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Jess Tauber wrote:
It is another property of tetrahedra of close packed spheres that once you establish a core of 20 in a tetrahedral arrangement (20 being a tetrahedral number) every surrounding 'jacket' uses the same formula for count, being the sum of two contiguous squares of even integers. That is, 20 is 4+16, and the next jacket of 100 has 36+64. This relates directly to tetrahedral diagonal numbers, but doesn't stacked triangular layers.
Thu Nov 17 17:52:15 UTC 2011
Valery Tsimmerman wrote:
Explanation. Post 4 of 3.
So, to make it even more simple and to avoid references to the Pascal triangle, following procedure can be used: In order to build geometric representation of atomic sequences in the periodic system, buy bunch of red and green marbles. Take one red marble, write H and He on it and put it on top of table, then take two green spheres and put them next to the red to form a triangle. Take another red sphere and write Li and Be and put it on top of the triangle. Take three more red spheres and put them on the table next to two green. Write B, C and N on them, then write O an the marble marked with B, F on the marble marked with C and Ne on the marble marked with N (this is to account for Hunds rule). Take two green spheres and place them next to red sphere in second row. Then, take one one red sphere, put it on top to complete the tetrahedron. Mark it with Na and Mg____ Start new layer by putting 4 green spheres on top of the table next to three red spheres. Put three red spheres on top of green in the second layer and write Al, Si, P and then write S next to Al, Cl next to Si and Ar next to P. Place two green spheres in the third layer on top of three red and then put one green sphere on top of third layer to complete the tetrahedron. Mark last red sphere with K and Ca. Continue this procedure for the rest of the elements of the periodic system. I am sure you will enjoy it.__ Best, Valery.
Thu Nov 17 17:35:03 UTC 2011
Folded Mendeleev's Line
Some time back I announced on the Yahoo tetrahedronT3 group that I had found a way to fold up an unbroken sequence of elements into a tetrahedron of close-packed spheres, as a variation of the skew-rhombi theme of the T3 model. That version was a bit off-kilter in terms of placement of elements within the figure, but on the positive side used an entirely consistent procedure, resulting in the distribution of the s-block element around the tetrahedral 'equator'. I've now found another way to fold the line, with a procedure that uses mirror image folding. This new model first completes a 20-sphere tetrahedron and then starts a new jacket of 100 around that completing a 120-sphere tetrahedron.
Thu Nov 17 17:34:52 UTC 2011
Couple of years ago Jess noticed similarity between the tetrahedral numbers 1,4,10,20,35,56,84,120...and alkaline earth atomic numbers: Z=4,12,20,38,56,88,120.____ Few years before that, in 2007, I published on internet my ADOMAH Periodic Table. That web site also includes page on tetrahedral character of the periodic table and the system of the quantum numbers n, l, ml and ms that shows my tetrahedral constructions using spheres of two colors: red and green. I built my tetrahedron alternating red and green spheres so that red spheres comprise always odd rows and green comprise even rows.____I came up with three dimensional Aufbau diagram that shows not only n and l, but also ml. Later I noticed that such two-colored tetrahedron can be represented by suppressing even numbers in the second sequence of the Pascal, that is instead natural numbers 1,2,3,4,5,6,7... I would write down 1,0,3,0,5,0,7,0,.... in the second sequence. Then using same Pascal procedure as described above I got 1,1,4,4,9,9,16,16..., instead of triangular numbers. Those numbers represent number of red spheres in triangular layers of the tetrahedron, if multiplied by 2 they also represent lengths of periods in Janet's LSPT: 2,2,8,8,18,18,32,32.____ Next row, corresponding to the tetrahedral numbers would be 1,2,6,10,19,28,44,60 - number of red only spheres in the tetrahedrons. If multiplied by two they will give you alkaline earth atomic numbers (including He): 2,4,12,20,38,56,88,120.___ The question is why to multiply by two? Because each element is represented by an electron-proton pair and electrons, as all particles of matter, have spin s=1/2, us was confirmed by Paul Dirac in 1928. Therefore, each red sphere in my tetrahedrons represents a pair of elements, not one element. This is where I and Jess disagree. Jess decided that it does not look right and started to search for another system where each sphere would represent one element and there would be no green, as he calls them, "empty" spheres. I argue that my tetrahedral system represents nature more accurately by pairing the elements to account for 1/2 spin. The green spheres in my system represent atoms of antimatter that could exist, but are not with us because of apparent CPT symmetry violation.____ I hope this will give you good head start and you would go to www.PerfectPeriodicTable.com/novelty for more information.
Thu Nov 17 17:32:48 UTC 2011
Explanation. Post 1 of 3.
As promised, I would like to explain you briefly the idea behind the statement that I made earlier: "Sequences of the atomic numbers of the elements in the periodic table can be derived mathematically without any regard to the properties of the elements."___I would like to start with Pascal Triangle that was known to ancients. First, write down the row of 1's like this: 1,1,1,1,1,1,1,1,...This represents zero dimensional space mathematically. Then start the second row with 1 also. To get the second member of second row, add number above it (that is 1) to the first member of the second row (also 1). You will get 2. Repeat this to find the third member: add 1 that is above to the 2 that is on the left and you will get 3. Repeat this procedure to get 4,5,6. So the second row will be natural number sequence: 1,2,3,4,5,6,7..... Mathematically this represents single dimension. ___Start third row with 1. The second number of the third row will be derived by adding number above to the number on the left: 2+1=3. The third number of the third row will be: 3+1=4 and so on. The third row is the sequence of so-called triangular numbers: 1,3,6,10,15,21,28,36. Mathematically they represent two- dimensional space.___ The reason why they are called triangular is because if you try to form triangles using coins, for example, you will get 1 coin for the basic triangle, 3 coins per triangle, 6 coins per triangle, 10 coins, 15, 21......____ Now, start forth row of the numbers with 1. To get the second member of the forth row add number from the third row above to the number on the left: 3+1=4. To get the third member of the forth row again add number above to the number on the left: 6+4=10. By doing that you will generate sequence of so-called Tetrahedral Numbers: 1,4,10,20,35,56,84,120. This sequence mathematically represents 3D space. You can come up with such sequence if you build tetrahedrons using marbles for example. One marble represents basic tetrahedron, then you can build another tetrahedron with 3 marbles in base layer and one marble on top, then you can add three marbles to the base layer, two marbles to the second layer and one marble on top. You will get the tetrahedron with 10 marbles. And so on to get the tetrahedrons with 20,35,56,84,120 marbles.__ (to be continued)
Thu Nov 17 17:30:19 UTC 2011
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