all within the tetrahedral symmetry.

Buckminster Fuller almost figured it out himself. A few days ago I discovered the pages at: http://www.rwgrayprojects.com/. The following figure (http://www.rwgrayprojects.com/synergetics/s04/figs/f1701.html) of the tetrahedron of 120 close-packed spheres, cut in half, is relevant to the new periodic representation, as such half cuts are natural borders within the system.

Ironically, just below the text this is introduced in (http://www.rwgrayprojects.com/synergetics/s04/p1600.html#417.00), is section 418 where he talks about the filling of atomic nuclei by protons and neutrons.

The problem, I think, is that he was researching and writing at a time when not enough was known about the periodic relation. The actinide (recently renamed actinoid) elements were being newly synthesized- some of his writing reflects serious misinformation about these, and had been controversial when first reemplaced under the the f-orbital lanthanides by Glenn Seaborg (who had done this against all advice by the experts). And nucleosynthesis had not yet produced many (any?) of the following 6d and 7p elements we see now. Terminal 8s syntheses are still in the works, or not yet confirmed. Whether anything past atomic number 120 is possible to make we can't yet know, but 120 is the end of this major chunk. The next elements will involve an unattested g orbital.

In any case, Fuller had discussed interesting properties relating to the 20 and 120 tetrahedra- the first actually will fit inside another single-sphere thick shell to create the latter. Also, element 20 is the last one in the periodic table that has, in stable isotopes, the same number of neutrons as protons. This is probably very important. 20 and 120 show up all the time in his geometrical work relating solids and mappings to spheres. Also these tetrahedral shells appear to be created by sums of squares of even numbers:

sq2+sq4= 4+16=20; sq6+sq8+ 36+64=100. The next out jacket of spheres is 244, or 100+144= sq10 +sq12. You see the pattern- it goes on forever. But after having calculated very far out, I've found that only the 120 tetrahedron (so far) has an outer jacket that itself is a sum of even squares and a square itself. Dunno if this is significant.

The close-packed sphere tetrahedron has other properties relating to squares- if placed on its base, and counting from a zero sphere on top downwards, the sum of spheres of any two layers is a square- 0+1, 1+3, 3+6, 6+10, 10+15, 15+21, 21+28, and 28+36. And the tetrahedron has its own diagonal in Pascal's triangle. So there is a lot going on.

I'd love to hear comments, either here or via email. Thanks.

Jess Tauber
phonosemantics@earthlink.net