Tuesday, January 10, 2012
Lexicographic Stringy Trivial Sudoku Triality
Lexicographic Stringy Trivial Sudoku Triality L. Edgar Otto Jan. 10, 2012
You would think that the idea as in the names of numbers is rather constant- but numbers have shifted a lot for me from being uncertain things to living things, boring things, and ultimately mysterious things. It occurred to me to read the Sudoku sets of three numbers as if a literal decimal number. For example 153 does not matter if we shift the digits and take the cube and sum, we get 153. This reminds me of the first nuclear idea of strings of which it could not be determined which of three quarks were in the center of a string segment of two quark ends.
The same thing applies to our averaging or mixing say the masses of some particles over their generations and all such phase averaging or mixing ideas. The triviality then is the asymmetry involved in the values to be (quasically) explained. Of course such asymmetry is most likely fundamental in nature if not ultimately so.
Well, whatever my sense of integer numbers they danced around a bit as I thought about them- and I chose to use the calculator for a sense of accurate accounting. Thus the permutations of these Sudoku games in a row of cells of them 3x3 summed to 4995.
Now there may be properties of numbers and formulas of which I am not aware making my education in them rather naive (what is casting out nines?) but in this dancing some rather interesting numbers and numerology came out usually involving group theory values- (years ago in the search for those with theories like mine which may have shown me I am just reinventing the wheel and a rather stone one to ride on at that- I only saw Clifford's algebra as close to my system- but alas in a different way as to how I understood dimension and what is finite or not.) One thing for sure these Sukodu concepts in the low dimensions (composite theory again- and after all 10 is a composite number) helps by the nonagon diagram of division above explain why in such systems of space taken literally some asymmetry and discreteness can arise.
4995-1152 =3840 + 3 = 32 x 120 +3 ; 6^3 = 216, 216^2 = 12^3 x 3^3 = 46656 ;
405 = 45 (the sum of 1 to 9) x 111 (7 in binary), and 111 = 3 x 37; 4559 /4 x 1152 = 384 +3; 27 x 192 eventually reduces to 10; The factors 5 x 27 x 37 as 2^5D times 3
expands to (3 x 5)x(3^4)x (3 x 37) = 2^8D. (9^3)^3 = 19683^2... In a sense the divisions and permutations here involve at least elliptic space curve equivalents.
I also looked to find higher analogs for the Sukodu but had little luck or time, things like 5 by 5 ones (looking on google I see only 4 by 4 ones and some other unusual ones considered but not 5 by 5 or any other odd number considerations. Of course 5 x 81 is 405.
Sudoku has a vast internet literature and some interesting papers that use it. But for me the odd numbers I intuit as important in these permutations, 7 applies to three space, the diatonic scale, and 9 to four space (we imagine the natural dimension as doubled and add one, thus 15 natural dimensions would have 31 lexicographic numbers in its parallels of planes and lines etc... These moreover are knight functions and do relate to the idea of the Greaco-latin squares and thus the restrictions of such freedom in the lower dimensions- or if you will the global restrictions of the so called primacy of string theory. (So Lubos we stand again as Godel and Cohen- but take this as humorous in the spirit you posted on this- with our roles perhaps reversed, that we cannot prove or disprove string theory either! for now anyway just as with the continuum hypothesis in its variations.)
Interestingly, Lubos motl's programmed sudoku example (and we can find very many done by brute force to exhaustion by the computers by algorithms on line I saw some of them today) appears close to the top of google image search, in top ten or so in relation to minimum hint solutions. (I should look at this too for it may be even lower I sense by other methods, after all seeing from four space is more powerful) But in matters of Clifford algebra in a search for the same things listed as searched for topics on this blog that found me, I am not on the first page of the gods, some of the hopeful and suffering god, of theory, but the second page where perhaps I am most everywhere- a matter I suspect of the terminology or some human judgement as to the relevance of my topics. Yes, the gods of theory live high up on the food chain from the working physicists and students- they control the lightning so approach them with respect and caution. Laugh only at their came to late but publish or perish papers. It may be just a matter of prestige for China to reach the moon- but at least they are doing something that will give the world benefits- at least they are doing something.
So, as Lubos says, Sudoku was invented in the Czech republic by the king and taken up by the Japanese- I am trying to show why the matrix I posted yesterday was closely related to the one he gave (considering all permutations and sub-permutations) but I am not posting the method explicitly although the clues are certainly here, without some formal dialog. Let us hope the town and gown poets and gods are not wasting each others and their own time when there is much to be done!
It is clear also that my recreations with n-ominos and the 15 cubes of 6 colors prepared me for some rapid understanding of what can be done with a freer definition of space and numbers. But I had no idea it was relevant to physics at the time or it was such a very hard problem before the fact. For economic reasons my technology has been the paper and pen- somewhere along the way Conway and I parted company on the use of computers over physical models- each has advantages and now I feel we are pretty much talking about the same thing and methods when it comes to unified physics.
Last night after posting I watched a taxi-driver poet soon to teach at the tech here work his Sukodu in the newspaper and he answered several questions including as to why the game fascinated him- but I also watch him play his crosswords. Such games, like the classical puzzle of many parts and the same shape I tend to find rather excessively boring. If that means anything.
* * *
This an interesting arrangement. Note that the number of cells in this array is 153. from Daniel here
* * *
I should add for those of you enamored of the number 160 in the theoretical particle physics world... if we see this interesting configuration, with four fold wings, there are five more cells of 9 numbers between them... in this respect we add another 9 doubled or superimposed in the center for a total of 153 + 9 for 162 which of course is a doubling of all the cells in the triality sense that it = 2x 81.
I can hardly wait to take this game to the next level to see what is there in the doubling and higher generations of the irreducible's and field expansions so to speak. I hope others share my enthusiasm. of the four sets of 9 cells of 9 this adds to 108 which many have computed to be the rotations of the hypercube counting from reflections from a corner- but is it not half of 6^3? For me there is still a little more possible than the composite concepts of particles, I hope, worth exploring. But is this not somehow a unity of groups involving sums of cubes and the unexpected patterns that share symmetry and asymmetry on generational levels?
But Kea said we need to find the lower trialities first... and thank you Lubos for the link and predicted values of such particles. Certainly I see that our views of supersymmetry may have to be reconciled for a deeper understanding of number and nature.
* * * * *