L.. Edgar Otto
01 May, 2012
That which came before and will so come does so now and never did in that edge of the world made all of edge...
As I ended my last post for April this sentence strikes me as simple but more philosophical and intuitive than the geometry I find more concrete after reading tonight a link on newscientist.com concerning a problem of space solved by fractal like considerations which John Nash considered.
Let us not dismiss the wisdom of John Nash despite his diagnosis as what one would be called, well, "crackpot" seems to be the blog term these days, nor that he was a rather angry or mean person as reported by the son of his doctor.
On the news a couple of days ago the mini cupcakes and other small snacks was said to have become very popular in this calorie conscious society, It mentioned also that the constant snacking of the smaller things may be more than a large portion- the example as a little boy who while he did not have a doughnut he filled up nevertheless on whole packages of doughnut holes.
Still, if one takes this as a model for the universe, with due consideration for other principles I have set down, or for the structure of a particle, or issues of symmetry breaking alternatives and so on, one can see quite further than if someone will eventually consider this as a model as I have developed it herein.
This is the simple idea that a square region of space, I have called a quasic pixel or even a brane in a sense a quasifinite entity, also a dihedron at the zero point in such polyhedral kaleidoscope graphs We know from Coxeter that such tori are flat despite how they are described. I know also these need not be orthogonal but as the count of the sub pixels in my illustration above these are so, and the dynamic between the triangular figures and the orthogonal are of course important as a general principle. Note also that the description of such a square (or circle) is generally of some indefinite size or extent by Cantor. What the article point out is that if we take such a surface literally the vertical or the horizontal direction are subject to the same type of problems we have as in the spherical mapping into a plane, Riemann et al.
Such a difference may lead to divergent theories if taken literally this way (while the article suggest 3D printing of such objects they still cannot imagine this as a 4D object that would require better understanding of such printing in higher space.). Let us recall the view of things can be finite in the application of group theory counting that we agree corresponds to the continuous groups. How else could we imagine the Feynman diagram as rotated thru a right angle or even more some generalization of the diagram?
In the gene code, and the structures of things like the Dirac operators in general, we focus on the four elements in the matrix component. For example in quasic space, the article suggests is fractal like although far from generalized to higher dimensions as visualizing space proves rather difficult among such dimensions, four base as in the gene code GAUC can logically be represented as the point of the self dual simplest Platonic polyhedron, the tetrahedron. The point being I have a better way to state my case within this new topological model. In some ways we can generalize also as in the visions of Pitkanen upon such generalizations from the continuous approach of which the looping of such systems can attach to the physical environs as the finite region is after all capable of total description and connections in the ideal as one organism and the dynamics between such connections.
In the illustration above I take four regions of symmetry (just as I am growing tired of this way of drawing that seems trivial and pointless or boring after awhile) so to put them on my screen saver that they connect as if a torus spaces. These can be treated as separate or distinct regions in the count as I did with the notation using the 9 men Morris game, the connections I imagine as something akin to Pitkanen's space of generalized wormholes and mouths, at least in the abstract- this issue too of the tree plus one or two plus two formalism.
Clearly we have in the basic 4 by 4 grid six of the cells as if electromagnetic and ten as at rest in Einstein's sense of this magic square like matrix, in which case to extend the boundaries symmetrically that a torus is presented should they tile and match in the span we divide the sub regions into four sets of ten colors for forty regions and within it the 24 as if a hole in the center. This of course as if a antiorthogonal matrix comes from the permutation of the complimentary colors from the six of the sixteen, the four things taken three at a time. 40 by the way the close packing of five dimensional spheres or as not a closed matrix the twenty and so on. I style these the anti-Conway matrix. Note I used the #rrggbb reductions for the color safe colors for the sets of the tens with inverse and so on all very intelligible in the geometrical (finite and Euclidean counting).
Quasicity important then in the application of groups for such generalizations that now can be expressed as such tori at least in three and four dimensions.
With these principles and view of such supersymmetries in mind I wonder what those who will start to consider this geometric object make of ideas of the compacted six dimensional spaces. One simply cannot unify the sub vectors without at each point of a 64 Clifford like dimensional space without the information of four coordinates axes. Now this is a simple low dimensional structure hard to see but it is not that much harder to imagine say active fractal sequences that interlace for the explicit generalization. We may also ask from the quasifinite view if the expansion beyond the eight arms of their diagram relates to natural dimensions, that is if the complexity of spaces or manifolds increase as we peel the layers of atoms to finer and finer region of the center as evidently in quantum theory the values jump wildly. In the symmetry breaking also we relate the five regions in the small for the Tarski division and reassembly of spheres.
Lastly we imagine the dualism in the five levels of Coxerters polytope delta n which is a plane or brane of the successive symmetries five layered. But on any actual plane where things do not necessarily connect as one universe locally it is clear that there is some disorder in possible actual design in this sense hidden and condensed rather than compactified. I feel it a mistake or error of intuition even with this model to rigidly distinguish the count of universes as one or many as this is both a quasic issue of similarity and of higher than four hidden dimensions and symmetries even in one brane of which superimposition is a given, and signals or relations between cells where the sense is non local and this as if instantaneous we begin with Penrose's quantanglemenet requiring eight and not just for bits- and yes, even sixteen and so on in the higher spaces.
But is this not close to the idea of singularity points as envisioned by postivie only and not generally torus space of the hidden dimensions of compactification? In matters of entropy perhaps we need to develop an analog to qubits we could style quasibits... Surely we can do more with the math we have in these matters but we have to go beyond what we think is the most complicated and only spaces imagined today- configuration, Hilbert and so on... It is not that great a step to generalize a little more in these directions, but theory should show the details and finer definition of such structures as this torus. Let us consider too other types of tori in the quasifinite views such as the volume of a tesseract for example. I wonder if this simplifies genus as applied to knot theory and how things switch handedness at some point out into a looping infinity. Have some not imagined the brane as a super wall or edge of the universe... perhaps in these religious like cosmologies man's achievement in architecture after exploring the Dante like levels of heaven and hell is the taking of the Euclidean plane and standing it on edge as in the Islamic view of Heaven, for a start...
This sort of modeling of which things can move within the boundaries of the region relative to some central position can also be the way to model several of the subsets of general theories and explain their unification and divergences (such a torus universe can be conceived as expanding as if there is central pressure as in inflationary models, or can be thought of as cyclic ones and so on from some view where nature seems to balance such physical phenomena or their analogs. The wider universe as a series of tori can be seen as open or closed paradoxically. But these theoretical ideas on the structure of particles and the cosmos are within the range of human comprehension with just a little more work and careful ordering of these multidimensional thinking intelligibility. Now for the true topologist, what happens if we ask the same thing of the two holed doughnut and so on? What is this saying about space? Let us not underestimate the zero polyhedron of Plato for that is also a breakthrough in culture as would be the discovery or invention of of the number concept. Let us also imagine the mirrors in space, even the relative motion independent in a region like the arrangement of rhombs in a decagon in Penrose tiles is not necessarily relevant to the rest of the plane as a fundamental principle to exclude other physics, nor that such boundaries or edges or shifting of axes necessarily relate to complex imaginary content as if the subspaces as null polytopes of filled vacua is directly a physics of the dark matter ideas.
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[Note: the flow of this post could have shown more continuity of thought or arrangement had not my roommate came home drunk and cooking in the kitchen, and complaining about his life and parents and so on... but I think I got it down only I will have to wait until morning to post it. His is now a very predictable pattern.]
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