Cheshire Grins and Other Generalizations of Euclidean Space

Cheshire Grins and Other Generalizations
of Euclidean Space

Well, maybe this is all we can see of the Higgs particle, the U of the Cheshire cat's smile.
Not quiet the feisty quantum cat with nine lives but a more subtle creature. In any case one might go down the white rabbit hole- time speeds up and the white rabbit races because he is late. maybe it is not that we wonder where dreams and dust come from as if nothing, a black hole of sorts, but from some mysterious source appears the white rabbit hole.

Now, sweet Alicia playing with her little cubes written on them Latin letters, does puzzles made by the old Oxford don logician- wondering what keys he holds in the bulge as he punts along the back of the Granta in Cambridgeshire, his caterpillar blowing smoke rings and letters around hidden truths and hidden dreams in which we are always at times the Mad Hatter celebrating the un-birthdays as the clocks and planets turn, capturing but the promise of a smile in photographic light- Alice does not know yet how to get thru the doors to that kingdom of hearts as she discovers not what he is smoking but how a few mushrooms might change her perception of scale.

Let the kids play while for appearances sake, that there be no scandal as the old Don dates the chancellor's wife and Alice's mother- or fill their heads with questions like looking glass written poems and lesser quantum fires on the other side of the mirror. Or running in place as so many do in dreams challenge them with the fundamental question- How is a raven like a writing desk?

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There is a fundamental principle of Intelligibility to which we explore some grounding theory or speculation- and things are not always what we expect, nor are they clearly a simple solution- or a unique one when all is understood. Of the 1100 or so areas of mathematics the great thing is when we find and understand parallels between our formulations- so to our theories of physics.

The occasion of this post is the deeper understanding- yes, something new to me but surely if we do not at least see an idea or equation how will we be able to comprehend it, even subconsciously as we read rapidly to digest the meaning later?
I now find the p-adic ideas deeper than my superficial reading and understanding thereby, thanks to a link with the ideas of TGD of Matti Pitkanen... one that was already incubating in the foundations of my mathematical pondering s on the basic nature of number itself which are issues in the background anyway to which there are many quotes and ideas held back in the teaching or presentations and publications. But, my approach is from quite a different geometry and the link seems to me not a link at some foundational place- but a link, much like the ideas behind some of the lecturers in theoretical physics I have encountered here lately- that have not accepted Pitkanen's deeper ideas, probably because like me I cannot risk the step into a purely mind description of reality without a little more data to prove it.

Of course I am not sure this theory does what I think it does from what little I have recently seen of it- although I have played in the background with these concepts and at times felt rather out of my league and my intuitions leaning more to the realm of fantasy or science fiction. Even with these links and understandings these ideas are in some ways still in their infancy, and are not sufficient alone in the complexity of merging new theories as the world's theorists seem to finally be going back to the drawing board. Even so, left to our own areas of research in which it seems few have come to recently, there are some deeper questions to be asked and resolved.

In fact, these speculations came by a small fortunate accident of drawing some grids and finding the picture and count a little off. Perhaps, this was the most rapid of transitions to my uncertainty about a theory to the sense it is a foregone conclusion to the discovery that most of the work done by others decades ago then to the realization that we in the present time are still ahead of the others in our understanding.

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The big question seems to be, and this a concern of p-adic theory, that there is in a sense a finite concern with things such as the diagonals of some squares and rectangles. I wonder then if p-adic notions for a calculus is the same as that for surreal calculus- I mean in a sense and as a grounding issue we ask if the square root of two can be seen as a rational number? But is the conclusion than the diagonal is equal to the edge of a square as a proof an error of what is the transfinite number of things? Would those who imagine gluons arising in strings, or some limit to the number of patterns of singularities not be making this error or in a way confirming some of the maths offered in the blogs, such as braiding theory and p-adics (let alone my quasic interpretations which connect many ways to those far better trained than I)?

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Let me say that the Ulam square of factored numbers is that which we mysteriously see something there, and he has put it into spirals of factoring things.

Is the p-adic physics, especially in relation to Gaussian and other primes of that nature, the same as the idea of congruence by Whitehead as a further generalization of the Euclidean plane- something that for awhile shed light on relativity theory?

Spirals, well, if we take the Ulam square we can extend it of course in the series of rectangles through higher dimensions across the diagonals, the square root of that number as if the spiral of such numbers made long ago.

We know that what evolves the galactic arms (from recent sci mag articles) is not as simple as say the theory of Lagrange and his probabilistic treatment that bypassed Newtons concerns that perhaps God sets the planets in all the same motion. In a sense we may ask of the particle world also what is the analog to this concept in the particle physics. Apparently statistical methods are not deep enough to explain things to the satisfaction of the reductionists. Thus, my long held cherished belief that there is a viewpoint I called the chiral theorem that explains the evolving structure of galaxies is also not good enough- it follows that the ideas of chirality, anti-matter and so on, to explain mass (as in the muons and weak force concepts of Rowlands) is not deep enough at the foundations. In fact this question of why matter and not more anti-matter is a big one in the sense that perhaps anti-matter exists like recent articles for an anti-helium nucleus- but this will be found as we set our experiments up to find it narrowing other areas and methods- I see no justification other than this design principle to assume that we get back to conditions of the big bang where such things were assumed symmetrically balanced.

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Consider then, the lattices of normal three space that are close to other objects and objects like the logical ones decomposing in many ways some plane, squares, matrices... for example: in a square of say 64 cubes these are equal to the volume of 8^3 tetrahedra. But seen as analogs in a cube to the 4D polytope as an octahedron we obvious count 64 x 16 or 1024 tetrahedra.

One interesting thing about this space is that we can imagine a labeling of the points in a cubic array and the coloring of them... but it would take seven colors not the five given us in the case of a plane in two space. It would be interesting to develop a higher arithmetical analog of say 10^3 with ten colors, and so on.

In Lewis Carroll's logic diagrams and his algebraic forms of them not all can be a square you see, some times there is the 8 fold problem or the odd dimensions to the powers of two... 32 128 512 and so on to which just doubling the regions does not seem as elegant in a plane as in at least a three space formulation.

Also, with the calyptic idea mentioned earlier there should be some rather ghostly particles (quasic grids to find particle resonances and hierarchies btw as they did in 95 faster than a computer could for my string theorist friend) I am not sure these do not exist in the literature (or perhaps the intuitive genius of Kea) but it does amount so some sort of extension or generalization of the Hessian polytopes and it was expected because of this even odd problem in the coloring of my quilt patterns of which there are 36 in calyptic cubes in which we make fundamental choices of patterns that if we counted them only would be in the same binary power.

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This relevant comment to Ulla on Pitkanen's blog:

Ulla,

Clearly when we view things from this p-adic physics of numbers and extend the spirals of factoring in some direction (where it is still important to look more deeply at the arrow of time as Matti does and some say is just a high level illusion) we find some numbers to high to analyze that can be shown not to be Mersene numbers. In which case the hex symmetry is intrinsic to number, math as superior in descriptions where we at least have no surprises today to current physics.

But I still bet memory in the structured vacuum as if in nothingness or what cannot be seen- in at least these mathematical ideas. Carbon or water it is the same geometry. It must be deeper than your link that imagines superconductive photons in a sort of Higgs condensate and so on...

The PeSla

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and... if we are not cluttering up his blog too much- I am wondering why he cut the value in half for the difference in 107 and 89 square roots- if not by my observation of the sort of odd logical diagrams of Carroll? And I would ask him why going down to 61 is important that it is a lesser number- then again these numbers are not always discovered in order of value- after all what do we do with a number who if seen as a coordinate is but a series of unity in all directions?

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http://zone-reflex.blogspot.com/2011/04/photons-are-superconducting.html

http://ishangobones.com/?p=1058

Reply to Ulla in an e-mail (nothing personal here) that may show some context of our thinking:

http://physics.aps.org/articles/v4/10

Your link is most interesting as is the tone of that you have posted from this author on your zone reflex blog. Wow, you have found the frontiers in which our little planet of thinkers in the blogosphere are still linking and continuing on despite the chaos around us.

Of course, while his attitude towards how we should be doing the sciences is right on, he still does not have the insights some of our bloggers have long developed. But to supply a proof of some of this- a justification perhaps of our concerns- that alone is quite impressive.

PeSla

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In these links we find deep echos of formulas and diagrams as used by Pitkanen and the work of Kea in her blog of updates.

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A couple of links for recent ideas mentioned at a distance in my blogs:
http://www.sciencedaily.com/releases/2011/04/110420111343.htm

http://www.sciencedaily.com/releases/2011/04/110418084005.htm

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Here are some relevant links I found today for these number issues:

http://mathworld.wolfram.com/GaussianPrime.html

http://en.wikipedia.org/wiki/Eisenstein_integer

http://www.newton.dep.anl.gov/newton/askasci/1995/math/MATH036.HTM

http://en.wikipedia.org/wiki/Cubic_reciprocity

http://en.wikipedia.org/wiki/Quartic_reciprocity

(Of course we find the square root of three symmetries in 3 space numbers reduced to the complex, and other factoring planes of many dimensions.)

Finally-

http://www.perfectperiodictable.com/

which discusses the magic numbers including from a quantum perspective for the nucleus- with a tetrahedral map- yet here they did not limit the elements to the 120 they describe as I long have done on this sheet of what makes the hidden and seen structures of such material objects. It is amazing that such things I found early on and rapidly, and it was so hard to connect to the information on it for so many years, perhaps even in academia. So now we need to look back on all such things in light of the new physics.

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