Tuesday, January 11, 2011

Abstract and Concrete Knot Notation

Abstract and Concrete Knot Notation

Back when the universe consisted only of our galaxy as an island in empty space, and Einstein dreamed, this sufficient in the entropy for the complexity of life to arise, one could travel to the far reaches of the universe at the speed of light and return with weird things happening to time, and somewhere where the transition occurs the left and right hand of things were interchanged.

I hold some paper on one side it says the statement on the other side of this paper is false. Turning it over it says the statement on the other side of this paper is true. But the thing is that with a wide enough overview we can see that the paper in question is on a strip which has only one side.

What can it mean if in a fractal all the cells in a flat space are covered in a knights tour or at right angles? That all of math can be modeled by something like the game of life with moving pixels that evolve. All the right angle case means is that given in some natural dimension a space coordinate we can find a certain sense of higher coordinate axes, a sort of abstract sense of perpendicularity by which nature also seems to arrange or derive order in the foamy chaos, even if in our insistence on reductionist interpretations we turn the white noise down and find a few stray photons in the darkness of the television screen.

Our equations, like some metaphor of a poem, may be descriptive, contain meaning and distilled information, a higher language of sorts to contain an idea of which it take training to speak, of which we may gloss over the symmetry that keeps the soul of words in the background, the vowels abstractly implied, subconscious mystery felt and embodied. Children also learn to speak as do adults a new language as well as deal with developmental design logic and theoretical systems.

We cannot say the situation, as when we see in complex space what amounts to a mirror object unto some dimension, especially 3, 4, and 5 of interest to our current era of exploration. This global situation may be asymmetric. Yet sometimes what we think as asymmetric on some intimate and immediate dimension proves not absolutely so. Whence the grounding and difference we may state in descriptive words and symbols that only loosely corresponds to the intelligibility of number and geometric regular forms?

Lampion 01-11-11a In the process of algebraic operations, the powers and roots, the information in the linear encoding contains all the information even if it is but one factored part of the space in question. Yet, in the smashing or reduction of such roots one can be far away from lost details in the changing of dimensions if our theories are not more complete.

Lampion 01-11-11b Such is the way of describing Knots as links and crossovers. For in a knot as a linear string in its natural dimension of three space we can early on interweave it such that from the view of space what seems is discerned as right and left handed senses of the knot viewed globally, there is no such distinction.

Lampion 01-11-11c Alternatively, we can imagine a one sided polyhedron of three intersecting squares as if half of the volume of a octahedron. Yet it only takes a rotation of 90 degrees to show that these inverted objects as chiral are the same.

Lampion 01-11-11d So we are dealing with regions of space in which we decide saying it is simply connected or not, that it contains a center, or some form of singularity. The convex mirror of such a sphere, and imaginary sphere, may in the abstract contain no such center while in the concrete the real sphere does. We can duplicate the abstract space, divide and reconstruct it to a volume greater than the original real sphere- but to do this in the real sphere may have structural restraints.

Lampion 01-11-11e In this sense, Peter Rowlands applies the orders of tetrahedra to describe certain symmetry which he applies to particles and DNA and to get the full 64 with intersperse two such, each without a real singularity center, to make the metatron stella octanga and from that derive the pentads of Dirac. But by he lampions above we might, in this twistor mini-revolution, Imagine the stella octanga alone can discover such number relations- that is an octahedron equals four tetrahedra so the total of the second order one is 21, 5 tets and 4 octs.

Lampion 01-11-11f Now, in this rarefied space that takes into account also the differences of dimensions intelligibly (a growing more so perhaps) we find the reduction into lower space of the five fold symmetry polyhedral objects where the ratio of the axes are golden reduced or as ovals smashed in such a way that the result is surprisingly a skewed four fold or square root of two symmetry. This four fold symmetry moreover may be grounded in four quasic grid quadrants or be part of the centered shell structure of all the quadrants for a cell like replication and multiplication of the original real polyhedra.

I now post my writing on this of last night where I ponder how one might label the topological invariants and loops with four things taken three at a time to describe just what from these abstract structures we can find intelligible codon correspondence.

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Counting Codons as Polyhedra

Just as with number theory some problems can be stated simply but are elusive in the proof, we can have that with some geometric concepts as intelligible too. Some things are only simple in retrospect after solutions.

I find Peter Rowlands' suggestion for the labeling of triplet faces on octagons a rather hard problem that I feel an unlikely answer but cannot dismiss off hand as easily as the details of how to solve it- even if I disagree with the general theory as to its metaphysical foundations as physics. But I too after all may be only offering suggests how to solve it rather than the solution to this problem.

Now my last post the number 32 comes up in the counting of things in the quasic field. I note also that in a wider sense other fractal curves do not fill what we suppose is all of space and its point lattices and grid(could two such fractal curves of different styles do this?). 4pi x 1/4 pi is the all important pi^2. Let us not get lost in the equation reductionism anymore than assume we can casually duplicate and interlace space in a complex number manner and it not have intelligible structure and counting as exact or perhaps arbitrary part of the big picture.

I considered my genetic chain code compass as 32 and 32 perhaps to find 2 or 4 paths for the codon base labeling of the metatron structure. (If DNA is read fractally what can it mean if molecular chirality is not present in 6D as Rowlands points out- a useful principle and observation or a brick wall bottleneck? I also note that 8x4=32 and 16x4=64 in the count of tetrahedra possible and base things on 24 from one space arrangement as 24 rather than 20 and apply the circles of chords as if the 12 tone music. (The classification of melody also solved as fractal distance I presume is in quasic space where the proximity seems more pleasing to our hearing.)

Still, another idea- take the 8 tetracubes and each tet with 4 base colors, see how these may combine into higher dimensional fractal like shapes (and of course the obvious cosmological principle analog of Crick and Watson invoked fir replication and doubling) In fact this fractal and orthogonal path of three of 4 axes LHC* which
by the way requires 21 in the 64 quasic cells for a triangular basis where some of them are not strictly in real space [ I may make this clearer later or see earlier postings on line ] which are in that space for the path realization and choice of real directions in the same dimensional real and abstract motion.

Of course a symmetrical shape, or an intelligible number count, suggests relationships and give conclusions. But arithmetic and geometry are higher level codes than the grounding informational "physics". Yet sometimes that information can intuitively be there and seen from the higher (description metaphor) level - and solutions can be sorted out there as well as theoretical models. But sometimes informational coordinates, arrayed in general spatial patterns, can see the abstract details of form with x-ray vision.

Alternatively, this semi regular 3D tiling with four color dimensions may form a general code space without local restraints and choices of labeling, but an adaptive and variable expression of such abstract motion. Let us call this Rowland code space and let us generalize it with restraints to the many concepts of dimension.
[I forgot to post this earlier due to interruption. What sort of fractal three D tiling did I have in mind: well, that fractal, orthogonal through a lattice of cubes such that 8x8x4 for example makes the 32 tetracubes- which I assume will be 8 and each of its tetrahedron (roughly what a tetracube can be seen as- the four colors in their various permutations even in each tetracube of which the whole can make analogous shapes of the individual ones to add a certain logic to abstract motion in three space with still a wider freedom of arbritrary collections of assemblies that can be contiguous of these tets. Thus, we see a mapping of 32 and presumbly another 32 for a cube of sixty for... In a sense then the gene code can map into this space and the paths traced with respect to three and an hidden dimension... This seems to suggest another way to do it as I will show in my next post Kea Explores Triality. I had intended to add an illustration of such a compex of tetracubes.]

Where such complex algebra applies it does so best for knot theory. But clearly we need better definitions of the scientific and philosophic terms such as: dimension, information, number (arithmetic), null, fractal, compactification, conformal folding, transcendental, and chirality for a start. And a clear grounding as to what we mean by symmetry breaking in all its forms such as the lampion on ontological symmetry breaking in the same natural dimensional level. What is the within and without of things, and yes in the all important differences or not of our deep notions of continuity, contiguity, and consecutive discreteness and where they mix- time's arrow and causality itself, and yes entropy... and maybe as we try to live in the world together what are our individual and society's expectations.

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