Tuesday, August 9, 2011
24-cell symmetry in the Deeper Conway Matrix
24-cell symmetry in the Deeper Conway Matrix
L. Edgar Otto Aug.9, 2011
The tables will take me a little time this morning and I do not want to post the tangled notes I scribbled on scraps of paper. Come back later, please.
Correction to the illustration: the count is misleading 6 orange points for the octahedron and 6 blue, but their are 12 for the cuboctahedcron =24... then again we do seem to have a problem when things double or half like this for we are in a world of hyperviriality or hyperduality where 4 dimensions are intimately related informationally to the 8 for that sort of intelligible symmetry breaking.
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Other things I explored but did not illustrate was the taking of the 8 points of the Malhoon set of thirty cubes which globally match say the standard cube from the set and reversing the notation such that at each of these we generate another subset of the 8 for a total of 64 and so on... Also the Depth in Depth table of the super Conway space matrices where the color exclusion is such a Conway table such that one can orient and match our choice of one of the six alphabetical cubes. On the main diagonal if we consider other and deeper matrices we are aided in the general counting of empty cells--- for example if there are in each diagonal cell 64 things and this looks linear despite the appearance of non-linearity in terms of information 6 x 64 = 384 the rigid rotations and inversion of the 4-space orthogon.
We mix the concept of 3 and 4 space transitive across this idea of depth matrices and this shows the idea of alternative arithmetical grids (my epsilon-delta and the relationship to cycles of 5 or 12 a natural as well as theoretical distinction). This is why phase space is not quite the correct term of the new Lee Smolin type models of the physics and why given this "news" many try to adapt their general system to it to some degree of congruent success.
I await what TGD and Lubos may have to say about it in reference to their frames.
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This morning I see an interesting post by Kea to which I make this comment, synchronously- again it will be posted here to save space and distraction from her blog. I keep thinking how Mdme Curie who discovered Radium could not afford to work with it in the later part of her career. This happens when I run out of paper too or ink and crave it more than food on a good hair day:
Such an octonionic plane can be mapped onto a general plane.
The 15 dimensional sphere can describe the analog to a rhombidodecahedron which is the triacontrahedron, and this relationship is the reason we find reductions from 24-cell tori to hypercubes and so on...
Quantum privileging while allowed can trivially show in a same 3D system we can substitute colors.
Thus the analogy from part of the geometry (which is not simply part of the algebra when we can see these as a system of one, as is the logic and math thought possibly the same language- both notions make a dynamic space)is that the algebra of the rhombidodecahedron is the analog to the 24-cell.
So of we allow bioctonions we may need in a more general physics bi-associahedra. (so not just one per natural dimension).
I will post tables for this today. But 31 dimensions if we have the same language is presently outside the span of my graph paper.
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Another Philosphic Interlude:
There is a certain intuitive state of mind where the general solution is satisfying and felt certain even when we do not know its factual solution details.
Sometimes a focused search in the sub-conscious background makes the state of awareness seem a little distracted or confused.
On the other hand if I want to solve the facts "counting, etc..." details, I have to take the steps constructing first the lesser cases in real enquiry time as we try to not stray too far in the span of paths to new ideas yet have faith in epiphany results at the end of your search, in your intuition.
Sometimes descriptive terms and symbols with math or logic rules aid this in its own proofs and inner evolving meanings, especially when pure and intelligible intuition is lacking and this the only available way to explore and describe a phenomenon.
Given the scope of a large stretch of time, our deepest intuitions say for a symmetrical geometric system such as Plato and the metacube or other core symbols used in the rituals and songs of thinkers and believers, these fade away in the presence of a few generations of deeper wisdom to even yet distant higher intuitions.
But along the way when the looping of dreams and thought comes together or we who want a center and concrete reference for the world and ourselves, for awhile the journey can seem pointless or impossible, endless and overwhelming, too much for the trivial statements and insights of our fantasy. The ideal points at infinity forever out of reach.
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comments today to Ulla, Pitkanen
Of course our human tendencies may misunderstand our intentions- I am sure there is no problem here- it is not that your challenges are taken as not helpful and they do have an effect- but they can be distracting at least in my case- not that I mind.
I am not sure I have that formal education myself in any language but it is commendable we want to find a way to whatever understanding there is.
I went a little further in the consideration of some of the higher and deeper uses of dimensions where they are after all rapidly complex even in the low numbers.
I am interested in what you might think of Lee Smolin's new book about what I think are theories we have played with all along.
See my pesla.blogspost com today on the 24 cell and depth Conway matrices mentioned asking this of your and other reference frames.
Odd thing for me is I have not attempted to promote my blog and its audience nor monetize it. You should look into this.
Why indeed would the exact mirror image of a large organic molecule have different energy? It is a good question, and in what context TGD?
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Footnote: the 2x2x3 = 12 cuboid is of use for the pentominos to replicate some of the 5 cube shapes so this connects well to some of these methods in context of physics. I left the enquiry today with the realization or hint that even in the case of the 24-cell we may have a certain triality or duplication of these structures for otherwise how does the 1152 of its group add up- it appears all the more remarkable to me this polytope has no natural analogs in dimensions above and below it. Also we justify the near or failed polyhedra with small looping differences that contribute to the "mass" of the whole via recursive and increasing dimensional conway depth fields- to answer but not in the standard way to view dimensions why say there are no 18 face deltahedra of say 11 points.
Ulla, perhaps the situation is this: that we have a core of chaos and randomness surrounded by one of absolute determinism and that in turn generalized to one of a sort of compromise and what seems freedom with form and purpose. New levels of notions as well of space are at one time a trivial reduction and multiplication of the the freedoms we may find as between the binary dimensions we do find there are wider views of expansion and contraction of space.
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