Alphanumeric Intuitions L. Edgar Otto August 10, 2011
Last night I read the preface to Peter Rowlands book and found some interesting philosophy of foundational physics. For one thing he pretty much agrees with what some of our alternative new physics bloggers have said in his critique of the current state of physics. We cannot by discrete algorithm alone find or even point toward a unified theory. Indeed, his is an insistence that such a theory should be most simple and the independent background is a sort of neutrality as far as science is concerned and more the simplicity of organizing things from "nothingness". He emphasizes that a total theory would at this level not insist on things being totally discrete or totally continuous. His points I have held as notions and theories for some time save that the potential infinite also is as if a creative framework- that and thus the idea of unity.
He seeks the field idea to include a natural order of things- as physics is to be founded free from the notions of number, thus a natural order or code. From this his book begins to explore on firmer ground some of our more heretofore metaphysical notions (and he says as far as quark fractional charges that is still need less vague analysis in his system as physics). There is a principle of arithmetic beyond the idea of a field that may contain an axiom of order which I should look up to see if and where it may fit in. By the way, a great part of this depends on our notions and systems of logic and in a sense proofs involving prime numbers if in the great landscape, for example the sting landscape as he mentions, we are to find uniqueness in the expression of physical laws. Yet from these foundational notions I offer the general principle of non-necessity which is ultimately paradoxical because in the end we can see that something unique and necessary in the landscape is not forbidden to be a matter of count and label order and information. I am surprised my work done so intuitively has fulfilled Rowlands criterion in the such for unified theories.
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I present a table of the Depth Conway idea, presumably will go beyond that to even higher ideas of dimensions for an aid to find the details of orientations of the color cubes for a better use of these ideas of dimensions. I have found in fact some intuitive doing of things with the color cubes now have a firm basis beyond my intuitive sense of sorting, extending, and arranging them as just recreational mathematics, in fact my era of intense puzzle building proved most relevant today despite how much of an amateur I was in the beginning of explorations.
I did other things last night, among them the realization that with such representations as to the nature of objects in space if seen as shadows in the higher dimensions, inversions and centers or not, that the supersymmetry breaking and merging idea of the virial doubling of things applies (symmetry alone is not enough either Rowlands says), that is the doubling of a set of binary numbers in a linear sequence to show patterns as in that done for the golden irrational, or if we have a two base sequence the doubling to four base, then to the 8 base- an informational view really that leads from quaternions to octonions and so on...
that we find a representation much like the point centered hypercube in the above illustration, but it suggests analogs in conway space of the 24-cell. It is always a good idea to consider duality of points and lines and where these interchange. It follows that those who have thought about lattices of the 24 cell (composed of 24 octahedra by the way, not an analog to the 16 cell polytope a hyperdeltahedron of sorts composed of tetrahedra) should see them in such depth spaces of higher lattices.
I noticed just now that Kea has a series of posts on the simplicity but not the boredom of the subject- and I certainly agree although the work can be tedious and especially the way I do it long shunned as a method- for example Hinton, or Conway himself suggesting that he still saw the shape of the 3rd tetranomnio as "green".
Of course in the reductions or expansions of the matrix or space of the chart above we can immediately see the close connection to the simplest of ideas or matrices that involve Kea's sort of methods, the Arcadian, to which the rest of humanity if it can even bother with it at all strives to see three space let alone four and above... That is we can reduce things even further than the 15 groups where these are surprisingly complex for such a low number.
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I also, in this wealth of new symmetries to count and explore, found several areas of interest and inter connections among the notions and methods. Albeit more an informal exploration.
For one thing the exclusion of the main diagonal, without making a rigid concept of matrices or numbers as having to be square or even magic squares, or even in the sense the so called oblong figurate numbers. That the conway matrix is a sort of squarish matrix of two triangular numbers with a diagonal shift between them- all of this of course expressible in algebra or number theory formulas and for all I know has been known for a long time. The result of this is the realization in views of such systems where a polyhedron is a shadow but of not a specific dimension among the free possibilities over an intelligible range- that we can see some things as simple as the difference between 20 or 30 points or edges, or twelve, as a symbolic reduced notation for such symmetries. What then is the structure akin to the point centered hypercube of 14 points with 2 in the center in the 20 and 30 case which of course can be assigned shifts of say 5 or ten tetrahedra. In the directional vectors of say the diamond triaconahedron could I put them into an array as well in the stacking and color code relative to inversion? In a sense I color coded the edges and faces of the dodecahedron with the 30 cubes long ago, (the example I told Conway which he remarked was very elegant and invited me to dinner if I were ever in Princeton) but I have not explored things from this greater depth of space.
One consequence is that we can represent some of the faces of this point centered hypercube as five sub-objects such as any of the three axes of a colored cube. What this means in the logic of finite aspects of the topology is that the number 5 comes up again and the ensemble of things can stand in for the 4-space simplex, that is sorted into 5 four dimensional objects.
Another interesting aspect of the ordered counting are sequences involving the knight moves wherein one can make an array of five squares to fill a plane of something centered on one of the five colors or in three space 7 objects or cubes or in four space 9 and so on for the odd numbers. In the dimensions 1 4 13 40 121 we find such abstract motions (and as Rowlands points out a good unified theory has to be based on abstraction) as 3 9 27 81 243 and so on. Or in the quasic like conception of symmetry viriality, 1 2 4 8 16 32 ... "dimensions" becomes 3 5 9 17 31 65 129 which of course shows the grid generations or states of quasicity or the openly obscure idea of 2n+1 as if a trivial count to be disambiguated.
Given this view of labels and counting we can take the illustration and count many things, even predict which count of the pixels in the top superconway table have a certain relevance of intelligibility as to structures, even of the excluded cells. Along the main diagonal we see there are 6x6 or 36 diagonals or 20 x 6 or 120 other cells. That is if we set the cells to the depth of 64, and of course 6 x 64 is 384 or again the rotations and inversions of a hypercube. (or the lines of the days of the I Ching, again that which came as pattern awakening for the binary system and that which quasically implies the observed four base gene encoding as possible as well the observed epi-genome idea of 8 codon bases... and perhaps more things if we can show that on this level of complexity things do not somehow trivially loop and precipitate out as in the case of some notions of fractals.
I did not find an interesting connection between quasic numbering to replace the 1 to 15 values A to O with it, 7 8 19 20 23 = ABCDE and so on (but their may be a good idea herein somewhere.
Of the 15 LHC cube notations between any two of the letters we have the combination's of units of 15, thus 15 30 45 90 180 all relevant to total counts and the very thing the superconway grid hopes to sort out happily and coincidentally alphabetically. What order of point labels do we put on a point centered hypercbe compared with the cube centered for example? This itself a global problem of possibilities in the freedom of greater symmetries and parities.
3^15 = 27^5, obvious but relevant. Consider in the count of pixels in the lead grid here things like: 64 x 30 = 1920 ; 30 x 30 = 900; 900 + 180 = 720 = 6! ; 720 - 666 = 54 = 2 x 27... 180 - 144 = 36 and so on... 15 + 12 = 27 ...
Now, in the 3 x 3 matrix of the quarter of the 6x6 board these arrangements of the three LHC type letters centered on say one of them, H has six sides (also 6x5 for 30 letters on a labeled cube) Now between any two of the 3 sides and three mirror sides we generate 9 of the 15 color cubes of which all 15 are found but ony four have the same name between them, which I need to explore this interesting direction further. See the second illustration for this. And remember the alphabetical label represents two color combinations (or say red red, even rrr and rrx, x a different color that lies on the higher chart diagonals where such values cancel to gray.
While we can say of 6 objects 4 or 2 of them taken at a time, and the all important 20 in the center of 3 at a time, has a certain symmetry of such label description, that is we can color tetrahedra with say two letters of 4 colors.
Well, so much for recreating the wheel, in fact so long from my earlier writings I have to recreate my own wheels within the wheels of that known in the literature.
And yesterday, well, I was almost ready to get off the grid of my openness on the social media- but any time I stop it is likely as much as I needed to say. The outline was there long ago and it is gravy on the theory to have the time to fill in the colors to our paint by number dreams.
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