Quadratic Primal Space (Conjectures)
I am not sure the few things I looked at last night are anything more than some vague conjectures. That said, regardless of interpretations I found some interesting graphs and ideas in relation to numbers. I think I am near to understanding Pitkanen's vision and the reply to me is below. I understand they rarity of such Mersene primes and that it is not clear they continue indefinitely.
I was more interested in the detail in the short run and lower dimensions and numbers. But with this reply of a topology nature I do not feel my speculations are that strange to think about- yet, to me it feels more vague and complicated than the last post with orthogonal space abstract motions. Of course I am examining the groups involving knight moves- so I can say there is at least one thing original here or rather that I found years before it published in the Journal of Recreational Mathematics. It involved placing the queens on a chessboard and before I made one it was not considered by the chess players at least that one could be in a corner. The article there wondered why no one had noticed these groups before. Perhaps I if nothing else have an answer for that- it was beyond the scope of our more general new physics.
Essentially, I regard the places in a grid as isolated or connected to other places or cells. That is in a quasic cell we can interpret it as a torus which can be self contained or part of a larger group of them. A singularity or wildcard cell in space is not clearly one or the other. In the same location we can have variations that either isolate the knight like moves or the queen like moves. How we separate them for various descriptions (for example alternative dual tessellations) is guaranteed by this primitive concept on a deeper level than any application or explanation.
In the illustration I have a grid of 5 x 5 cells which is colored by 5n+4 knight moves also. (I have other drawings to copy to paint with the other patterns that are after all part of these Recondite systems).
Clearly some form of topology and n-dimensional geometry is in play here.
The pairs of primes seem to be involved within a recursive binary square region. In that sense they are close to the golden ratio. In this golden ratio we find the sum of the 11th Fibonacci number related to 89 where the decimal sums them. We note also that 11 repeats the sequences of the binary representation of the ratio.
I have used the knight moves- and recall that 11 is some sort of limit to the coloring of squares and also 10 was a big deal on the cover of Scientific American in the coloring once for those sorts of squares- to match the axes in spaces of so many dimensions for the purposes of coloring various dice.
I questioned my assumptions last night- five colors for two space, seven for three space, but maybe 11 instead of 9 for four space. There may be something we do not see there.
But the question of quadratic angles in at least Euclidean space is important. I make the conjecture then that- in two space where the square root of two is involved that in three space it has to involve the cube root, and in four space the fourth root (but why not the fifth root?)
In any case in 4D the essential ratio is the quadratic involving the square root of five or the golden ratio. In the 5D we expect then the square root of 11 as the ratio. Now, considering Ramanujan have we reached the end to such prime patterns of numbers?
In an isolated group of so many queens across the vertical and horizontal axes of the quasic grid the patterns there read in pairs or if we make the recondite space four fold "extending it beyond the ideas of a barrier or boundary at the edges where these tori can so merge but perhaps to change the sign as we can read such pairs we have a sort of symmetric pattern arise akin to the vectors of the 24 group as co-variant, contra-variant and mixed. What moreover in a closed torus is not a symmetric pattern in the directions of the knight moves involving the square root of two or in three space such division as the square roots of three and so on in tessellations of subdivisions of a cube, when it is open within a tori cell region we do find symmetric patterns along the quasic diagonal axis of the grid. It seems permissible therefore to collect negative or positive numbers freely beyond the boundaries if we do it with intelligible local arithmetic.
Sorry I did not have time to make this difficult presentation clearer even when it contains deceptively simple words. I may come back here again with words and pictures.
Interestingly from the Recondite Systems approach 11 x 8 +1 is of course 89 and that is 11n+6 two groups of knight moves x the recondite four.
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12:03 AM, Blogger Matti Pitkanen said...
TGD based explanation of family replication phenomenon does not relate to space-time dimension but the identification of particles as 2-D wormhole throats. Orientable 2-D surfaces are characterized topologically by genus, the number of handles attached to sphere to get them. Sphere, torus, sphere with two handles are very special from the point of view of conformal invariance in that they allow global Z_2 conformal symmetry and this explains why three lowest genera are light/can be interpreted as elementary particles.
This is one of the signature of TGD and the prediction is flavor changing neutral currents for which recent anomaly gives support.
What is essential that prime 89 defines Mersenne prime: M_89 = 2^89 -1. Mersenne primes are very rare and it turned out that the most important p-adic primes correspond to Mersenne primes.
I can look your posting later. Just now I have a little bit hurry!
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Number Theory 101 -
Before I casually looked for such patterns I considered an overview of what number theory was- for example, I recall reading that 36 was related to 666 which was a triangular number. From the quadratic treatment and things like completing the squares I understood this, internalized the theorem. I did not arrive there by the usual methods but in doing so I find a great sense of satisfaction as the ability to see such things and yes, a great sense of beauty. I begin to wonder if in general and for the masses we have not conveyed this sense of arithmetic to young students (in fact news yesterday said that the boys were falling behind as they lose the math abilities- and that this was not so long ago true of the ladies. The segment showed women in the universities complaining of a shortage of males, a 40 to 60 ratio at least in the USA. The future critical to have all our citizens upgraded to the sciences and math. Oddly, the show still said that where the women find a job they are paid still 78% of their male counterparts.) But in understanding this simple number theory I realize that in my side hobby of tutoring algebra if I had known things a little better this way I would have made a much better teacher.
In considering the role of 7 in patterns in relation to the powers of 2 and 4 (in general where I have mentioned some numbers like primes in such sequences my posts were not meant to be an exhaustive enumeration- that is in such numbers we simply say 2^N + or - k, where N is a binary power such as the Mersene Primes.) I note first that 15 + 21 = 36. And that 15^2 + 21^2 = 666. 36^2 - 21^2 - 25^2 = 630, and
630 + 36 = 666. I looked a little at the complete square of 17, that is 136 + 120 = 256. Of course 36 x 37 / 2 is 666. Note: the first seven primes squared (an independent discovery) summed = 666.
But in my conjectural mode I thought about how this seems to relate, at a vague distance to the Pythagorean theorem. In particular, Fermat's last theorem to which I am convinced when he wrote the word cube in the margins he was talking geometry and not just the algebra of it. So, fancifully, I imagined squares that sum to a triangular number like 666 or better cubes that sum to a tetrahedron of sorts- maybe this direction of exploration could tell us some things- not maybe a simple proof but the state of mind Fermat was in in relation to these number ideas to which Gauss gave a more professional treatment. After all the shape between the legs of such triangles can be a triangle or most anything.
In thinking of the location or pattern of cells as isolated it reminds me of the issue of what is discrete and what is continuous and how we approach these problems. Certainly, once we have the intuitionist discrete down we again, on firmer foundations will see the more general beauty of the mathematics of the infinite and continuous. (But what does Pitkanen mean by infinite primes? If he means a hierarchy of Mersene primes and that is in question then I would suggest that in the general pattern of things there is a sort of logic to the thought that in some cases these can be argued as having a limit and in some not and both systems be applicable to topology. This a sort of inverse idea to if topological shapes can be opened or closed or not so as to said to be open or isolated, discrete or continuous systems. But I do not object at all with the term infinite primes nor as used by Rowlands a sort of interitial or infinite motion.
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Well, I almost just posted pictures perhaps worth the thousand words- some more here shortly to which the imagination of the viewer may supply the commentary:
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I should have related the Fibonacci primes this way: there is a certain asymmetry in the Fibonacci number and if it is prime as to what can show the case or not. In a sense the application of these binary powers as such an asymmetry. As I understand it the asymmetry idea is important in the loop like theories, that is nature can be seen as fundamentally asymmetric in the dust if not the shadows. Of this I agree but not that it implies something like the forms of an inflation universe. The CPT can as Rowlands does be mapped to the issues of Dirac algebras- but I feel we have to take a further leap to actually define what we mean by chirality and negativity and so on. The idea of the symmetries involve here or not is as ill defined as of information and dimension save for the concrete working definition on some level of enquiry. If we find broken symmetry much of it can be explained on the generational level and not just the properties of Fibonacci numbers. If this is the case it is a little more complex than we see if we rigidly separate the wider ideas of dimension strictly. As these issues are dynamic in the larger view to need constants of adjustment or even those we imagine can vary we should have a wider view of what is discrete. Certainly we can imagine small differences in the mirror things which are reflected on the molecular level of things that fits well in quantum asymmetry. But can we not imagine an idea anti-particle that does not distinguish its right or left and that the world in application of structure so makes the distinction? Can we not imagine analogs to this sort of duality and still the asymmetry applies as the background of things on all intelligible scales and levels? Not that we multiply the applications of the right and left as much as generalize the whole concept in what appears in the near notions as a hierarchy of such discernible's.
With the quasic background (and that also interestingly touches on the surreal number multiplication) the various loop quantum like ideas seem to me at the moment to be the most likely of the total vision game of physics in town. But where it may be eventually surpassed as error these are still intuitive matters and interpretative matters of philosophy. The anti-particles, even the anti-neutral ones may in a sense double again what is the actual and the shadow images. Can we have a multiverse that made of strings in some sense implies recondite anti-strings?
If erect a theory on partial facts can we expect it to be a total theory? The business then does require a stance towards such problems or their solutions.
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http://neutel11.wordpress.com/2011/03/05/m-d-sheppeard-cpt-violation-and-the-minos-experiment/
I post this most excellent dialog for reference. Now just what is a general co-homogeneity theory hmmmmmm, part of the unified picture still longed for I imagine?:
All you scientist at the philosophychatforum com and some who work for Fermilab take note of this article if you come here from your new facebook presence. And Marshall, thank you for thinking my recreations interesting and scientific enough to post there early on- I begin to see your excitement with the quantum loop and such ideas.
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