Friday, March 25, 2011
Through the Mirror (and PeSla's Cube)
Through the Mirror (and PeSla's Cube)
Yesterday, I considered some fundamental and general principles concerning the properties of the Golden Section to which I will post more today here. I found an interesting Rubik's Cube puzzle, probably as a consequence of trying to figure out the various steradidan crown and star point designs.
In my illustration of yesterday I posted an earlier photo I called Breakthrough in honor of Pitkanen's statement of one- but I cannot say I find enough in his links to truly understand his applications- and in honor of Kea for I think she too has made a breakthrough- I imagine these sorts of things, like my recreational mathematics cube before physics interpretations, is a matter of higher generalization or a wider world view. For me the exploration into such new realizations at the frontier can be a sort of vertigo while our intensity of concentration and free association in which case one might doubt or feel some solid achievement is made later- in effect we doubt our work and selves especially if circumstances is exhausting and we may not see errors or false paths in the moment. In a sense what is to be unified here when we understand each others abstractions and language is the generalization where it is real of the connections between us as we all are surprised as to on what level we have used but not been aware until the sense of breakthrough of subtle mathematical connections within the methods and notions of our own work. There is a lot here in this theme of mirrors to write- but on the way here I may have got a glimpe by dreaming of just what some of these braid twistor ideas mean as if we have to deal with each others theories in such subtle ways to learn them as we do our own internalization and creativity of the body of learning.
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Synchronously I see that some of the considerations for sketching the Steradian crowns and stars from basic Geometry looks similar to Kea's link today:
http://en.wikipedia.org/wiki/Weierstrass_substitution. In it I am considering the square abstract quasic region, the boundaries for certain 8pt stars from four directions as if (on the mirror) these are actually straight and consequently the quarter edge seems to be divided into three while the quarter circle circumscribed around the square is 4 x pi. In many ways the irrational numbers seem intelligibly concrete and integral if we treat them as geometrical objects. And the tau or phi ratio cube above is certainly such a case.
But let us consider again what happens if we take a 5-cell (four space analog to the tetrahedron with 5 points determining that space) and explode (popcorn vector) it out such that we have one tetrahedron in the center and four on each face. After expanding out from four space into three space and since the center point has the coordinate of w x y z each equal tau, the four tetrahedra on the edges of the central tetrahedron all are Golden tetrahedra. This general concept applies to the analog of the octahedra (and stella octanga) also and recently is popular again as a fundamental object of foundational physics study. Recall that the tau relations as if linear are actually "on the mirror".
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The Pesla cube can be arranged into octants each a Pesla cube and so on. Good luck on designing the mechanism for such a cube- my best guess is some hard shapes with that soft glue to hold them together while turning- a sense of coherency awhile anyway. This model is like the brick or integral idea of the mass of particles and the mortar of fine adjustments between them. This generalization suggests to me that there can be further structures to our concepts of black holes. Black Holes can have black holes within them, at least over a certain ascending or descending generational scale of things into generalized singularity dimensions. Clearly we might have a hierarchy of Planck like volumes to consider on and through the mirror.
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Guess, I am backlogged on ideas so I will just make summary points as the recollections come back to me- forgive duplication if any.
*1 Generally I have regarded physics as a branch of biology- something that seems to go back and forth. With these newer ideas it comes back to physics again- but thinking about this there may be a place or time, or times when these are really the same question.
*1a. Looking at the trees just before the budding of spring I am reminded that the fractal trees are really but an approximation of real trees although these do express growth and energy by Fibonacci growth. I recall also that in the structures of chemistry that the world of Buckyballs is also but an approximation, the 108 or 109.54.54 angles are a little fuzzy, perhaps this difference accounts for some of the aromatic and superconductive properties we observed in such carbon structures which more or less are based on geodesic domes (92 the element limit in three space).
*1b. There seems to be a major distinction between plants and animals
*2 In a sense this is a question of art also. I was checking out the art store and for forms of clay for mould making and in looking at the spectrum of acrylic paints on a finer detail I realized that there is no good reason for buying materials only as good as the student grade. Essentially, without a thin under-painting or with latex gesso, or some pigments like zinc oxide white, in a century paint cracks. I did find the right sized prepared canvass to make paintings for my 1024 x 512 illustrations here so to paint them then photograph them for a series of subjects one vaguely the world of this idea of brading.
*2a This leaves me with a general idea- I wanted a modular and standard spectrum, one that moreover translates well into the ideal of internet color values. Yet it depends so much on the actual physical pigment- and that in turn is best with the human eye and artistic temperament of those who mix the colors. So we compromise with what we find in our reality on a less defined or precise scale. After all in the presence of strong gravitational fields the colors can change.
*2b So in a way our artistic vision of physics in relation to light is the idea of what is this variable and discrete spectrum. We use it to decide what is the stuff of stars for example. Color then is of a higher qualitative dimension than just light and dark, and that over what is substance and space in a quantized hierachy of things. In the other direction the more subjective and mental concepts on which we colorize the world with such human notions, even generalize the feeling to great systems of magic and metaphysics beyond the actual facts, perhaps, of our reality. In the matters of pixels of dark and light we gaze into the cosmic background to make reductionist conclusions about the general process and shape of the universe. And here we discuss the metaphysical fundamentals of something and nothingness. In the other direction it certainly seems that on some threshold of emergence that mind and life can be considered a foregone given and perhaps an ultimate creative foundation.
*3 Considering the abstract structure of the Pesla Cube, Rowland's took up formally the consideration of a Rubik's cube corner twists and related to plus and minus fractional charges. What then does this ability to break the cube into binary octants mean for this idea of fractional charges (and as a generational-creative thing)?
*3a This question of scale as the uncertainty of a constellation of particles when considered as finite is a source of unity as well as the more general diversity of paths of topological structures possible. We note the unit cube as we can see any constellation or class of subparticles as part of own substance or grounding unity of discrete measure.
*3b Clearly with such a hierarchy or fractal like series of such cubes while we may twist the whole we may not do so in the largest binary (quasic) division while it is in a constellation of such cubes (and perhaps the twists and turns of cells within them across the internal peslacube mirrors. In a sense there is no center to this constellation as part of the concrete substance of it all.
*3c We can use the F-of-n abstract quasic motion notation to list the possibilities of what happens if we start with such an octant then place it into a constellation in which its relations between its smallest and greatest tau cubes is relatively coherently fixed. In a sense a unit cube can be in a corner and the tau^6, which here I suggest along the lines of Clifford algebra we have the six dimensional compactified structures of which clearly in such a hierachy and constellation the complexity of such compactified structures is much more general then we have said is too vast yet to pin down as to which applies to the universe's physics. So in relation to the unity the tau^6 can be in three of the motion corners thru a face, three thru and edge, one thru the digonal (a point) and abstractly the tau^3 and unity cube may be superimposed on the same point unto the dimension in question and in a sense at rest.
3d. If we consider the unit cube as the continuum aleph we also then can consider the tau^6 cube as 2^n continuum which is a fundamental idea of the subsets of R somehow greater than itself. In this sense the unit cube is my iota particle.
3e. So at the abstract quasic motion function of zero and a change of four coordinates in four space we have n^3 + 3n which = 4, 14, 36, 76, 140 ... of which 14 reminds me of associahedra vaguely, but in any case these can be divided by two
3f. That tau is involved here shows the slowest irrational expansion in 4D and the idea is that octants are not trivial but are important to the more general coordinate system contrary to the usual of only considering one such octant.
4. In the expansion of the irrationals to a spiral where we erect a right angle length of unity for he series the square roots of the natural numbers which leads to 17 as the ancient Greek knew was a limit or it simply filled up the circle of 360 or so degrees- let us note that the unity to which we erect the two dimensional steradian base of the square root of two is also a root. It is that we exclude this radius unity and the origin in consideration of the 2pi jumps or an iota model of string like things (in my theory) as the beginning discontinuity and singularity.
4a. The jumps between octants if relevant may be useful to describe equally well our ideas of charge, gravity, mass, volume and so on as a generalization method.
4b. In the unit cube, within cubes with a center or not and with the self dual inversion of old structures like Kepler's stella octanga or Plato's metatron cube in consideration again we can envison candles of stacked stars in four space that have spherical symmetry in three space. In general the tau^n defines the natural notion of dimensions as the six colors of the rubiks faces for example are self fixed.
4c. Sometimes the popcorn vector or expansion of what is measurable within when it comes to the without is of a fixed angle in reference to linear tessellation grids so we can observe in say two space these same general principles of rotocenter symmetries and exhaustive lattice possibilities. In particular in the stacking of such hyper-stars we do not necessary just alternate them shifting a half angle as in three space.
5. In consideration of steradian crowns and stars (I may post an illustration along with the fano candle of seven hexes stacked in the 12 color diatonic notes). when we compress or expand angles we can imagine them as quasic polygon straight limits at infinity (not necessarily a fixed idea of hyperspheres that can compress to some point from some imagined maximum equator) such that an equator around a polytope may be a discrete loop of these abstract boundaries.
5a. Let us imagine a loop around a discrete or continuous spherical object; it may have one or several twists, when it is not twisted it is "on the mirror" but even with one twist in one direction or the other in the stacking as if stars, it will stand out in a sense (and in a way I am not sure I understand in depth for these things any more than when they twist hands somewhere between there and back again across some supposed intrinsically curved circumference of the universe- one that moreover these quasic notions justify when some ask if the universe is expanding where does it expand into or does the universe have some sort of a wall.)And that standing out seems to have the concept of charge, especially the idea of chirality and handedness as charge as subtle difference it is on either side of the mirror. [but this is from of what I could make out in the hard to forget dream of last night.] Certainly when these things cross in higher space they are subject to the same wider freedoms we experience in say permutations of axes woven in space.